In what follows, we assume that \(h_{1},h_{2}:J\to \mathbb{R}\) are two nonnegative and nonzero functions, with \(h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) \neq 0\), \(\sigma = ( \sigma _{1},\sigma _{2} ) \), \(\rho = ( \rho _{1},\rho _{2} ) \), \(\eta = ( \eta _{1},\eta _{2} ) \), and \(\omega = ( \omega _{1},\omega _{2} )\) with \(\rho _{1},\rho _{2},\eta _{1},\eta _{2}\in [0,+\infty )\) and \(\omega _{1},\omega _{2}\in \mathbb{R}\).
Theorem 3.1
Let \(f:\Delta \to \mathbb{R}\) be an integrable and \(( l_{1},h_{1} ) \)-\(( l_{2},h_{2} ) \)-convex function on coordinates on Δ. Then we have
$$\begin{aligned}& \frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad = \frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } f_{g} \biggl( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2}, \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr) \\& \quad \leq \frac{1}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {} \times \bigl\{ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \bigr\} \\& \quad \leq \frac{ ( f ( \chi _{1},\chi _{3} ) +f ( \chi _{1},\chi _{4} ) +f ( \chi _{2},\chi _{3} ) +f ( \chi _{2},\chi _{4} ) ) }{\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {} \times \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{ \eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {} \times \bigl( h_{1} (\xi _{1} ) +h_{1} ( 1- \xi _{1} ) \bigr) \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \,d\xi _{2}\,d\xi _{1}, \end{aligned}$$
where \(f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) ) \) with \(g_{1} ( x ) =x^{\frac{1}{l_{1}}}\) and \(g_{2} ( y ) =y^{\frac{1}{l_{2}}}\).
Proof
It is easy to see that
$$ \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{ \frac{1}{l_{1}}}= \biggl[ \frac{ ( (\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} ) ^{\frac{1 }{l_{1}}} ) ^{l_{1}}+ ( ( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1}\chi _{2}^{l_{1}} ) ^{\frac{1}{l_{1}}} ) ^{l_{1}}}{2} \biggr] ^{ \frac{1}{l_{1}}} $$
(3.1)
and
$$ \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}}= \biggl[ \frac{ ( (\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} ) ^{\frac{1 }{l_{2}}} ) ^{l_{2}}+ ( ( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2}\chi _{4}^{l_{2}} ) ^{\frac{1}{l_{2}}} ) ^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}}. $$
(3.2)
Making use of (3.1) and (3.2), and the fact that f is \((l_{1},h_{1} ) \)-\(( l_{2},h_{2} ) \)-convex on the coordinates, we have
$$\begin{aligned}& f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad \leq h_{1} \biggl( \frac{1}{2} \biggr) h_{2} \biggl( \frac{1}{2} \biggr) \bigl\{ f \bigl( \bigl(\xi _{1} \chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{ 1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{ l_{2}}} \bigr) \\& \qquad {}+f \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2} \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\& \qquad {}+f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\& \qquad {}+ f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl( ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \bigr\} . \end{aligned}$$
(3.3)
Multiplying on both sides of (3.3) by
$$ \xi _{1}^{\eta _{1}-1}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl(\chi _{4}^{l_{2}} -\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr], $$
and then integrating the resulting inequality with respect to \((\xi _{1},\xi _{2} ) \) on \([ 0,1 ] ^{2}\), we get
$$\begin{aligned}& \frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \qquad {}\times \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{ \eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \,d\xi _{2}\,d\xi _{1} \\& \quad \leq \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \\& \qquad {}\times f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{ \frac{1}{l_{1}}}, \bigl[ \xi _{2} \chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \\& \qquad {}\times f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{ \frac{1}{l_{1}}}, \bigl[ ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2}^{ \rho _{2}} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) \xi _{2}^{ \rho _{2}} \bigr] \\& \qquad {}\times f \bigl( \bigl[ ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr] ^{ \frac{1}{l_{1}}}, \bigl[ \xi _{2} \chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \\& \qquad {}\times f \bigl( \bigl[ ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr] ^{ \frac{1}{l_{1}}}, \bigl[ ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1}. \end{aligned}$$
(3.4)
By making a change of variables in (3.4), we obtain
$$\begin{aligned}& \frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad \leq \frac{1}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} )} \\& \qquad {}\times \biggl\{ \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\eta _{1}-1} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\rho _{1}} \bigr] \\& \qquad {}\times\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\rho _{2}} \bigr] f \bigl( x^{ \frac{1}{l_{1}} },y^{\frac{1}{l_{2}}} \bigr) \,dy\,dx \\& \qquad {}+ \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\eta _{1}-1} \bigl( y-\chi _{3}^{l_{2}} \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\rho _{1}} \bigr] \\& \qquad {}\times\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( y-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}} \bigr] f \bigl( x^{ \frac{1}{l_{1}} },y^{\frac{1}{l_{2}}} \bigr) \,dy\,dx \\& \qquad {}+ \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( x-\chi _{1}^{l_{1}} \bigr) ^{\eta _{1}-1} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( x-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}} \bigr] \\& \qquad {}\times\mathfrak{F} _{ \rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2}^{\rho _{2}} \bigl( \chi _{4}^{l_{2}}-y \bigr) \bigr] f \bigl( x^{ \frac{1}{l_{1}}},y^{\frac{1}{l_{2}}} \bigr) \,dy\,dx \\& \qquad {}+ \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( x-\chi _{1}^{l_{1}} \bigr) ^{\eta _{1}-1} \bigl( y-\chi _{3}^{l_{2}} \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( x-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}} \bigr] \\& \qquad {}\times\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( y-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}} \bigr] f \bigl( x^{ \frac{1}{ l_{1}}},y^{\frac{1}{l_{2}}} \bigr) \,dy\,dx \biggr\} \\& \quad = \frac{1}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} )} \\& \qquad {}\times \bigl\{ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\& \qquad {}+\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}}, \chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{ \sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \bigr\} . \end{aligned}$$
(3.5)
Since f is \(( l_{1},h_{1} ) \)-\(( l_{2},h_{2} ) \)-convex on the coordinates, we have
$$\begin{aligned}& \begin{aligned}[b] &f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{\frac{1}{ l_{1}}}, \bigl[ \xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr] ^{\frac{1}{l_{2} }} \bigr) \\ &\quad \leq h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{1},\chi _{3} ) +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{1},\chi _{4} ) +h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{2},\chi _{3} ) \\ &\qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{2},\chi _{4} ) , \end{aligned} \end{aligned}$$
(3.6)
$$\begin{aligned}& \begin{aligned}[b] &f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{\frac{1}{ l_{1}}}, \bigl[ ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2} \chi _{4}^{l_{2}} \bigr] ^{\frac{1}{l_{2} }} \bigr) \\ &\quad \leq h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{1},\chi _{3} ) +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{1},\chi _{4} ) +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{2},\chi _{3} ) \\ &\qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{2},\chi _{4} ) , \end{aligned} \end{aligned}$$
(3.7)
$$\begin{aligned}& \begin{aligned}[b] &f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\ &\quad \leq h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{1},\chi _{3} ) +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{1},\chi _{4} ) +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{2}, \chi _{3} ) \\ &\qquad {}+h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{2},\chi _{4} ), \end{aligned} \end{aligned}$$
(3.8)
and
$$\begin{aligned} &f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2} \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\ &\quad \leq h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{1},\chi _{3} ) +h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{1},\chi _{4} ) \\ &\qquad {}+h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{2},\chi _{3} )+h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{2},\chi _{4} ) . \end{aligned}$$
(3.9)
Adding inequalities (3.6)–(3.9), multiplying the resulting inequality by
$$ \xi _{1}^{\eta _{1}-1}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr], $$
and then integrating the result with respect to \((\xi _{1},\xi _{2} ) \) on \([ 0,1 ] ^{2}\), we get
$$\begin{aligned}& \int^{1}_{0} \int^{1}_{0}\xi _{1}^{ \eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{ \sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{ \rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {}\times \bigl\{ f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1- \xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{\frac{1}{l_{1}}}, \bigl[ \xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \\& \qquad {}+f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{\frac{1}{ l_{1}}}, \bigl[ ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2} \chi _{4}^{l_{2}} \bigr] ^{\frac{1}{l_{2} }} \bigr) \\& \qquad {}+f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\& \qquad {}+ f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl( ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \bigr\} \,d\xi _{2}\,d\xi _{1} \\& \quad \leq \bigl( f ( \chi _{1},\chi _{3} ) +f ( \chi _{1}, \chi _{4} ) +f ( \chi _{2},\chi _{3} ) +f ( \chi _{2},\chi _{4} ) \bigr) \\& \qquad {}\times \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{ \eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {}\times \bigl( h_{1} (\xi _{1} ) h_{2} ( \xi _{2} ) +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) +h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \bigr) \,d\xi _{2}\,d\xi _{1}. \end{aligned}$$
Making use of the change of variables and multiplying the result by
$$ \frac{1}{\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) }, $$
we obtain
$$\begin{aligned}& \frac{1}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1}, \eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {} \times \bigl\{ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \bigr\} \\& \quad \leq \frac{ ( f ( \chi _{1},\chi _{3} ) +f ( \chi _{1},\chi _{4} ) +f ( \chi _{2},\chi _{3} ) +f ( \chi _{2},\chi _{4} ) ) }{\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {} \times \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{ \eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {} \times \bigl( h_{1} (\xi _{1} ) +h_{1} ( 1- \xi _{1} ) \bigr) \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \,d\xi _{2}\,d\xi _{1}. \end{aligned}$$
This rearranges to the proof of Theorem 3.1. □
Remark 3.1
Theorem 3.1 with \(l_{1}=l_{2}=1\) and \(h_{1} (\xi _{1} ) =h_{2} (\xi _{1} ) =\xi _{1}\) becomes Theorem 2.1 in [39].
Remark 3.2
Theorem 3.1 with \(l_{1}=l_{2}=1\), \(\eta _{1}=\eta _{2}=\alpha \), \(\sigma _{1}(0) =\sigma _{2} ( 0 ) =1\), \(\omega _{1}=\omega _{2}=0\), and \(h_{1} (\xi _{1} )=h_{2} (\xi _{1} ) =\xi _{1}\) becomes Theorem 3 in [33].
Remark 3.3
Theorem 3.1 with \(\eta _{1}=\eta _{2}=1\), \(\sigma _{1} ( 0 ) =\sigma _{2} ( 0 ) =1\), and \(\omega _{1}=\omega _{2}=0\) becomes Theorem 2.1 in [40].
Remark 3.4
Theorem 3.1 with \(l_{1}=l_{2}=1\), \(\eta _{1}=\eta _{2}=1\), \(\sigma _{1} ( 0 ) = \sigma _{2} ( 0 ) =1\), \(\omega _{1}=\omega _{2}=0\), and \(h_{1} (\xi _{1} ) =h_{2} (\xi _{1} )= h(\xi _{1})\) becomes Theorem 7 in [35].
Theorem 3.2
Let \(f:\Delta \to \mathbb{R}\) be an integrable and \(( l_{1},h_{1} ) \)-\(( l_{2},h_{2} ) \)-convex function on coordinates on Δ. Then we have
$$\begin{aligned} &f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\ &\quad \leq \frac{h_{1} ( \frac{1}{2} ) ( \Im _{\rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{\sigma _{1}}f_{g} ( \chi _{2}^{l_{1}},\frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ) +\Im _{\rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{\sigma _{1}}f_{{g}} ( \chi _{1}^{l_{1}},\frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ) ) }{2 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} )^{\rho _{1}} ) } \\ &\qquad {}+ \frac{h_{2} ( \frac{1}{2} ) ( \Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}f_{g} ( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2},\chi _{4}^{l_{2}} ) +\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}f_{{g}} ( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2},\chi _{3}^{l_{2}} ) ) }{2 ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\ &\quad \leq h_{1} \biggl( \frac{1}{2} \biggr) \frac{f ( \chi _{1}, [ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ] ^{\frac{1}{l_{2}}} ) +f ( \chi _{2}, [ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ] ^{\frac{1}{l_{2}} } ) }{2\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) } \\ &\qquad {}\times \int^{1}_{0}\xi _{1}^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho 1} \bigr] \bigl( h_{1} (\xi _{1} ) +h_{1} ( 1- \xi _{1} ) \bigr) \,d\xi _{1} \\ &\qquad {}+h_{2} \biggl( \frac{1}{2} \biggr) \frac{f ( [ \frac{ \chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{\frac{1}{l_{1}}},\chi _{3} ) +f ( [ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{\frac{1}{l_{1}}},\chi _{4} ) }{ 2\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\ &\qquad {}\times \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \,d\xi _{2}, \end{aligned}$$
where \(f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) ) \) with \(g_{1} ( x ) =x^{\frac{1}{l_{1}}}\) and \(g_{2} ( y ) =y^{\frac{1}{l_{2}}}\).
Proof
Since f is an \(( l_{1},h_{1} ) \)-\(( l_{2},h_{2} ) \)-convex function on coordinates on Δ, the partial mapping \(f_{x}: [ \chi _{3},\chi _{4} ] \to \mathbb{R}\) defined by \(f_{x} ( v ) =f ( x,v ) \) is \((l_{2},h_{2} ) \)-convex with respect to v on \([ \chi _{3},\chi _{4} ] \), and \(f_{y}: [ \chi _{1},\chi _{2} ] \to \mathbb{R}\) defined by \(f_{y} ( u ) =f ( u,y ) \) is \((l_{1},h_{1} ) \)-convex with respect to u on \([ \chi _{1},\chi _{2} ]\). So, we have
$$\begin{aligned}& f_{x} \biggl( \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \\& \quad =f_{x} \biggl( \biggl[ \frac{ ( (\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} ) ^{\frac{1}{l_{2}}} ) ^{l_{2}}+ ( ( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2}\chi _{4}^{l_{2}} ) ^{\frac{1}{l_{2}}} ) ^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \\& \quad \leq h_{2} \biggl( \frac{1}{2} \biggr) \bigl( f_{x} \bigl( \bigl( \xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) +f_{x} \bigl( \bigl( ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2} \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{ l_{2}}} \bigr) \bigr) \\& \quad \leq h_{2} \biggl( \frac{1}{2} \biggr) \bigl( h_{2} (\xi _{2} ) f_{x} ( \chi _{3} ) +h_{2} ( 1-\xi _{2} ) f_{x} ( \chi _{4} ) + h_{2} ( 1-\xi _{2} ) f_{x} ( \chi _{3} ) +h_{2} (\xi _{2} ) f_{x} ( \chi _{4} ) \bigr) \\& \quad =h_{2} \biggl( \frac{1}{2} \biggr) \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1-\xi _{2} ) \bigr) \bigl( f_{x} ( \chi _{3} ) +f_{x} ( \chi _{4} ) \bigr). \end{aligned}$$
(3.10)
From (3.10), we get
$$\begin{aligned} \frac{1}{h_{2} ( \frac{1}{2} ) }f_{x} \biggl( \biggl[ \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \leq& f_{x} \bigl( \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) +f_{x} \bigl( \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \\ \leq& \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \bigl( f_{x} ( \chi _{3} ) +f_{x} ( \chi _{4} ) \bigr). \end{aligned}$$
(3.11)
Multiplying (3.11) by \(\xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}}\xi _{2}^{\rho _{2}} ] \) and then integrating the resulting inequalities with respect to \(\xi _{2}\) on \([ 0,1 ]\), we obtain
$$\begin{aligned}& \frac{1}{h_{2} ( \frac{1}{2} ) }f_{x} \biggl( \biggl[ \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2}, \eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \,d\xi _{2} \\& \quad =\frac{1}{h_{2} ( \frac{1}{2} ) }f_{x} \biggl( \biggl[ \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl( \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{ \rho _{2}} \bigr) \\& \quad \leq \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] f_{x} \bigl( \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}} } \bigr) \,d\xi _{2} \\& \qquad {}+ \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{ \rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] f_{x} \bigl( \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2} \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \,d\xi _{2} \\& \quad = \frac{1}{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{4}^{l_{2}}-w \bigr) ^{\eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-w \bigr) ^{\rho _{2}} \bigr] f_{x} ( w ) \,dw \\& \qquad {}+ \frac{1}{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( w-\chi _{3}^{l_{2}} \bigr) ^{\eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( w-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}} \bigr] f_{x} ( w ) \,dw \\& \quad = \frac{1}{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}} \bigl( \Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+}, \omega _{2}}^{\sigma _{2}}f_{x} \bigl( \chi _{4}^{l_{2}} \bigr) + \Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega _{2}}^{\sigma _{2}}f_{x} \bigl( \chi _{3}^{l_{2}} \bigr) \bigr) \\& \quad \leq \bigl( f_{x} ( \chi _{3} ) +f_{x} ( \chi _{4} ) \bigr) \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2}, \eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {}\times \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1-\xi _{2} ) \bigr) \,d\xi _{2}. \end{aligned}$$
(3.12)
This implies that
$$\begin{aligned}& \frac{1}{h_{2} ( \frac{1}{2} ) }f \biggl( x, \biggl[ \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \\& \quad \leq \frac{1}{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {}\times \bigl( \Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}f_{g} \bigl( x^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega _{2}}^{\sigma _{2}}f_{{g}} \bigl( x^{l_{1}}, \chi _{3}^{l_{2}} \bigr) \bigr) \\& \quad \leq \frac{f ( x,\chi _{3} ) +f ( x,\chi _{4} ) }{\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {}\times \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1-\xi _{2} ) \bigr) \,d\xi _{2}. \end{aligned}$$
(3.13)
Put \(x= [ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{ \frac{1}{l_{1}}}\) into (3.13) to get
$$\begin{aligned}& f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad \leq \frac{h_{2} ( \frac{1}{2} ) }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {}\times \biggl( \Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}f_{g} \biggl( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ,\chi _{4}^{l_{2}} \biggr) + \Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega _{2}}^{\sigma _{2}}f_{{g}} \biggl( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ,\chi _{3}^{l_{2}} \biggr) \biggr) \\& \quad \leq h_{2} \biggl( \frac{1}{2} \biggr) \frac{f ( [ \frac{ \chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{\frac{1}{l_{1}}},\chi _{3} ) +f ( [ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{\frac{1}{l_{1}}},\chi _{4} ) }{ \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {}\times \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \,d\xi _{2}. \end{aligned}$$
(3.14)
Similarly, we can deduce
$$\begin{aligned}& f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad \leq \frac{h_{1} ( \frac{1}{2} ) }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) } \\& \qquad {}\times \biggl( \Im _{\rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{\sigma _{1}}f_{g} \biggl( \chi _{2}^{l_{1}}, \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr) + \Im _{\rho _{1}, \eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{ \sigma _{1}}f_{{g}} \biggl( \chi _{1}^{l_{1}}, \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr) \biggr) \\& \quad \leq h_{1} \biggl( \frac{1}{2} \biggr) \frac{f ( \chi _{1}, [ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ] ^{\frac{1}{l_{2}}} ) +f ( \chi _{2}, [ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ] ^{\frac{1}{l_{2}} } ) }{\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) } \\& \qquad {}\times \int^{1}_{0}\xi _{1}^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho 1} \bigr] \bigl( h_{1} (\xi _{1} ) +h_{1} ( 1- \xi _{1} ) \bigr) \,d\xi _{1}. \end{aligned}$$
(3.15)
By adding (3.14) and (3.15) together, and then multiplying the result by \(\frac{1}{2}\), we get the desired result. Thus we get the proof of Theorem 3.2. □
Remark 3.5
Theorem 3.2 with \(l_{1}=l_{2}=1\) and \(h_{1} (\xi _{1} ) =h_{2} (\xi _{1} ) =\xi _{1}\) becomes Theorem 2.2 in [39].
Lemma 3.1
Let \(f:\Delta \to \mathbb{R}\) be a partial differentiable function on Δ. If \(\frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}}\in L ( \Delta ) \), then we have
$$\begin{aligned}& \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}} \chi _{4}^{1-l_{2}}}-A \\& \qquad {}+ \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times \bigl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }\chi _{1}^{l_{1}-1} \chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma }\chi _{1}^{l_{1}-1} \chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\& \qquad {}+ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma }\chi _{2}^{l_{1}-1} \chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }\chi _{2}^{l_{1}-1} \chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \bigr) \\& \quad =\frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times \biggl( \int^{1}_{0} \int^{1}_{0} \mathcal{B} (\xi _{1}, \xi _{2} ) \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1} \chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl( ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1- \xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \mathcal{B} (\xi _{1}, \xi _{2} ) \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2} \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}} } \bigr) \,d\xi _{1}\,d\xi _{2} \biggr) , \end{aligned}$$
where
$$ \mathcal{B} (\xi _{1},\xi _{2} ) =\xi _{1}^{\eta _{1}}\xi _{2}^{ \eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] $$
(3.16)
and
$$\begin{aligned} A =& \frac{1}{4 ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \biggl( \frac{\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}\chi _{3}^{l_{2}-1}f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}} \\ &{}+ \frac{\Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}\chi _{4}^{l_{2}-1}f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}} + \frac{\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}\chi _{3}^{l_{2}-1}f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}} \\ &{}+ \frac{\Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}\chi _{4}^{l_{2}-1}f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}} \biggr) + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] } \\ &{}\times \biggl( \frac{\Im _{\rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{\sigma _{1}}\chi _{1}^{l_{1}-1}f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{3}^{1-l_{2}}}+ \frac{\Im _{\rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{\sigma _{1}}\chi _{1}^{l_{1}-1}f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{4}^{1-l_{2}}} \\ &{}+ \frac{\Im _{\rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{\sigma _{1}}\chi _{2}^{l_{1}-1}f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{3}^{l_{2}-1}}+ \frac{\Im _{\rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{\sigma _{1}}\chi _{2}^{l_{1}-1}f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{4}^{1-l_{2}}} \biggr), \end{aligned}$$
(3.17)
and \(f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) ) \) with \(g_{1} ( x ) =x^{\frac{1}{l_{1}}}\) and \(g_{2} ( y ) =y^{\frac{1}{l_{2}}}\).
Proof
Set
$$ \hbar :=\hbar _{1}-\hbar _{2}- \hbar _{3}+\hbar _{4}, $$
(3.18)
where
$$\begin{aligned}& \begin{aligned} \hbar _{1}:={} & \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}} \xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}} + ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2}; \end{aligned} \\& \begin{aligned} \hbar _{2}:={}& \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}} \xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2} \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2}; \end{aligned} \\& \begin{aligned} \hbar _{3}:={}& \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}} \xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2}; \end{aligned} \\& \begin{aligned} \hbar _{4}:={}& \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}} \xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2}. \end{aligned} \end{aligned}$$
Integrating by parts \(\hbar _{1}\), we have
$$\begin{aligned} \hbar _{1}={}& \int^{1}_{0}\xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \biggl( \int^{1}_{0}\xi _{1}^{\eta _{1}} \mathfrak{F} _{\rho _{1}, \eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{1} \biggr) \,d\xi _{2} \\ ={}& \frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [\omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ (\chi _{1}^{l_{1}} -\chi _{2}^{l_{1}} ) \chi _{1}^{1-l_{1}}} \int^{1}_{0}\xi _{2}^{\eta _{2}} \mathfrak{F} _{ \rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[\omega _{2} \bigl( \chi _{4}^{l_{2}} -\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times\frac{\partial f}{\partial \xi _{2}} \bigl( \chi _{1}, \bigl(\xi _{2} \chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{2} \\ &{}- \int^{1}_{0} \frac{l_{1}\xi _{1}^{\eta _{1}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}}\xi _{1}^{\rho _{1}} ] }{ ( \chi _{1}^{l_{1}}-\chi _{2}^{l_{1}} ) (\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} ) ^{\frac{1}{l_{1}}-1}} \biggl( \int^{1}_{0} \xi _{2}^{\eta _{2}} \mathfrak{F} _{ \rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial f}{\partial \xi _{2}} \bigl( \bigl( \xi _{1}\chi _{1}^{l_{1}} + ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2} \chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{2} \biggr) \,d\xi _{1} \\ ={}& \frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) } \frac{l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{1},\chi _{3} ) }{\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) \chi _{1}^{1-l_{1}}} \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \\ &{}\times \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{1-\frac{1}{ l_{2}}}f \bigl( \chi _{1}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) \chi _{3}^{1-l_{2}}} \int^{1}_{0}\xi _{1}^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \\ &{}\times \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1- \xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{1-\frac{1}{ l_{1}}}f \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1 }{l_{1}}},\chi _{3} \bigr) \,d\xi _{1} \\ &{}+ \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \int^{1}_{0} \int^{1}_{0}\xi _{1}^{\eta _{1}-1} \mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \\ &{}\times \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \bigl(\xi _{1} \chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{1-\frac{1 }{l_{1}}}\bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{1-\frac{1}{ l_{2}}} \\ &{}\times f \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1 }{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{ l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1}. \end{aligned}$$
By the change of variables, we get
$$\begin{aligned} \hbar _{1} ={}&\frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) } \frac{l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{1},\chi _{3} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}\chi _{1}^{1-l_{1}}} \\ &{}\times \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\eta _{2}-1}\mathfrak{F} _{ \rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\rho _{2}} \bigr] y^{1-\frac{1}{l_{2}}}f_{g} \bigl( \chi _{1}^{l_{1}},y \bigr) \,dy \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) \chi _{3}^{1-l_{2}}} \\ &{}\times\int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\rho _{1}} \bigr] x^{1- \frac{1}{l_{1}}}f_{g} \bigl( x,\chi _{3}^{l_{2}} \bigr) \,dx \\ &{}+ \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\rho _{1}} \bigr] \\ &{}\times \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\rho _{2}} \bigr] x^{1- \frac{1}{l_{1}}}y^{1-\frac{1}{l_{2}}}f_{g} ( x,y ) \,dy\,dx. \end{aligned}$$
(3.19)
Making use of Definition 2.3 in (3.19), we get
$$\begin{aligned} \hbar _{1} ={}&\frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{1},\chi _{3} ) }{\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}}\chi _{1}^{l_{1}-1}\Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}\chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }\chi _{3}^{l_{2}-1}\Im _{ \rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{ \sigma _{1}}\chi _{2}^{l_{1}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\ &{}+ \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }\chi _{2}^{l_{1}-1} \chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr). \end{aligned}$$
(3.20)
Likewise, we can deduce
$$\begin{aligned}& \begin{aligned}[b] \hbar _{2} ={}&{-} \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{1},\chi _{4} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \\ &{}+ \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}}\chi _{1}^{l_{1}-1}\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}\chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\ &{}+ \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }\chi _{4}^{l_{2}-1}\Im _{ \rho _{1},\eta _{1},a ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{ \sigma _{1}}\chi _{2}^{l_{1}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \\ &{}- \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma }\chi _{2}^{l_{1}-1} \chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr), \end{aligned} \end{aligned}$$
(3.21)
$$\begin{aligned}& \begin{aligned}[b] \hbar _{3}={}&{-} \frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) } \frac{l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{2},\chi _{3} ) }{ \chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \\ &{}+ \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}}\chi _{2}^{l_{1}-1}\Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}\chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \\ &{}+ \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }\chi _{3}^{l_{2}-1}\Im _{ \rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{ \sigma _{1}}\chi _{1}^{l_{1}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\ &{}- \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }\chi _{1}^{l_{1}-1} \chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr), \end{aligned} \end{aligned}$$
(3.22)
and finally
$$\begin{aligned} \hbar _{4} ={}&\frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) } \frac{l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{2},\chi _{4} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}}\chi _{2}^{l_{1}-1}\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}\chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }\chi _{4}^{l_{2}-1}\Im _{ \rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{ \sigma _{1}}\chi _{1}^{l_{1}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \\ &{}+ \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma }\chi _{1}^{l_{1}-1} \chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr). \end{aligned}$$
(3.23)
Making use of (3.20)–(3.23) in (3.18) and then multiplying by
$$ \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }, $$
we arrive at the desired result. Thus we get the proof of Lemma 3.1. □
Theorem 3.3
Let \(f:\Delta \to \mathbb{R}\) be a partial differentiable function on Δ. If \(\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert \) is an \((l_{1},h_{1} ) \)-\(( l_{2},h_{2} ) \)-convex function on coordinates on Δ, then we have
$$\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} -A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \bigl( h_{1} ( 1-\xi _{1} ) +h_{1} (\xi _{1} ) \bigr) \bigl( h_{2} ( 1- \xi _{2} ) +h_{2} (\xi _{2} ) \bigr) \,d\xi _{1}\,d\xi _{2} \biggr) \\& \qquad {} \times \biggl( \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{3} ) \biggr\vert + \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{4} ) \biggr\vert \biggr) , \end{aligned}$$
where \(\mathcal{B} (\xi _{1},\xi _{2} ) \) and A are as in (3.16) and (3.17), respectively, and \(f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) ) \) with \(g_{1} ( x ) =x^{\frac{1}{l_{1}}}\) and \(g_{2} ( y ) =y^{\frac{1}{l_{2}}}\).
Proof
By Lemma 3.1 and the properties of modulus, we have
$$\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} -A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \\& \qquad \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \\& \qquad \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \\& \qquad \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B} (\xi _{1}, \xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \\& \qquad \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \biggr) . \end{aligned}$$
Using the \(( l_{1},h_{1} ) \)-\(( l_{2},h_{2} )\)-convexity of \(\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert \) on coordinates, we obtain
$$\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}}-A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \biggl( h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert \\& \qquad {}+h_{1} ( \xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert + h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert \biggr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl( h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert \\& \qquad {} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert + h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert \\& \qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert \biggr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl( h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert + h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert \\& \qquad {}+h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert \biggr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl( h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert + h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert \\& \qquad {}+h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert \biggr) \,d\xi _{1}\,d\xi _{2} \biggr) \\& \quad = \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \bigl( h_{1} ( 1-\xi _{1} ) +h_{1} (\xi _{1} ) \bigr) \bigl( \bigl( h_{2} ( 1-\xi _{2} ) +h_{2} (\xi _{2} ) \bigr) \bigr) \,d\xi _{1}\,d\xi _{2} \biggr) \\& \qquad {} \times \biggl( \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{3} ) \biggr\vert + \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{4} ) \biggr\vert \biggr). \end{aligned}$$
This completely ends the proof of Theorem 3.3. □
Corollary 3.1
Theorem 3.3with \(\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert \leq K\) gives the new inequality:
$$\begin{aligned} & \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{3} ) }{4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}}-A \\ &\qquad {}+ \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} (\chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} )^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\ &\qquad {}\times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} (\chi _{1}^{l_{1}},\chi _{4}^{l_{2}} )}{\chi _{1}^{1-l_{1}} \chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{ \sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\ &\qquad {}+ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, (\chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} (\chi _{2}^{l_{1}}, \chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\ &\quad \leq \frac{K ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) (\chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} )^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}} -\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\ &\qquad {}\times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \bigl( h_{1} ( 1-\xi _{1} )+h_{1} (\xi _{1} ) \bigr) \bigl( h_{2} ( 1- \xi _{2} )+h_{2} ( \xi _{2} ) \bigr) \,d\xi _{1}\,d\xi _{2} \biggr). \end{aligned}$$
Remark 3.6
Theorem 3.3 with \(l_{1}=l_{2}=1\) and \(h_{1} (\xi _{1} ) =h_{2} (\xi _{1} ) =\xi _{1}\) becomes Theorem 3.2 in [39].
Remark 3.7
Theorem 3.3 with \(l_{1}=l_{2}=1\), \(\eta _{1}=\eta _{2}=\alpha \), \(\sigma _{1} ( 0 ) =\sigma _{2} ( 0 ) =1\), \(\omega _{1}=\omega _{2}=0\), and \(h_{1} (\xi _{1} ) =h_{2} (\xi _{1} )=\xi _{1}\) becomes Theorem 3 in [33].
Theorem 3.4
Let \(f:\Delta \to \mathbb{R}\) be a partial differentiable function on Δ. If \(\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert ^{q} \) is an \(( l_{1},h_{1} ) \)-\(( l_{2},h_{2} ) \)-convex function on coordinates on Δ, then for \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), we have
$$\begin{aligned} & \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} -A \\ &\qquad {}+ \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\ &\qquad {}\times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\ &\qquad {}+ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\ &\qquad {}\times \biggl( \int^{1}_{0} \int^{1}_{0} \bigl[\mathcal{B} (\xi _{1},\xi _{2} ) \bigr]^{p} \,d\xi _{1}\,d\xi _{2} \biggr)^{\frac{1}{p}} \biggl[ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\ &\qquad {}+h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert ^{q}+ h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\ &\qquad {} +h_{1} ( 1- \xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\ &\qquad {}+ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q} \\ &\qquad {}+ h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\ &\qquad {}+h_{1} ( 1- \xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{ \frac{1}{q}} \\ &\qquad {}+ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\ &\qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q}+ h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\ &\qquad {} +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{ \frac{1}{q}} \\ &\qquad {}+ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\ &\qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q}+ h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\ &\qquad {} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\mathcal{B} (\xi _{1},\xi _{2} ) \) and A are defined as in (3.16) and (3.17), respectively, and \(f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) ) \) with \(g_{1} ( x ) =x^{\frac{1}{l_{1}}}\) and \(g_{2} ( y ) =y^{\frac{1}{l_{2}}}\).
Proof
Making use of Lemma 3.1 and the properties of modulus, we get
$$\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} -A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \\& \qquad \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \\& \qquad \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \\& \qquad \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B} (\xi _{1}, \xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \\& \qquad \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \biggr) . \end{aligned}$$
Making use of the \(( l_{1},h_{1} ) \)-\(( l_{2},h_{2} ) \)-convexity of \(\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert ^{q} \) on coordinates and Hölder’s inequality, we obtain
$$\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}}-A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times\biggl( \int^{1}_{0} \int^{1}_{0} \bigl[\mathcal{B} (\xi _{1},\xi _{2} ) \bigr]^{p} \,d\xi _{1}\,d\xi _{2} \biggr)^{\frac{1}{p}} \\& \qquad {} \times \biggl( \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \biggr\vert ^{q} \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert ^{q} \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1}\chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert ^{q} \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1}\chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \biggr\vert ^{q} \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \biggr) \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times\biggl( \int^{1}_{0} \int^{1}_{0} \bigl[\mathcal{B} (\xi _{1},\xi _{2} ) \bigr]^{p} \,d\xi _{1}\,d\xi _{2} \biggr)^{\frac{1}{p}}\biggl[ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\& \qquad {} +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert ^{q}+ h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\& \qquad {} +h_{1} ( 1- \xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q} \\& \qquad {} + h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\& \qquad {}+h_{1} ( 1- \xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{ \frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\& \qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q} \\& \qquad {} + h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{ \frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\& \qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q}+ h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\& \qquad {} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \biggr]. \end{aligned}$$
This rearranges to the proof of Theorem 3.4 □
Corollary 3.2
Theorem 3.4with \(\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert ^{q}\leq K\), give the new inequality:
$$\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}} \chi _{4}^{1-l_{2}}}-A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{K ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times\biggl( \int^{1}_{0} \int^{1}_{0} \bigl[\mathcal{B} (\xi _{1},\xi _{2} ) \bigr]^{p} \,d\xi _{1}\,d\xi _{2} \biggr)^{\frac{1}{p}} \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0} \bigl( h_{1} ( 1-\xi _{1} ) +h_{1} (\xi _{1} ) \bigr) \bigl( h_{2} ( 1-\xi _{2} ) +h_{2} (\xi _{2} ) \bigr) \,d\xi _{1}\,d\xi _{2} \biggr)^{ \frac{1}{q}}. \end{aligned}$$