We denote
$$\begin{aligned}& \begin{aligned} & \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \\ &\quad = \bigl( \mathcal{R} (x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a }} \bigl( \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }}, \end{aligned} \end{aligned}$$
(5)
$$\begin{aligned}& \mathcal{I} ( a,b,x;\mathcal{R} ) = \textstyle\begin{cases} ( ( {}_{a}I_{\ast }^{\delta }\mathcal{R} ) ( \frac {a+b}{2} ) ( {}_{\ast }I_{b}^{\delta }\mathcal{R} ) ( \frac {a+b}{2} ) ) ^{ \frac{2^{\delta -1}}{ ( b-a ) ^{\delta -1}}}&\text{for } x=a, \\ ( ( {}_{\ast }I_{x}^{\delta }\mathcal{R} ) ( a ) ( {}_{(a+b-x)}I_{\ast }^{\delta }\mathcal{R} ) ( b ) ) ^{ \frac{1}{ ( x-a ) ^{\delta -1}}} & \\ \quad{} \times ( ( _{x}I_{\ast }^{\delta }\mathcal{R} ) ( \frac {a+b}{2} ) ( {}_{\ast }I_{(a+b-x)}^{\delta }\mathcal{R} ) ( \frac {a+b}{2} ) ) ^{ \frac{2^{\delta -1}}{ ( a+b-2x ) ^{\delta -1}}} & \text{for } a< x< \frac {a+b}{2}, \\ ( ( {}_{\ast }I_{\frac{a+b}{2}}^{\delta }\mathcal{R} ) ( a ) ( _{ \frac{a+b}{2}}I_{\ast }^{\delta }\mathcal{R} ) ( b ) ) ^{\frac{2^{\delta -1} }{ ( b-a ) ^{\delta -1 }}}& \text{for } x=\frac {a+b}{2}, \end{cases}\displaystyle \end{aligned}$$
(6)
$$\begin{aligned}& \Theta ( \varrho ,\delta ,s ) =\textstyle\begin{cases} \frac {2\delta ( 1-\varrho ) ^{\frac{\delta +s+1}{\delta }} -\delta + ( \delta +s+1 ) \varrho}{ ( \delta +s+1 ) ( s+1 ) }& \text{for } 0\leq \varrho \leq 1, \\ \frac {\varrho ( \delta +s+1 ) -\delta }{ ( \delta +s+1 ) ( s+1 ) }&\text{for } \varrho >1,\end{cases}\displaystyle \end{aligned}$$
(7)
and
$$ \mathcal{N} ( \varrho ,\delta ,s ) =\textstyle\begin{cases} \frac {1-\varrho }{s+1} ( 1-2 ( 1- ( 1-\varrho ) ^{\frac{1}{\delta }} ) ^{s+1} ) -\mathcal{G}_{ ( 1-\varrho ) ^{\frac{1}{\delta }}} ( \delta +1,s+1 ) &\text{for } 0\leq \varrho \leq 1, \\ \frac{\varrho -1}{s+1}+B ( s+1,\delta +1 ) &\text{for } \varrho >1,\end{cases} $$
(8)
with
$$ \mathcal{G}_{\rho } ( u,v ) =B_{\rho } ( u,v ) -B_{1- \rho } ( v,u ) , $$
where B and \(B_{\rho }\) are beta and incomplete beta functions, respectively.
To substantiate our findings, it is imperative to invoke the following lemma.
Lemma 3.1
Assuming that \(\mathcal{R}\) is a positive multiplicatively differentiable function on \([ a,b ]\) and \(\mathcal{R}^{\ast}\) is multiplicatively integrable over \([a,b]\), then for \(\delta >0\), \(\varrho \geq 0\), and \(x\in [a,\frac {a+b}{2} ]\), we have the following fractional multiplicative integral identity:
$$\begin{aligned} &\mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{ b-a}} \\ &\quad = \biggl( \int_{0}^{1} \bigl( \bigl( \mathcal{R}^{ \ast } \bigl( ( 1-\gamma ) a+ \gamma x \bigr) \bigr) ^{ \gamma ^{\delta }} \bigr) ^{d\gamma} \biggr) ^{\frac{ ( x-a ) ^{2}}{b-a}} \\ &\qquad{} \times \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) x+\gamma \frac {a+b}{2} \biggr) \biggr) ^{1-\varrho - ( 1-\gamma ) ^{\delta }} \biggr) ^{d\gamma} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }} \\ &\qquad{} \times \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) \frac {a+b}{2}+ \gamma ( a+b-x ) \biggr) \biggr) ^{\gamma ^{\delta }- ( 1-\varrho ) } \biggr) ^{d\gamma} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }} \\ &\qquad{} \times \biggl( \int_{0}^{1} \bigl( \bigl( \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) ( a+b-x ) +\gamma b \bigr) \bigr) ^{- ( 1-\gamma ) ^{ \delta }} \bigr) ^{d\gamma} \biggr) ^{ \frac{ ( x-a ) ^{2}}{b-a}}, \end{aligned}$$
(9)
where \(\mathcal{Q}\) and \(\mathcal{I}\) are defined in (5) and (6), respectively.
Proof
Let
$$\begin{aligned}& I_{1} = \biggl( \int_{0}^{1} \bigl( \bigl( \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) a+ \gamma x \bigr) \bigr) ^{\gamma ^{\delta }} \bigr) ^{d\gamma} \biggr) ^{ \frac{ ( x-a ) ^{2}}{b-a}}, \\& I_{2} = \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) x+\gamma \frac {a+b}{2} \biggr) \biggr) ^{1-\varrho - ( 1-\gamma ) ^{\delta }} \biggr) ^{d\gamma} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }}, \\& I_{3} = \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) \frac {a+b}{2}+ \gamma ( a+b-x ) \biggr) \biggr) ^{\gamma ^{\delta }- ( 1-\varrho ) } \biggr) ^{d\gamma} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }}, \end{aligned}$$
and
$$ I_{4}= \biggl( \int_{0}^{1} \bigl( \bigl( \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) ( a+b-x ) +\gamma b \bigr) \bigr) ^{- ( 1-\gamma ) ^{ \delta }} \bigr) ^{d\gamma} \biggr) ^{ \frac{ ( x-a ) ^{2}}{b-a}}. $$
First, let us consider the case \(x\in (a,\frac {a+b}{2} )\).
Using Lemma 2.6, for \(I_{1}\), we have
$$\begin{aligned} I_{1} & = \biggl( \int_{0}^{1} \bigl( \bigl( \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) a+ \gamma x \bigr) \bigr) ^{\gamma ^{\delta }} \bigr) ^{d\gamma} \biggr) ^{ \frac{ ( x-a ) ^{2}}{b-a}} \\ & = \int_{0}^{1} \bigl( \bigl( \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) a+ \gamma x \bigr) \bigr) ^{ \frac{ ( x-a ) ^{2}}{b-a} ( \gamma ^{\delta } ) } \bigr) ^{d\gamma} \\ & = \frac { ( \mathcal{R} ( x ) ) ^{\frac{x-a}{b-a}}}{1} \frac {1}{\int_{0}^{1} ( ( \mathcal{R} ( ( 1-\gamma ) a+\gamma x ) ) ^{\delta \frac{x-a}{b-a}\gamma ^{\delta -1}} ) ^{d\gamma}} \\ & = \bigl( \mathcal{R} ( x ) \bigr) ^{\frac{x-a}{b-a}} \frac {1}{\exp \{ \delta \frac{x-a}{b-a}\int_{0}^{1} ( \gamma ^{\delta -1} ( \ln \mathcal{R} ( ( 1-\gamma ) a+\gamma x ) ) ) \,d\gamma \} } \\ & = \bigl( \mathcal{R} ( x ) \bigr) ^{\frac{x-a}{b-a}} \frac {1}{\exp \{ \delta \frac{ ( x-a ) ^{1-\delta }}{b-a}\int^{x}_{a} ( ( u-a ) ^{\delta -1} ( \ln \mathcal{R} ( u ) ) ) \,du \} } \\ & = \bigl( \mathcal{R} ( x ) \bigr) ^{\frac{x-a}{b-a}} \frac {1}{ ( \exp \{ ( \frac{1}{\Gamma ( \delta ) } \int^{x}_{a} ( ( u-a ) ^{\delta -1} ( \ln \mathcal{R} ( u ) ) ) \,du ) \} ) ^{ ( x-a ) ^{1-\delta }\frac{\Gamma ( \delta +1 ) }{b-a}}} \\ & = \bigl( \mathcal{R} ( x ) \bigr) ^{\frac{x-a}{b-a}} \bigl( \bigl( {}_{\ast }I_{x}^{\delta }\mathcal{R} \bigr) ( a ) \bigr) ^{- \frac{ ( x-a ) ^{1-\delta }\Gamma ( \delta +1 ) }{b-a}}. \end{aligned}$$
(10)
In the same vein,
$$\begin{aligned}& \begin{aligned}[b] I_{2} & = \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) x+\gamma \frac {a+b}{2} \biggr) \biggr) ^{1-\varrho - ( 1-\gamma ) ^{\delta }} \biggr) ^{d\gamma} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }} \\ & = \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{ \ast } \biggl( ( 1-\gamma ) x+\gamma \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( 1-\varrho - ( 1- \gamma ) ^{\delta } ) } \biggr) ^{d\gamma} \biggr) \\ & = \frac { ( \mathcal{R} ( \frac{a+b}{2} ) ) ^{\frac{ ( 1-\varrho ) ( a+b-2x ) }{2 ( b-a ) }}}{ ( \mathcal{R} ( x ) ) ^{-\frac{\varrho ( a+b-2x ) }{2 ( b-a ) }}} \frac {1}{\int_{0}^{1} ( ( \mathcal{R} ( ( 1-\gamma ) x+\gamma \frac{a+b}{2} ) ) ^{\delta \frac{ ( a+b-2x ) }{2 ( b-a ) } ( 1-\gamma ) ^{\delta -1}} ) ^{d\gamma}} \\ & = \frac { ( \mathcal{R} ( x ) ) ^{\frac{\varrho ( a+b-2x ) }{2 ( b-a ) }} ( \mathcal{R} ( \frac{a+b}{2} ) ) ^{\frac{ ( 1-\varrho ) ( a+b-2x ) }{2 ( b-a ) }}}{\exp \{ \delta \frac{ ( a+b-2x ) }{2 ( b-a ) }\int_{0}^{1} ( 1-\gamma ) ^{\delta -1}\ln ( \mathcal{R} ( ( 1-\gamma ) x +\gamma \frac{a+b}{2} ) ) \,d\gamma \} } \\ & = \frac { ( \mathcal{R} ( x ) ) ^{\frac{\varrho ( a+b-2x ) }{2 ( b-a ) }} ( \mathcal{R} ( \frac{a+b}{2} ) ) ^{\frac{ ( 1-\varrho ) ( a+b-2x ) }{2 ( b-a ) }}}{\exp \{ \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( a+b-2x ) ^{\delta -1} ( b-a ) } ( \frac{1}{\Gamma ( \delta ) }\int^{\frac{a+b}{2}}_{x} ( \frac{a+b}{2}-u ) ^{\delta -1}\ln ( \mathcal{R} ( u ) ) \,du ) \} } \\ & = \bigl( \mathcal{R} ( x ) \bigr) ^{ \frac{\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{2 ( b-a ) }} \biggl( \bigl( {}_{x}I_{\ast }^{\delta }\mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( a+b-2x ) ^{\delta -1} ( b-a ) }}, \end{aligned} \end{aligned}$$
(11)
$$\begin{aligned}& \begin{aligned}[b] I_{3} & = \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) \frac {a+b}{2}+ \gamma ( a+b-x ) \biggr) \biggr) ^{\gamma ^{\delta }- ( 1-\varrho ) } \biggr) ^{d\gamma} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }} \\ & = \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) \frac {a+b}{2}+\gamma ( a+b-x ) \biggr) \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \gamma ^{ \delta }- ( 1-\varrho ) ) } \biggr) ^{d\gamma} \\ & = \frac { ( \mathcal{R} ( a+b-x ) ) ^{\frac{\varrho ( a+b-2x ) }{2 ( b-a ) }}}{ ( \mathcal{R} ( \frac{a+b}{2} ) ) ^{-\frac{ ( 1-\varrho ) ( a+b-2x ) }{2 ( b-a ) }}} \frac {1}{\int_{0}^{1} ( ( \mathcal{R} ( ( 1-\gamma ) \frac{a+b}{2}+\gamma ( a+b-x ) ) ) ^{\delta \frac{ ( a+b-2x ) }{2 ( b-a ) }\gamma ^{\delta -1}} ) ^{d\gamma}} \\ & = \frac { ( \mathcal{R} ( \frac{a+b}{2} ) ) ^{\frac{ ( 1-\varrho ) ( a+b-2x ) }{2 ( b-a ) }} ( \mathcal{R} ( a+b-x ) ) ^{\frac{\varrho ( a+b-2x ) }{2 ( b-a ) }}}{\exp \{ \delta \frac{ ( a+b-2x ) }{2 ( b-a ) }\int_{0}^{1}\gamma ^{\delta -1}\ln ( \mathcal{R} ( ( 1-\gamma ) \frac{a+b}{2}+\gamma ( a+b-x ) ) ) \,d\gamma \} } \\ & = \frac { ( \mathcal{R} ( \frac{a+b}{2} ) ) ^{\frac{ ( 1-\varrho ) ( a+b-2x ) }{2 ( b-a ) }} ( \mathcal{R} ( a+b-x ) ) ^{\frac{\varrho ( a+b-2x ) }{2 ( b-a ) }}}{ ( \exp \{ \frac{1}{\Gamma ( \delta ) }\int^{a+b-x}_{\frac{a+b}{2}} ( u-\frac{a+b}{2} ) ^{\delta -1}\ln ( \mathcal{R} ( u ) ) \,du \} ) ^{\frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ( a+b-2x ) ^{\delta -1}}}} \\ & = \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{2 ( b-a ) }} \bigl( \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{\varrho ( a+b-2x ) }{2 ( b-a ) }} \\ &\quad{} \times \biggl( \bigl( {}_{\ast }I_{(a+b-x)}^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ( a+b-2x ) ^{\delta -1}}}, \end{aligned} \end{aligned}$$
(12)
and
$$\begin{aligned} I_{4} & = \biggl( \int_{0}^{1} \bigl( \bigl( \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) ( a+b-x ) +\gamma b \bigr) \bigr) ^{- ( 1-\gamma ) ^{ \delta }} \bigr) ^{d\gamma} \biggr) ^{ \frac{ ( x-a ) ^{2}}{b-a}} \\ & = \biggl( \int_{0}^{1} \bigl( \bigl( \mathcal{R}^{ \ast } \bigl( ( 1-\gamma ) ( a+b-x ) + \gamma b \bigr) \bigr) ^{-\frac{ ( x-a ) ^{2}}{b-a} ( 1-\gamma ) ^{\delta }} \bigr) ^{d \gamma} \biggr) \\ & = \frac {1}{ ( \mathcal{R} ( a+b-x ) ) ^{-\frac{x-a}{b-a}}} \frac {1}{\int_{0}^{1} ( ( \mathcal{R} ( ( 1-\gamma ) ( a+b-x ) +\gamma b ) ) ^{\frac{\delta ( x-a ) }{b-a} ( 1-\gamma ) ^{\delta -1}} ) ^{d\gamma}} \\ & = \bigl( \mathcal{R} ( a+b-x ) \bigr) ^{\frac{x-a}{b-a}} \frac {1}{\exp \{ \delta \frac{x-a}{b-a}\int_{0}^{1} ( 1-\gamma ) ^{\delta -1}\ln ( \mathcal{R} ( ( 1-\gamma ) ( a+b-x ) +\gamma b ) ) \,d\gamma \} } \\ & = \bigl( \mathcal{R} ( a+b-x ) \bigr) ^{\frac{x-a}{b-a}} \frac {1}{\exp \{ \frac{ ( x-a ) ^{1-\delta }\Gamma ( \delta +1 ) }{b-a} ( \frac{1}{\Gamma ( \delta ) } \int^{b}_{a+b-x}( b-u ) ^{\delta -1}\ln ( \mathcal{R} ( u ) ) \,du ) \} } \\ & = \bigl( \mathcal{R} ( a+b-x ) \bigr) ^{\frac{x-a}{b-a}} \bigl( \bigl( {}_{(a+b-x)}I_{\ast }^{\delta }\mathcal{R} \bigr) ( b ) \bigr) ^{- \frac{ ( x-a ) ^{1-\delta }\Gamma ( \delta +1 ) }{b-a}}. \end{aligned}$$
(13)
Multiplying equalities (10)–(13), we get the desired result.
Now, for the cases where \(x=a\) and \(x=\frac{a+b}{2}\), it suffices to follow the same demonstration principle as above, taking into account that when \(x=a\), we have \(I_{1}=I_{4}=1\), and when \(x=\frac{a+b}{2}\), \(I_{2}=I_{3}=1\), and equation (9) will be reduced:
-
(1)
For \(x=a\),
$$\begin{aligned} & \bigl( \mathcal{R} ( a ) \bigr) ^{\frac{\varrho }{2}} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{1- \varrho } \bigl( \mathcal{R} ( b ) \bigr) ^{ \frac{\varrho }{2}} \biggl( \bigl( {}_{a}I_{\ast }^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl({}_{\ast }I_{b}^{\delta }\mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 )}{ ( b-a ) ^{\delta}}} \\ &\quad = \biggl[ \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) a+\gamma \frac {a+b}{2} \biggr) \biggr) ^{1-\varrho - ( 1-\gamma ) ^{\delta }} \biggr) ^{d\gamma} \biggr) \\ &\qquad{}\times \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) \frac {a+b}{2}+\gamma b \biggr) \biggr) ^{\gamma ^{\delta }- ( 1-\varrho ) } \biggr) ^{d \gamma} \biggr) \biggr]^{\frac{b-a}{4}}; \end{aligned}$$
-
(2)
For \(x=\frac {a+b}{2}\),
$$\begin{aligned} &\mathcal{R} \biggl( \frac {a+b}{2} \biggr) \bigl( \bigl( {}_{\ast }I_{ \frac{a+b}{2}}^{\delta }\mathcal{R} \bigr) ( a ) \bigl( {}_{\frac{a+b}{2}}I_{\ast }^{\delta }\mathcal{R} \bigr) ( b ) \bigr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \\ &\quad = \biggl[ \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) a+\gamma \frac {a+b}{2} \biggr) \biggr) ^{\gamma ^{\delta }} \biggr) ^{d \gamma} \biggr) \\ &\qquad{}\times \biggl( \int_{0}^{1} \biggl( \biggl( \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) \frac {a+b}{2} +\gamma b \biggr) \biggr) ^{- ( 1-\gamma ) ^{\delta }} \biggr) ^{d\gamma} \biggr) \biggr] ^{\frac{b-a}{4}}. \end{aligned}$$
The proof is completed. □
Theorem 3.2
Let \(\mathcal{R}:[a,b]\rightarrow \mathbb{R}^{+}\) be an increasing multiplicatively differentiable function on \([a,b]\). If \(\mathcal{R}^{\ast}\) exhibits multiplicative s-convexity on this interval, then for any \(x\in [a, \frac{a+b}{2} ]\), \(\varrho \geq 0\), and \(\delta > 0\), we have
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{b-a}B ( \delta +1,s+1 ) } \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) }\mathcal{N} ( \varrho ,\delta ,s ) } \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }\Theta ( \varrho ,\delta ,s ) + \frac{ ( x-a ) ^{2}}{ ( \delta +s+1 ) ( b-a ) }}, \end{aligned}$$
where \(\mathcal{Q}\), \(\mathcal{I}\), Θ, and \(\mathcal{N}\) are defined in (5)–(8), and B is the beta function.
Proof
From Lemma 3.1 by the properties of multiplicative integrals we have
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \int^{1}_{0} \gamma ^{\delta }\ln \bigl( \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) a+\gamma x \bigr) \bigr) \,d\gamma \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) }\int_{0}^{1} \bigl\vert 1-\varrho - ( 1- \gamma ) ^{\delta } \bigr\vert \ln \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) x+\gamma \frac {a+b}{2} \biggr) \,d\gamma \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) }\int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1- \varrho ) \bigr\vert \ln \mathcal{R}^{\ast } \biggl( ( 1- \gamma ) \frac {a+b}{2}+ \gamma ( a+b-x ) \biggr) \,d\gamma \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \int^{1}_{0}( 1-\gamma ) ^{\delta }\ln \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) ( a+b-x ) +\gamma b \bigr) \,d \gamma \biggr) . \end{aligned}$$
From the multiplicative s-convexity of \(\mathcal{R}^{\ast }\) we have
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ & \quad \leq \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \int^{1}_{0} \gamma ^{\delta } \bigl( ( 1- \gamma ) ^{s}\ln \mathcal{R}^{\ast } ( a ) +\gamma ^{s}\ln \mathcal{R}^{ \ast } ( x ) \bigr) \,d\gamma \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) }\\ &\qquad{}\times \int_{0}^{1} \bigl\vert 1-\varrho - ( 1- \gamma ) ^{\delta } \bigr\vert \biggl( ( 1-\gamma ) ^{s}\ln \mathcal{R}^{\ast } ( x ) +\gamma ^{s} \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) \,d \gamma \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) }\\ &\qquad{} \times \int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1- \varrho ) \bigr\vert \biggl( ( 1-\gamma ) ^{s} \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) +\gamma ^{s}\ln \mathcal{R}^{\ast } ( a+b-x ) \biggr) \,d\gamma \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \int^{1}_{0} ( 1-\gamma ) ^{\delta } \bigl( ( 1- \gamma ) ^{s} \ln \mathcal{R}^{\ast } ( a+b-x ) + \gamma ^{s}\ln \mathcal{R}^{\ast } ( b ) \bigr) \,d\gamma \biggr) \\ &\quad =\exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \int^{1}_{0} \gamma ^{\delta } ( 1-\gamma ) ^{s}\,d \gamma \biggr) \ln \mathcal{R}^{\ast } ( a ) + \frac { ( x-a ) ^{2}}{b-a} \biggl( \int^{1}_{0} \gamma ^{\delta +s}\,d\gamma \biggr) \ln \mathcal{R}^{\ast } ( x ) \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \biggl( \int^{1}_{0} \bigl\vert 1-\varrho - ( 1-\gamma ) ^{\delta } \bigr\vert ( 1-\gamma ) ^{s}\,d \gamma \biggr) \ln \mathcal{R}^{\ast } ( x ) \\ &\qquad{} + \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \biggl( \int_{0}^{1} \bigl\vert 1-\varrho - ( 1-\gamma ) ^{\delta } \bigr\vert \gamma ^{s}\,d\gamma \biggr) \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \biggl( \int^{1}_{0} \bigl\vert \gamma ^{\delta }- ( 1-\varrho ) \bigr\vert ( 1-\gamma ) ^{s}\,d \gamma \biggr) \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \\ &\qquad{} + \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \biggl( \int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1-\varrho ) \bigr\vert \gamma ^{s}\,d\gamma \biggr) \ln \mathcal{R}^{\ast } ( a+b-x ) \,d \gamma \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \int^{1}_{0} ( 1-\gamma ) ^{\delta +s}\,d \gamma \biggr) \ln \mathcal{R}^{\ast } ( a+b-x ) \\ &\qquad{} + \frac { ( x-a ) ^{2}}{b-a} \biggl( \int^{1}_{0} ( 1-\gamma ) ^{\delta }\gamma ^{s}\,d\gamma \biggr) \ln \mathcal{R}^{\ast } ( b ) \biggr) \\ &\quad = \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{b-a}B ( \delta +1,s+1 ) } \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) }\mathcal{N} ( \varrho ,\delta ,s ) } \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }\Theta ( \varrho ,\delta ,s ) + \frac{ ( x-a ) ^{2}}{ ( \delta +s+1 ) ( b-a ) }}, \end{aligned}$$
where we have used (7), (8), and the equalities
$$\begin{aligned}& \int_{0}^{1}\gamma ^{\delta } ( 1-\gamma ) ^{s}\,d\gamma = \int^{1}_{0} ( 1-\gamma ) ^{\delta }\gamma ^{s}\,d \gamma =B ( \delta +1,s+1 ), \end{aligned}$$
(14)
$$\begin{aligned}& \int_{0}^{1}\gamma ^{\delta +s}\,d\gamma = \int^{1}_{0} ( 1-\gamma ) ^{\delta +s}\,d \gamma = \frac {1}{\delta +s+1}, \end{aligned}$$
(15)
$$\begin{aligned}& \int_{0}^{1} \bigl\vert 1-\varrho - ( 1- \gamma ) ^{\delta } \bigr\vert ( 1-\gamma ) ^{s}\,d \gamma = \int_{0}^{1} \bigl\vert 1-\varrho - \gamma ^{\delta } \bigr\vert \gamma ^{s}\,d\gamma =\Theta ( \varrho , \delta ,s ), \end{aligned}$$
(16)
and
$$ \int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1- \varrho ) \bigr\vert ( 1-\gamma ) ^{s}\,d\gamma = \int_{0}^{1} \bigl\vert 1-\varrho - ( 1-\gamma ) ^{\delta } \bigr\vert \gamma ^{s}\,d\gamma =\mathcal{N} ( \varrho , \delta ,s ). $$
(17)
The proof is completed. □
Corollary 3.3
Taking \(x=a\) and \(\varrho \in [ 0,1 ] \) in Theorem 3.2, we get
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( a ) \bigr) ^{ \frac{\varrho }{2}} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{1-\varrho } \bigl( \mathcal{R} ( b ) \bigr) ^{ \frac{\varrho }{2}} \biggl( \bigl( {}_{a}I_{\ast }^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{b}^{ \delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac { ( \delta +s+1 ) \varrho -\delta +2\delta ( 1-\varrho ) ^{\frac{\delta +s+1}{\delta }}}{ ( s+1 ) ( \delta +s+1 ) }}\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2\mathcal{T} ( \varrho ,\delta ,s ) } \biggr) ^{\frac{b-a}{4}} \end{aligned}$$
with
$$\begin{aligned} \mathcal{T} ( \varrho ,\delta ,s ) & = \frac {1-\varrho }{s+1} \bigl( 1-2 \bigl( 1- ( 1-\varrho ) ^{\frac{1}{\delta }} \bigr) ^{s+1} \bigr) \\ &\quad{} -B_{ ( 1-\varrho ) ^{\frac{1}{\delta }}} ( \delta +1,s+1 ) + B_{1- ( 1-\varrho ) ^{\frac{1}{\delta }}} ( s+1,\delta +1 ) . \end{aligned}$$
(18)
Corollary 3.4
By setting \(\varrho =1\) in Corollary 3.3we obtain the following trapezium inequality:
$$\begin{aligned} & \biggl\vert \sqrt{\mathcal{R} ( a ) \mathcal{R} ( b ) } \biggl( \bigl( {}_{a}I_{\ast }^{\delta }\mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{b}^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac {1}{\delta +s+1}}\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2B ( s+1,\delta +1 ) } \biggr) ^{\frac{b-a}{4}}. \end{aligned}$$
Corollary 3.5
Taking \(\varrho =\frac{1}{2}\) in Corollary 3.3, we obtain the following Bullen inequality:
$$\begin{aligned} & \biggl\vert \biggl( \mathcal{R} ( a ) \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2}\mathcal{R} ( b ) \biggr) ^{\frac{1}{4}} \biggl( \bigl( {}_{a}I_{\ast }^{ \delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{ \ast }I_{b}^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{s+1-\delta +2\delta ( \frac{1}{2} ) ^{\frac{s+1}{\delta }}}{2 ( s+1 ) ( \delta +s+1 ) }} \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2\mathcal{T}_{1} ( \delta ,s ) } \biggr) ^{\frac{b-a}{4}} \end{aligned}$$
with
$$\begin{aligned} \mathcal{T}_{1} ( \delta ,s ) & = \frac {1}{2 ( s+1 ) } \biggl( 1-2 \biggl( 1- \biggl( \frac {1}{2} \biggr) ^{\frac{1}{\delta }} \biggr) ^{s+1} \biggr) \\ &\quad{} -B_{ ( \frac{1}{2} ) ^{\frac{1}{\delta }}} ( \delta +1,s+1 ) + B_{1- ( \frac{1}{2} ) ^{\frac{1}{\delta }}} ( s+1,\delta +1 ) . \end{aligned}$$
Corollary 3.6
Taking \(\varrho =\frac{1}{3}\) in Corollary 3.3, we obtain the following Simpson inequality:
$$\begin{aligned} & \biggl\vert \biggl( \mathcal{R} ( a ) \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{4}\mathcal{R} ( b ) \biggr) ^{\frac{1}{6}} \biggl( \bigl( {}_{a}I_{\ast }^{ \delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{ \ast }I_{b}^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{s+1-2\delta +4\delta ( \frac{2}{3} ) ^{\frac{s+1}{\delta }}}{3 ( s+1 ) ( \delta +s+1 ) }} \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2\mathcal{T}_{2} ( \delta ,s ) } \biggr) ^{\frac{b-a}{4}} \end{aligned}$$
with
$$\begin{aligned} \mathcal{T}_{2} ( \delta ,s ) & = \frac {2}{3 ( s+1 ) } \biggl( 1-2 \biggl( 1- \biggl( \frac {2}{3} \biggr) ^{\frac{1}{\delta }} \biggr) ^{s+1} \biggr) \\ &\quad{} -B_{ ( \frac{2}{3} ) ^{\frac{1}{\delta }}} ( \delta +1,s+1 ) + B_{1- ( \frac{2}{3} ) ^{\frac{1}{\delta }}} ( s+1,\delta +1 ) . \end{aligned}$$
Corollary 3.7
Taking \(\varrho =\frac{7}{15}\) in Corollary 3.3, we obtain the following corrected Simpson inequality:
$$\begin{aligned} & \biggl\vert \biggl( \bigl( \mathcal{R} ( a ) \bigr) ^{7} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{16} \bigl( \mathcal{R} ( b ) \bigr) ^{7} \biggr) ^{ \frac{1}{30}} \biggl( \bigl( {}_{a}I_{\ast }^{\delta }\mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{b}^{ \delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{7s+7-8\delta +16\delta ( \frac{8}{15} ) ^{\frac{s+1}{\delta }}}{15 ( s+1 ) ( \delta +s+1 ) }} \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2\mathcal{T}_{3} ( \delta ,s ) } \biggr) ^{\frac{b-a}{4}} \end{aligned}$$
with
$$\begin{aligned} \mathcal{T}_{3} ( \delta ,s ) & = \frac {8}{15 ( s+1 ) } \biggl( 1-2 \biggl( 1- \biggl( \frac {8}{15} \biggr) ^{ \frac{1}{\delta }} \biggr) ^{s+1} \biggr) \\ &\quad{} -B_{ ( \frac{8}{15} ) ^{\frac{1}{\delta }}} ( \delta +1,s+1 ) +B_{1- ( \frac{8}{15} ) ^{ \frac{1}{\delta }}} ( s+1,\delta +1 ) . \end{aligned}$$
Corollary 3.8
Taking \(\varrho =0\) in Corollary 3.3, we obtain the following midpoint inequality:
$$\begin{aligned} & \biggl\vert \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggl( \bigl( {}_{a}I_{\ast }^{\delta }\mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{b}^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{\delta }{ ( s+1 ) ( \delta +s+1 ) }}\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2 ( \frac{1}{s+1}-B ( \delta +1,s+1 ) ) } \biggr) ^{ \frac{b-a}{4}}. \end{aligned}$$
Corollary 3.9
Taking \(x=a\) and \(\varrho >1\) in Theorem 3.2, we get
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( a ) \bigr) ^{ \frac{\varrho }{2}} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{1-\varrho } \bigl( \mathcal{R} ( b ) \bigr) ^{ \frac{\varrho }{2}} \biggl( \bigl( {}_{a}I_{\ast }^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{b}^{ \delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{\varrho ( \delta +s+1 ) -\delta }{ ( s+1 ) ( \delta +s+1 ) }}\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2 ( \frac{\varrho -1}{s+1}+B ( s+1, \delta +1 ) ) } \biggr) ^{\frac{b-a}{4}}. \end{aligned}$$
Corollary 3.10
Taking \(\varrho =\frac{4}{3}\) in Corollary 3.9, we obtain the following Milne inequality:
$$\begin{aligned} & \biggl\vert \biggl( \bigl( \mathcal{R} ( a ) \bigr) ^{2} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{-1} \bigl( \mathcal{R} ( b ) \bigr) ^{2} \biggr) ^{ \frac{1}{3}} \biggl( \bigl( {}_{a}I_{\ast }^{\delta }\mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{b}^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{\delta +4s+4}{3 ( s+1 ) ( \delta +s+1 ) }}\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2 ( \frac{1}{3 ( s+1 ) }+B ( s+1,\delta +1 ) ) } \biggr) ^{\frac{b-a}{4}}. \end{aligned}$$
Corollary 3.11
Taking \(x=\frac{3a+b}{4}\) and \(\varrho \in [ 0,1 ] \) in Theorem 3.2, we get
$$\begin{aligned} & \biggl\vert \biggl( \mathcal{R} \biggl( \frac {3a+b}{4} \biggr) \biggr) ^{\frac{1+\varrho }{4}} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{\frac{1-\varrho }{2}} \biggl( \mathcal{R} \biggl( \frac {a+3b}{4} \biggr) \biggr) ^{ \frac{1+\varrho }{4}} \bigl( \mathcal{D} ( \mathcal{R} ) \bigr) ^{- \frac{4^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{B ( \delta +1,s+1 ) }\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2\mathcal{T} ( \varrho ,\delta ,s ) } \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {3a+b}{4} \biggr) \mathcal{R}^{\ast } \biggl( \frac {a+3b}{4} \biggr) \biggr) ^{ \frac { ( \delta +s+1 ) \varrho -\delta +2\delta ( 1-\varrho ) ^{\frac{\delta +s+1}{\delta }}}{ ( s+1 ) ( \delta +s+1 ) }+ \frac{1}{\delta +s+1}} \biggr) ^{\frac{b-a}{16}}, \end{aligned}$$
where \(\mathcal{T}\) is defined in (18), and
$$ \mathcal{D} ( \mathcal{R} ) = \biggl( \bigl( {}_{\ast }I_{ \frac{3a+b}{4}}^{\delta } \mathcal{R} \bigr) ( a ) \bigl( {}_{\frac{a+3b}{4}}I_{\ast }^{\delta } \mathcal{R} \bigr) ( b ) \bigl( {}_{\frac{3a+b}{4}}I_{\ast }^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{ \frac{a+3b}{4}}^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) . $$
(19)
Corollary 3.12
Taking \(x=\frac{3a+b}{4}\) and \(\varrho >1\) in Theorem 3.2, we get
$$\begin{aligned} & \biggl\vert \biggl( \mathcal{R} \biggl( \frac {3a+b}{4} \biggr) \biggr) ^{\frac{1+\varrho }{4}} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{\frac{1-\varrho }{2}} \biggl( \mathcal{R} \biggl( \frac {a+3b}{4} \biggr) \biggr) ^{ \frac{1+\varrho }{4}} \bigl( \mathcal{D} ( \mathcal{R} ) \bigr) ^{- \frac{4^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{B ( \delta +1,s+1 ) }\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2 ( \frac{\varrho -1}{s+1}+B ( s+1,\delta +1 ) ) } \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {3a+b}{4} \biggr) \mathcal{R}^{\ast } \biggl( \frac {a+3b}{4} \biggr) \biggr) ^{ \frac{\varrho ( \delta +s+1 ) -\delta }{ ( s+1 ) ( \delta +s+1 ) }+ \frac{1}{\delta +s+1}} \biggr) ^{\frac{b-a}{16}}, \end{aligned}$$
where \(\mathcal{D}\) is given by (19).
Corollary 3.13
Taking \(\varrho =\frac{5}{3}\) in Corollary 3.12, we obtain the following dual-Simpson inequality:
$$\begin{aligned} & \biggl\vert \biggl( \biggl( \mathcal{R} \biggl( \frac {3a+b}{4} \biggr) \biggr) ^{2} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{-1} \biggl( \mathcal{R} \biggl( \frac {a+3b}{4} \biggr) \biggr) ^{2} \biggr) ^{\frac{1}{3}} \bigl( \mathcal{D} ( \mathcal{R} ) \bigr) ^{- \frac{4^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{B ( \delta +1,s+1 ) }\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{2 ( \frac{2}{3 ( s+1 ) }+B ( s+1, \delta +1 ) ) } \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {3a+b}{4} \biggr) \mathcal{R}^{\ast } \biggl( \frac {a+3b}{4} \biggr) \biggr) ^{ \frac{2\delta +8s+8}{3 ( s+1 ) ( \delta +s+1 ) }} \biggr) ^{\frac{b-a}{16}}, \end{aligned}$$
where \(\mathcal{D}\) is given by (19).
Remark 3.14
By setting \(\varrho =\frac{17}{15}\) in Corollary 3.12 we recover the result established in Theorem 3.2 from [33] concerning the corrected dual-Simpson formula.
Corollary 3.15
Taking \(x=\frac{5a+b}{6}\) and \(\varrho \in [ 0,1 ] \) in Theorem 3.2, we get
$$\begin{aligned} & \biggl\vert \biggl( \mathcal{R} \biggl( \frac {5a+b}{6} \biggr) \biggr) ^{\frac{1+2\varrho }{6}} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{\frac{2 ( 1-\varrho ) }{3}} \biggl( \mathcal{R} \biggl( \frac {a+5b}{6} \biggr) \biggr) ^{\frac{1+2\varrho }{6}} \bigl( \mathcal{M} ( \mathcal{R} ) \bigr) ^{- \frac{3^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{B ( \delta +1,s+1 ) }\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{8\mathcal{T} ( \varrho ,\delta ,s ) } \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {5a+b}{6} \biggr) \mathcal{R}^{\ast } \biggl( \frac {a+5b}{6} \biggr) \biggr) ^{4 \frac { ( \delta +s+1 ) \varrho -\delta +2\delta ( 1-\varrho ) ^{\frac{\delta +s+1}{\delta }}}{ ( s+1 ) ( \delta +s+1 ) }+ \frac{1}{\delta +s+1}} \biggr) ^{\frac{b-a}{36}}, \end{aligned}$$
where \(\mathcal{T}\) is defined by (18), and
$$\begin{aligned} \mathcal{M} ( \mathcal{R} ) & = \biggl( \bigl( \bigl( {}_{\ast }I_{\frac{5a+b}{6}}^{\delta }\mathcal{R} \bigr) ( a ) \bigl({}_{ \frac{a+5b}{6}}I_{\ast }^{\delta }\mathcal{R} \bigr) ( b ) \bigr) ^{2^{\delta -1}} \\ &\quad{}\times \biggl( \bigl( {}_{\frac{5a+b}{6}}I_{\ast }^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{\frac{a+5b}{6}}^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) \biggr) . \end{aligned}$$
(20)
Corollary 3.16
Taking \(\varrho =\frac{5}{8}\) in Corollary 3.15, we obtain the following Maclaurin inequality:
$$\begin{aligned} & \biggl\vert \biggl( \biggl( \mathcal{R} \biggl( \frac {5a+b}{6} \biggr) \biggr) ^{3} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2} \biggl( \mathcal{R} \biggl( \frac {a+5b}{6} \biggr) \biggr) ^{3} \biggr) ^{\frac{1}{8}} \bigl( \mathcal{M} ( \mathcal{R} ) \bigr) ^{- \frac{3^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{B ( \delta +1,s+1 ) }\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{8\mathcal{T}_{4} ( \delta ,s ) } \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {5a+b}{6} \biggr) \mathcal{R}^{\ast } \biggl( \frac {a+5b}{6} \biggr) \biggr) ^{\frac{5s+5-3\delta +6\delta ( \frac{3}{8} ) ^{\frac{s+1}{\delta }}}{2 ( s+1 ) ( \delta +s+1 ) }+ \frac{1}{\delta +s+1}} \biggr) ^{\frac{b-a}{36}}, \end{aligned}$$
where \(\mathcal{M}\) is given by (20), and
$$\begin{aligned} \mathcal{T}_{4} ( \delta ,s ) & = \frac {3}{ ( s+1 ) } \biggl( 1-2 \biggl( 1- \biggl( \frac {3}{8} \biggr) ^{\frac{1}{\delta }} \biggr) ^{s+1} \biggr) \\ &\quad{} -B_{( \frac{3}{8} ) ^{\frac{1}{\delta }}} ( \delta +1,s+1 ) +B_{1- ( \frac{3}{8} ) ^{\frac{1}{\delta }}} ( s+1,\delta +1 ) . \end{aligned}$$
Corollary 3.17
Taking \(\varrho =\frac{41}{80}\) in Corollary 3.15, we obtain the following corrected Maclaurin inequality:
$$\begin{aligned} & \biggl\vert \biggl( \biggl( \mathcal{R} \biggl( \frac {5a+b}{6} \biggr) \biggr) ^{27} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{26} \biggl( \mathcal{R} \biggl( \frac {a+5b}{6} \biggr) \biggr) ^{27} \biggr) ^{\frac{1}{80}} \bigl( \mathcal{M} ( f ) \bigr) ^{- \frac{3^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta}}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{B ( \delta +1,s+1 ) }\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{8\mathcal{T}_{5} ( \delta ,s ) } \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {5a+b}{6} \biggr) \mathcal{R}^{\ast } \biggl( \frac {a+5b}{6} \biggr) \biggr) ^{ \frac{41s+41-39\delta +78\delta ( \frac{39}{80} ) ^{\frac{s+1}{\delta }}}{20 ( s+1 ) ( \delta +s+1 ) }+\frac{1}{\delta +s+1}} \biggr) ^{ \frac{b-a}{36}}, \end{aligned}$$
where \(\mathcal{M}\) is given by (20), and
$$\begin{aligned} \mathcal{T}_{5} ( \delta ,s ) & = \frac {39}{80 ( s+1 ) } \biggl( 1-2 \biggl( 1- \biggl( \frac {39}{80} \biggr) ^{ \frac{1}{\delta }} \biggr) ^{s+1} \biggr) \\ &\quad{} -B_{ ( \frac{39}{80} ) ^{\frac{1}{\delta }}} ( \delta +1,s+1 ) + B_{1- ( \frac{39}{80} ) ^{\frac{1}{\delta }}} ( s+1,\delta +1 ) . \end{aligned}$$
Corollary 3.18
Taking \(x=\frac{a+b}{2}\) in Theorem 3.2, we get
$$\begin{aligned} & \biggl\vert \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \bigl( \bigl( {}_{\ast }I_{\frac{a+b}{2}}^{\delta }\mathcal{R} \bigr) ( a ) \bigl( _{ \frac{a+b}{2}}I_{\ast }^{\delta }\mathcal{R} \bigr) ( b ) \bigr) ^{- \frac{2^{\delta -1 }\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{B ( \delta +1,s+1 ) } \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{\frac{2}{\delta +s+1}} \biggr) ^{\frac{b-a}{4}}. \end{aligned}$$
Corollary 3.19
Taking \(s=1\) and \(\varrho \in [ 0,1 ] \) in Theorem 3.2, we get
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( \delta +2 ) ( b-a ) }} \\ &\qquad{}\times \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \frac{2- ( 1-\varrho ) ( \delta +1 ) ( \delta +2 ) }{2 ( \delta +1 ) ( \delta +2 ) }+ \frac{2\delta }{\delta +1} ( 1-\varrho ) ^{ \frac{\delta +1}{\delta }}-\frac{\delta }{\delta +2} ( 1-\varrho ) ^{\frac{\delta +2}{\delta }} ) } \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } \frac{ ( \delta +2 ) \varrho -\delta +2\delta ( 1-\varrho ) ^{\frac{\delta +2}{\delta }}}{2 ( \delta +2 ) }+ \frac{ ( x-a ) ^{2}}{ ( \delta +2 ) ( b-a ) }}. \end{aligned}$$
Corollary 3.20
Taking \(s=1\) in Theorem 3.2, we get, for \(\varrho >1\),
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( \delta +2 ) ( b-a ) }B ( \delta +1,2 ) }\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \frac{\varrho -1}{2}+ \frac{1}{ ( \delta +1 ) ( \delta +2 ) } ) } \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( \varrho ( \delta +2 ) -\delta ) ( a+b-2x ) ^{2}}{8 ( \delta +2 ) ( b-a ) }+\frac{ ( x-a ) ^{2}}{ ( \delta +2 ) ( b-a ) }}. \end{aligned}$$
Corollary 3.21
Taking \(x=a\) in Corollary 3.19, we obtain
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( a ) \bigr) ^{ \frac{\varrho }{2}} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{1-\varrho } \bigl( \mathcal{R} ( b ) \bigr) ^{ \frac{\varrho }{2}} \biggl( \bigl( {}_{a}I_{\ast }^{\delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \bigl( {}_{\ast }I_{b}^{ \delta } \mathcal{R} \bigr) \biggl( \frac {a+b}{2} \biggr) \biggr) ^{- \frac{2^{\delta -1}\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{ ( \delta +2 ) \varrho -\delta +2\delta ( 1-\varrho ) ^{\frac{\delta +2}{\delta }}}{2 ( \delta +2 ) }} \\ &\qquad{} \times \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{ \frac{2- ( 1-\varrho ) ( \delta +1 ) ( \delta +2 ) }{ ( \delta +1 ) ( \delta +2 ) }+ \frac{4\delta }{\delta +1} ( 1-\varrho ) ^{ \frac{\delta +1}{\delta }}-\frac{2\delta }{\delta +2} ( 1-\varrho ) ^{\frac{\delta +2}{\delta }}} \biggr) ^{ \frac{b-a}{4}}. \end{aligned}$$
Remark 3.22
In Corollary 3.21, if we take
1) \(x=a\) and \(\varrho =\frac{1}{3}\), then we obtain Theorem 10 from [31],
2) \(x=a\) and \(\varrho =\frac{1}{2}\), then we obtain Theorem 5 from [30].
Corollary 3.23
By setting \(\delta =1\) in Theorem 3.2we get
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{ ( s+1 ) ( s+2 ) ( b-a ) }}\mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) }\widetilde{\mathcal{N}} ( \varrho ,s ) } \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } \widetilde{\Theta } ( \varrho ,s ) + \frac{ ( x-a ) ^{2}}{ ( s+2 ) ( b-a ) }}, \end{aligned}$$
where
$$ \widetilde{\Theta } ( \varrho ,s ) =\textstyle\begin{cases} \frac { ( s+2 ) \varrho -1+2 ( 1-\varrho ) ^{s+2}}{ ( s+1 ) ( s+2 ) }&\textit{for } 0\leq \varrho \leq 1, \\ \frac {\varrho ( s+2 ) -1}{ ( s+1 ) ( s+2 ) }&\textit{for } \varrho >1,\end{cases} $$
and
$$ \widetilde{\mathcal{N}} ( \varrho ,s ) =\textstyle\begin{cases} \frac{s+1-\varrho ( s+2 ) +2\varrho ^{s+2}}{ ( s+1 ) ( s+2 ) }&\textit{for } 0\leq \varrho \leq 1, \\ \frac{ ( s+2 ) ( \varrho -1 ) +1}{ ( s+1 ) ( s+2 ) }&\textit{for } \varrho >1.\end{cases} $$
Corollary 3.24
Taking \(\delta =s=1\) in Theorem 3.2, we get
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{6 ( b-a ) }}\mathcal{R}^{ \ast } \biggl( \frac {a+b}{2} \biggr) ^{ \frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) } \widehat{\mathcal{N}} ( \varrho ) } \\ &\qquad{}\times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } \widehat{\Theta } ( \varrho ) + \frac{ ( x-a ) ^{2}}{3 ( b-a ) }}, \end{aligned}$$
where
$$ \widehat{\Theta } ( \varrho ) =\textstyle\begin{cases} \frac {1-3\varrho +6\varrho ^{2}-2\varrho ^{3}}{6}&\textit{for } 0\leq \varrho \leq 1, \\ \frac {3\varrho -1}{6}&\textit{for } \varrho >1,\end{cases} $$
and
$$ \widehat{\mathcal{N}} ( \varrho ) =\textstyle\begin{cases} \frac{2-3\varrho +2\varrho ^{3}}{6}&\textit{for } 0\leq \varrho \leq 1, \\ \frac{3\varrho -2}{6}&\textit{for } \varrho >1.\end{cases} $$
Remark 3.25
In Corollary 3.24, if we take
1) \(x=\frac{a+b}{2}\), then we obtain Theorem 3.3 from [17];
2) \(x=a\) and \(\varrho =1\), then using the multiplicative convexity of \(\mathcal{R}^{\ast }\), we get Theorem 3.6 from [17];
3) \(x=a\) and \(\varrho =\frac{1}{3}\), then we obtain Corollary 3 from [31];
4) \(x=a\) and \(\varrho =\frac{1}{2}\), then we obtain Corollary 3 from [30];
5) \(x=\frac{5a+b}{6}\) and \(\varrho =\frac{5}{8}\), then we obtain Theorem 3.2 from [19];
6) \(x=\frac{3a+b}{4}\) and ϱ= \(\frac{5}{3}\), then we obtain Theorem 3.2 from [20].
Example 3.26
Consider the function \(\mathcal{R} (u )=2^{u^{s+1}}\) for \(s\in (0,1]\) with \(a=0\) and \(b=1\), the multiplicative derivative of this function is \(\mathcal{R}^{*} (u )=2^{(s+1)u^{s}}\), which exhibits multiplicative s-convexity on the interval \([0,1]\). Thus by Theorem 3.2, for \(0<\delta \leq 1\),
$$\begin{aligned} & \frac{2^{ [x^{s+1} (\frac {(2x+\varrho (1-2x))}{2} )+2^{-(s+1)} (1-\varrho )(1-2x)+(1-x)^{s+1} \frac {(2x+\varrho (1-2x))}{2} ]}}{e^{\delta \ln 2 [ ( \int^{x}_{0}u^{\delta +s} \,du +\int^{1}_{1-x}u^{s+1} ( 1-u ) ^{\delta -1}\,du )x^{1-\delta} + (\int^{\frac{1}{2}}_{x}u^{s+1} ( \frac{1}{2}-u )^{\delta -1}\,du+\int^{1-x}_{\frac{1}{2}}u^{s+1} ( u -\frac{1}{2} ) ^{\delta -1}\,du ) (\frac {1-2x}{2} )^{1-\delta} ]}} \\ &\quad \leq 2^{(s+1) [x^{2}B ( \delta +1,s+1 ) + \frac { ( 1-2x ) ^{2}}{2^{s+1}}\mathcal{N} ( \varrho , \delta ,s ) + (x^{s}+(1-x)^{s} ) ( \frac{ ( 1-2x ) ^{2}}{4 }\Theta ( \varrho ,\delta ,s ) +\frac{x^{2}}{ ( \delta +s+1 ) } ) ]}, \end{aligned}$$
where Θ and \(\mathcal{N}\) are defined by (7) and (8), and B is the beta function.
Since this result depends on four parameters, we will set \(\delta =1\) and \(x=a=0\) and then represent the result with respect to the remaining two. The outcome is illustrated in Fig. 1.
$$ 2^{ [2^{-(s+1)} (1-\varrho )+\frac {\varrho}{2}- \frac{1}{s+2} ]} \leq \textstyle\begin{cases} 2^{(s+1) [ \frac {s+1-\varrho (s+2)+2\varrho ^{s+2}}{2^{s+1}(s+1)(s+2)}+ \frac {(s+2)\varrho -1+2 (1-\varrho )^{s+2}}{4(s+1)(s+2)} ]}& \text{for } \varrho \in [0,1], \\ 2^{(s+1) [ \frac { (\varrho -1 )(s+2)+1}{2^{s+1}(s+1)(s+2)} + \frac {\varrho (s+2)-1}{4(s+1)(s+2)} ]}& \text{for } \varrho >1. \end{cases} $$
Based on the observations from Fig. 1, it is evident that the right-hand term consistently surpasses the left-hand term. This holds for all \(\varrho \in [0,2]\) and \(s\in (0,1]\), providing substantiation for the validity of our findings.
Theorem 3.27
Let \(\mathcal{R}:[a,b]\rightarrow \mathbb{R}^{+}\) be an increasing multiplicatively differentiable function on \([a,b]\). If for \(q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), \(( \ln \mathcal{R}^{\ast } ) ^{q}\) is s-convex on this interval, then for all \(x\in [a, \frac{a+b}{2} ]\), \(\varrho \geq 0\), and \(\delta > 0\), we have
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{b-a} ( \frac{1}{\delta p+1} ) ^{ \frac{1}{p}} ( \frac{1}{s+1} ) ^{\frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } ( x ) \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2}\mathcal{R}^{\ast } ( a+b-x ) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \frac{1}{s+1} ) ^{\frac{1}{q}} ( \Lambda ( \delta ,p,\varrho ) ) ^{ \frac{1}{p}}}, \end{aligned}$$
where \(\mathcal{Q}\) and \(\mathcal{I}\) are defined by (5) and (6), respectively, and
$$ \Lambda ( \delta ,p,\varrho ) =\textstyle\begin{cases} \frac{ ( 1-\varrho ) ^{p+\frac{1}{\delta }}}{\delta }B ( \frac{1}{\delta },p+1 ) + \frac{\varrho ^{p+\frac{1}{\delta }}}{\delta ( p+1 ) }\, _{2}\mathcal{F}_{1} ( 1-\frac{1}{\delta },1,p+2; \varrho ) &\textit{for } \varrho \in [ 0,1 ], \\ \varrho ^{p}\, _{2}\mathcal{F}_{1} ( -p,1,\frac{1}{\delta }+1; \frac{1}{\varrho } ) &\textit{for } \varrho \in [ 1,2 ], \\ \frac{ ( \varrho -1 ) ^{p}}{\delta }\ _{2}\mathcal{F}_{1} ( -p,\frac{1}{\delta },\frac{1}{\delta }+1;\frac{1}{1-\varrho } ) &\textit{for } \varrho >2, \end{cases} $$
(21)
with B and \({}_{2}\mathcal{F}_{1}\) the beta and hypergeometric functions, respectively.
Proof
By Lemma 3.1, applying the Hölder inequality, we get
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \int^{1}_{0} \gamma ^{\delta p}\,d\gamma \biggr) ^{\frac{1}{p}} \biggl( \int^{1}_{0} \bigl\vert \ln \bigl( \mathcal{R}^{\ast } \bigl( ( 1- \gamma ) a+\gamma x \bigr) \bigr) \bigr\vert ^{q}\,d \gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \biggl( \int^{1}_{0} \bigl\vert 1-\varrho - ( 1-\gamma ) ^{\delta } \bigr\vert ^{p}\,d\gamma \biggr) ^{ \frac{1}{p}} \\ &\qquad{}\times \biggl( \int^{1}_{0} \biggl\vert \ln \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) x+\gamma \frac {a+b}{2} \biggr) \biggr\vert ^{q}\,d\gamma \biggr) ^{ \frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \biggl( \int^{1}_{0} \bigl\vert \gamma ^{\delta }- ( 1-\varrho ) \bigr\vert ^{p}\,d\gamma \biggr) ^{ \frac{1}{p}} \\ &\qquad{}\times \biggl( \int^{1}_{0} \biggl\vert \ln \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) \frac {a+b}{2}+ \gamma ( a+b-x ) \biggr) \biggr\vert ^{q}\,d \gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \int^{1}_{0} ( 1-\gamma ) ^{\delta p}\,d \gamma \biggr) ^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int_{0}^{1} \bigl\vert \ln \mathcal{R}^{ \ast } \bigl( ( 1-\gamma ) ( a+b-x ) + \gamma b \bigr) \bigr\vert ^{q}\,d \gamma \biggr) ^{\frac{1}{q}} \biggr) . \end{aligned}$$
From the s-convexity of \(( \ln \mathcal{R}^{\ast } ) ^{q}\), we have
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta p+1} \biggr) ^{\frac{1}{p}} \biggl( \int_{0}^{1} \bigl( ( 1-\gamma ) ^{s} \bigl( \ln \bigl( \mathcal{R}^{ \ast } ( a ) \bigr) \bigr) ^{q}+\gamma ^{s} \bigl( \ln \bigl( \mathcal{R}^{\ast } ( x ) \bigr) \bigr) ^{q} \bigr) \,d\gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Lambda ( \delta ,p, \varrho ) \bigr) ^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int_{0}^{1} \biggl( ( 1-\gamma ) ^{s} \bigl( \ln \mathcal{R}^{\ast } ( x ) \bigr) ^{q}+\gamma ^{s} \biggl( \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{q} \biggr) \,d\gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Lambda ( \delta ,p, \varrho ) \bigr) ^{ \frac{1}{p}} \\ &\qquad{} \times \biggl( \int_{0}^{1} \biggl( ( 1-\gamma ) ^{s} \biggl( \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{q}+\gamma ^{s} \bigl( \ln \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{q} \biggr) \,d \gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta p+1} \biggr) ^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int_{0}^{1} \bigl( ( 1-\gamma ) ^{s} \bigl( \ln \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{q}+\gamma ^{s} \bigl( \ln \mathcal{R}^{\ast } ( b ) \bigr) ^{q} \bigr) \,d\gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\quad =\exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta p+1} \biggr) ^{\frac{1}{p}} \biggl( \frac {1}{s+1} \biggr) ^{\frac{1}{q}} \bigl( \bigl( \ln \bigl( \mathcal{R}^{\ast } ( a ) \bigr) \bigr) ^{q}+ \bigl( \ln \bigl( \mathcal{R}^{\ast } ( x ) \bigr) \bigr) ^{q} \bigr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Lambda ( \delta ,p, \varrho ) \bigr) ^{ \frac{1}{p}} \biggl( \frac {1}{s+1} \biggr) ^{\frac{1}{q}} \\ &\qquad{}\times \biggl( \bigl( \ln \mathcal{R}^{\ast } ( x ) \bigr) ^{q}+ \biggl( \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{q} \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Lambda ( \delta ,p, \varrho ) \bigr) ^{ \frac{1}{p}} \biggl( \frac {1}{s+1} \biggr) ^{\frac{1}{q}} \\ &\qquad{}\times \biggl( \biggl( \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{q}+ \bigl( \ln \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{q} \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta p+1} \biggr) ^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac {1}{s+1} \biggr) ^{ \frac{1}{q}} \bigl( \bigl( \ln \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{q}+ \bigl( \ln \mathcal{R}^{\ast } ( b ) \bigr) ^{q} \bigr) ^{\frac{1}{q}} \biggr) , \end{aligned}$$
(22)
where we have used that
$$ \int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1- \varrho ) \bigr\vert ^{p}\,d\gamma = \frac{ ( 1-\varrho ) ^{p+\frac{1}{\delta }}}{\delta }B \biggl( \frac{1}{\delta },p+1 \biggr) + \frac{\varrho ^{p+\frac{1}{\delta }}}{\delta ( p+1 ) }\ _{2} \mathcal{F}_{1} \biggl( 1-\frac{1}{\delta },1,p+2; \varrho \biggr) $$
for \(\varrho \in [ 0,1 ] \),
$$ \int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1- \varrho ) \bigr\vert ^{p}\,d\gamma =\varrho ^{p}\ _{2} \mathcal{F}_{1} \biggl( -p,1,\frac{1}{\delta }+1; \frac{1}{\varrho } \biggr) $$
for \(1<\varrho \leq 2\), and
$$ \int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1- \varrho ) \bigr\vert ^{p}\,d\gamma = \frac{ ( \varrho -1 ) ^{p}}{\delta }\ _{2} \mathcal{F}_{1} \biggl( -p,\frac{1}{\delta }, \frac{1}{\delta }+1; \frac{1}{1-\varrho } \biggr) $$
for \(\varrho >2\). Using the inequality \(\mathcal{X}^{q}+\mathcal{Y}^{q}\leq (\mathcal{X}+\mathcal{Y} )^{q}\) for \(\mathcal{X}\geq 0\), \(\mathcal{Y}\geq 0\), and \(q\geq 1\), (22) gives
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \bigr\vert \\ &\quad \leq \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta p+1} \biggr) ^{\frac{1}{p}} \biggl( \frac {1}{s+1} \biggr) ^{ \frac{1}{q}} \bigl( \ln \mathcal{R}^{\ast } ( a ) +\ln \mathcal{R}^{ \ast } ( x ) \bigr) \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Lambda ( \delta ,p, \varrho ) \bigr) ^{ \frac{1}{p}} \biggl( \frac {1}{s+1} \biggr) ^{\frac{1}{q}} \biggl( \ln \mathcal{R}^{\ast } ( x ) +\ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Lambda ( \delta ,p, \varrho ) \bigr) ^{ \frac{1}{p}} \\ &\qquad{}\times \biggl( \frac {1}{s+1} \biggr) ^{\frac{1}{q}} \biggl( \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) +\ln \mathcal{R}^{\ast } ( a+b-x ) \biggr) \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta p+1} \biggr) ^{\frac{1}{p}} \biggl( \frac {1}{s+1} \biggr) ^{ \frac{1}{q}} \bigl( \ln \mathcal{R}^{\ast } ( a+b-x ) +\ln \mathcal{R}^{\ast } ( b ) \bigr) \biggr) \\ &\quad = \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac { ( x-a ) ^{2}}{b-a} ( \frac {1}{\delta p+1} ) ^{\frac{1}{p}} ( \frac {1}{s+1} ) ^{\frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } ( x ) \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2}\mathcal{R}^{\ast } ( a+b-x ) \biggr) ^{\frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \frac {1}{s+1} ) ^{\frac{1}{q}} ( \Lambda ( \delta ,p,\varrho ) ) ^{ \frac{1}{p}}}. \end{aligned}$$
The proof is completed. □
Corollary 3.28
By setting \(s=1\) in Theorem 3.27we get
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{b-a} ( \frac{1}{\delta p+1} ) ^{ \frac{1}{p}} ( \frac{1}{2} ) ^{\frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } ( x ) \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2}\mathcal{R}^{\ast } ( a+b-x ) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \frac{1}{2} ) ^{\frac{1}{q}} ( \Lambda ( \delta ,p,\varrho ) ) ^{ \frac{1}{p}}}, \end{aligned}$$
where \(\mathcal{Q}\), \(\mathcal{I}\), and Λ are defined by (5), (6), and (21), respectively.
Corollary 3.29
By setting \(\delta =1\) in Theorem 3.27we get, for \(\varrho \in [ 0,1 ] \),
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{b-a}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } ( x ) \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2} \mathcal{R}^{\ast } ( a+b-x ) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( ( 1-\varrho ) ^{p+1}+\varrho ^{p+1} ) ^{ \frac{1}{p}}} \biggr) ^{ ( \frac{1}{p+1} ) ^{\frac{1}{p}} ( \frac{1}{s+1} ) ^{ \frac{1}{q}}}. \end{aligned}$$
Corollary 3.30
Taking \(\delta =1\) and \(\varrho >1\) in Theorem 3.27, we get
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{b-a}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } ( x ) \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2} \mathcal{R}^{\ast } ( a+b-x ) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \varrho ^{p+1}- ( \varrho -1 ) ^{p+1} ) ^{ \frac{1}{p}}} \biggr) ^{ ( \frac{1}{p+1} ) ^{\frac{1}{p}} ( \frac{1}{s+1} ) ^{\frac{1}{q}}}. \end{aligned}$$
Corollary 3.31
Taking \(\delta =s=1\) in Theorem 3.27, we get, for \(\varrho \in [ 0,1 ]\),
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{b-a}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } ( x ) \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2} \mathcal{R}^{\ast } ( a+b-x ) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( ( 1-\varrho ) ^{p+1}+\varrho ^{p+1} ) ^{ \frac{1}{p}}} \biggr) ^{ ( \frac{1}{p+1} ) ^{\frac{1}{p}} ( \frac{1}{2} ) ^{ \frac{1}{q}}}. \end{aligned}$$
Corollary 3.32
Taking \(\delta =s=1\) in Theorem 3.27, we get, for \(\varrho >1\),
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ &\quad \leq \biggl( \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{b-a}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } ( x ) \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{2} \mathcal{R}^{\ast } ( a+b-x ) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \varrho ^{p+1}- ( \varrho -1 ) ^{p+1} ) ^{ \frac{1}{p}}} \biggr) ^{ ( \frac{1}{p+1} ) ^{\frac{1}{p}} ( \frac{1}{2} ) ^{\frac{1}{q}}}. \end{aligned}$$
Theorem 3.33
Let \(\mathcal{R}:[a,b]\rightarrow \mathbb{R}^{+}\) be an increasing multiplicatively differentiable function on \([a,b]\). If \(( \ln \mathcal{R}^{\ast } ) ^{q}\) is s-convex on this interval for \(q>1\), then for all \(x\in [a, \frac{a+b}{2} ]\), \(\varrho \geq 0\), and \(\delta > 0\), we have
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( b-a ) } ( ( \delta +1 ) B ( \delta +1,s+1 ) ) ^{\frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( b-a ) } ( \frac{\delta +1}{\delta +s+1} ) ^{ \frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \Omega ( \delta ,\varrho ) ) ^{1-\frac{1}{q}} ( \Theta ( \varrho ,\delta ,s ) ) ^{ \frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \Omega ( \delta ,\varrho ) ) ^{1-\frac{1}{q}} ( \mathcal{N} ( \varrho ,\delta ,s ) ) ^{\frac{1}{q}}}, \end{aligned}$$
where \(\mathcal{Q}\), \(\mathcal{I}\), Θ, and \(\mathcal{N}\) are defined by (5)–(8), respectively, B is the beta function, and
$$ \Omega ( \delta ,\varrho ) =\textstyle\begin{cases} \frac {\varrho -\delta ( 1-\varrho ) }{\delta +1}+ \frac {2\delta }{\delta +1} ( 1-\varrho ) ^{ \frac{\delta +1}{\delta }}&\textit{for } \varrho \in [ 0,1 ] , \\ \frac { ( \delta +1 ) ( \varrho -1 ) +1}{\delta +1}& \textit{for } \varrho >1.\end{cases} $$
(23)
Proof
By Lemma 3.1, applying the power mean inequality, we deduce
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \int_{0}^{1} \gamma ^{\delta }\,d\gamma \biggr) ^{1-\frac{1}{q}} \biggl( \int^{1}_{0} \gamma ^{\delta } \bigl\vert \ln \bigl( \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) a+\gamma x \bigr) \bigr) \bigr\vert ^{q}\,d\gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \biggl( \int^{1}_{0} \bigl\vert 1-\varrho - ( 1-\gamma )^{\delta} \bigr\vert \,d\gamma \biggr) ^{1- \frac{1}{q}} \\ &\qquad{} \times \biggl( \int_{0}^{1} \bigl\vert 1- \varrho - ( 1-\gamma )^{\delta} \bigr\vert \biggl\vert \ln \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) x+\gamma \frac {a+b}{2} \biggr) \biggr\vert ^{q}\,d\gamma \biggr) ^{ \frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \biggl( \int^{1}_{0} \bigl\vert \gamma ^{\delta }- ( 1-\varrho ) \bigr\vert \,d \gamma \biggr) ^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl( \int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1-\varrho ) \bigr\vert \biggl\vert \ln \mathcal{R}^{\ast } \biggl( ( 1-\gamma ) \frac {a+b}{2}+\gamma ( a+b-x ) \biggr) \biggr\vert ^{q}\,d \gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \int^{1}_{0} ( 1-\gamma ) ^{\delta }\,d \gamma \biggr) ^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl( \int_{0}^{1} ( 1- \gamma ) ^{\delta } \bigl\vert \ln \mathcal{R}^{\ast } \bigl( ( 1-\gamma ) ( a+b-x ) +\gamma b \bigr) \bigr\vert ^{q}\,d\gamma \biggr) ^{\frac{1}{q}} \biggr). \end{aligned}$$
Since \(( \ln \mathcal{R}^{\ast } ) ^{q}\) is s-convex, we obtain
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta +1} \biggr) ^{1-\frac{1}{q}} \\ &\qquad{}\times \biggl( \int_{0}^{1}\gamma ^{\delta } \bigl( ( 1-\gamma ) ^{s} \bigl( \ln \bigl( \mathcal{R}^{\ast } ( a ) \bigr) \bigr) ^{q}+ \gamma ^{s} \bigl( \ln \bigl( \mathcal{R}^{\ast } ( x ) \bigr) \bigr) ^{q} \bigr) \,d\gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Omega ( \delta , \varrho ) \bigr) ^{1- \frac{1}{q}} \\ &\qquad{} \times \biggl( \int_{0}^{1} \bigl\vert 1- \varrho - ( 1-\gamma ) ^{\delta } \bigr\vert \biggl( ( 1-\gamma ) ^{s} \bigl( \ln \mathcal{R}^{\ast } ( x ) \bigr) ^{q}+\gamma ^{s} \biggl( \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{q} \biggr) \,d\gamma \biggr) ^{ \frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Omega ( \delta , \varrho ) \bigr) ^{1- \frac{1}{q}} \\ &\qquad{} \times \biggl( \int_{0}^{1} \bigl\vert \gamma ^{\delta }- ( 1-\varrho ) \bigr\vert \biggl( ( 1-\gamma ) ^{s} \biggl( \ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{q} \\ &\qquad{} +\gamma ^{s} \bigl( \ln \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{q} \biggr) \,d \gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta +1} \biggr) ^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl( \int_{0}^{1} ( 1- \gamma ) ^{\delta } \bigl( ( 1-\gamma ) ^{s} \bigl( \ln \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{q}+ \gamma ^{s} \bigl( \ln \mathcal{R}^{\ast } ( b ) \bigr) ^{q} \bigr) \,d\gamma \biggr) ^{\frac{1}{q}} \biggr) \\ &\quad =\exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta +1} \biggr) ^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl( \bigl( \bigl( B ( \delta +1,s+1 ) \bigr) ^{\frac{1}{q}}\ln \bigl( \mathcal{R}^{\ast } ( a ) \bigr) \bigr) ^{q}+ \biggl( \biggl( \frac {1}{\delta +s+1} \biggr) ^{ \frac{1}{q}}\ln \bigl( \mathcal{R}^{\ast } ( x ) \bigr) \biggr) ^{q} \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Omega ( \delta , \varrho ) \bigr) ^{1- \frac{1}{q}} \\ &\qquad{} \times \biggl( \bigl( \bigl( \widehat{\Theta } ( \varrho ,\delta ,s ) \bigr) ^{\frac{1}{q}}\ln \mathcal{R}^{ \ast } ( x ) \bigr) ^{q}+ \biggl( \bigl( \widehat{\mathcal{N}} ( \varrho ,\delta ,s ) \bigr) ^{ \frac{1}{q}}\ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{q} \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Omega ( \delta , \varrho ) \bigr) ^{1- \frac{1}{q}} \\ &\qquad{} \times \biggl( \biggl( \bigl( \widehat{\mathcal{N}} ( \varrho ,\delta ,s ) \bigr) ^{\frac{1}{q}}\ln \mathcal{R}^{ \ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{q}+ \bigl( \bigl( \widehat{\Theta } ( \varrho ,\delta ,s ) \bigr) ^{\frac{1}{q}}\ln \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{q} \biggr) ^{\frac{1}{q}} \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta +1} \biggr) ^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl( \biggl( \biggl( \frac {1}{\delta +s+1} \biggr) ^{\frac{1}{q}}\ln \mathcal{R}^{\ast } ( a+b-x ) \biggr) ^{q}+ \bigl( \bigl( B ( \delta +1,s+1 ) \bigr) ^{\frac{1}{q}} \ln \mathcal{R}^{\ast } ( b ) \bigr) ^{q} \biggr) ^{ \frac{1}{q}} \biggr) , \end{aligned}$$
(24)
where we have used the equalities
$$ \int_{0}^{1} \bigl\vert 1-\varrho - ( 1- \gamma ) ^{\delta } \bigr\vert \,d\gamma = \int_{0}^{1} \bigl\vert 1-\varrho -\gamma ^{ \delta } \bigr\vert \,d \gamma =\Omega (\delta ,\varrho ). $$
Substituting (14)–(17) and using the inequality \(\mathcal{X}^{q}+\mathcal{Y}^{q}\leq (\mathcal{X}+\mathcal{Y} )^{q}\) for \(\mathcal{X}\geq 0\), \(\mathcal{Y}\geq 0\), and \(q\geq 1\), (24) gives
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{ ( b-a ) ^{\delta }}} \bigr\vert \\ &\quad \leq \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta +1} \biggr) ^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl( \bigl( B ( \delta +1,s+1 ) \bigr) ^{\frac{1}{q}}\ln \bigl( \mathcal{R}^{\ast } ( a ) \bigr) + \biggl( \frac {1}{\delta +s+1} \biggr) ^{\frac{1}{q}}\ln \bigl( \mathcal{R}^{\ast } ( x ) \bigr) \biggr) \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Omega ( \delta , \varrho ) \bigr) ^{1- \frac{1}{q}} \\ &\qquad{} \times \biggl( \bigl( \Theta ( \varrho ,\delta ,s ) \bigr) ^{\frac{1}{q}}\ln \mathcal{R}^{\ast } ( x ) + \bigl( \mathcal{N} ( \varrho ,\delta ,s ) \bigr) ^{\frac{1}{q}}\ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( a+b-2x ) ^{2}}{4 ( b-a ) } \bigl( \Omega ( \delta , \varrho ) \bigr) ^{1- \frac{1}{q}} \\ &\qquad{} \times \biggl( \bigl( \mathcal{N} ( \varrho ,\delta ,s ) \bigr) ^{\frac{1}{q}}\ln \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) + \bigl( \Theta ( \varrho ,\delta ,s ) \bigr) ^{\frac{1}{q}}\ln \mathcal{R}^{\ast } ( a+b-x ) \biggr) \biggr) \\ &\qquad{} \times \exp \biggl( \frac { ( x-a ) ^{2}}{b-a} \biggl( \frac {1}{\delta +1} \biggr) ^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl( \biggl( \frac {1}{\delta +s+1} \biggr) ^{ \frac{1}{q}}\ln \mathcal{R}^{\ast } ( a+b-x ) + \bigl( B ( \delta +1,s+1 ) \bigr) ^{\frac{1}{q}}\ln \mathcal{R}^{\ast } ( b ) \biggr) \biggr) \\ &\quad = \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( b-a ) } ( ( \delta +1 ) B ( \delta +1,s+1 ) ) ^{\frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( b-a ) } ( \frac{\delta +1}{\delta +s+1} ) ^{ \frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \Omega ( \delta ,\varrho ) ) ^{1-\frac{1}{q}} ( \Theta ( \varrho ,\delta ,s ) ) ^{ \frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \Omega ( \delta ,\varrho ) ) ^{1-\frac{1}{q}} ( \mathcal{N} ( \varrho ,\delta ,s ) ) ^{\frac{1}{q}}}. \end{aligned}$$
The proof is completed. □
Corollary 3.34
Taking \(s=1\) in Theorem 3.33, we get, for \(\varrho \in [ 0,1 ] \),
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( b-a ) } ( \frac{1}{ ( \delta +2 ) } ) ^{\frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( b-a ) } ( \frac{\delta +1}{\delta +2} ) ^{ \frac{1}{q}}+ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) }}\mathcal{V} ( \varrho ,\delta ) \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \frac{\varrho -\delta ( 1-\varrho ) +2\delta ( 1-\varrho ) ^{\frac{\delta +1}{\delta }}}{\delta +1} ) ^{1-\frac{1}{q}} ( \mathcal{W} ( \varrho ,\delta ) ) ^{\frac{1}{q}}} \end{aligned}$$
with
$$ \mathcal{V} ( \varrho ,\delta ) = \biggl( \frac {\varrho -\delta ( 1-\varrho ) +2\delta ( 1-\varrho ) ^{\frac{\delta +1}{\delta }}}{\delta +1} \biggr) ^{1-\frac{1}{q}} \biggl( \frac { ( \delta +2 ) \varrho -\delta +2\delta ( 1-\varrho ) ^{\frac{\delta +2}{\delta }}}{2 ( \delta +2 ) } \biggr) ^{\frac{1}{q}} $$
and
$$ \mathcal{W} ( \varrho ,\delta ) = \frac {2- ( 1-\varrho ) ( \delta +1 ) ( \delta +2 ) }{2 ( \delta +1 ) ( \delta +2 ) }+\frac {2\delta }{\delta +1} ( 1- \varrho ) ^{\frac{\delta +1}{\delta }}- \frac {\delta }{\delta +2} ( 1-\varrho ) ^{ \frac{\delta +2}{\delta }}, $$
where \(\mathcal{Q}\) and \(\mathcal{I}\) are defined by (5) and (6), respectively.
Corollary 3.35
Taking \(s=1\) in Theorem 3.33, we get, for \(\varrho >1\),
$$\begin{aligned} & \bigl\vert \mathcal{Q} ( a,b,x,\varrho ;\mathcal{R} ) \bigl( \mathcal{I} ( a,b,x;\mathcal{R} ) \bigr) ^{- \frac{\Gamma ( \delta +1 ) }{b-a}} \bigr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( b-a ) } ( \frac{1}{ ( \delta +2 ) } ) ^{\frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{ ( \delta +1 ) ( b-a ) } ( \frac{\delta +1}{\delta +2} ) ^{ \frac{1}{q}}+ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \frac{ ( \delta +1 ) ( \varrho -1 ) +1}{\delta +1} ) ^{1-\frac{1}{q}} ( \frac{ ( \varrho ( \delta +2 ) -\delta ) }{8 ( \delta +2 ) } ) ^{\frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{\frac{ ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \frac{ ( \delta +1 ) ( \varrho -1 ) +1}{\delta +1} ) ^{1-\frac{1}{q}} ( \frac{\varrho -1}{2}+ \frac{1}{ ( \delta +1 ) ( \delta +2 ) } ) ^{\frac{1}{q}}}, \end{aligned}$$
where \(\mathcal{Q}\) and \(\mathcal{I}\) are defined by (5) and (6), respectively.
Corollary 3.36
Taking \(\delta =1\) in Theorem 3.33, we get, for \(\varrho \in [ 0,1 ] \),
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{2 ( b-a ) } ( \frac{2}{ ( s+1 ) ( s+2 ) } ) ^{\frac{1}{q}}} \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{2 ( b-a ) } ( \frac{2}{s+2} ) ^{\frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \frac{1-2\varrho +2\varrho ^{2}}{2} ) ^{1-\frac{1}{q}} ( \frac{ ( s+2 ) \varrho -1+2 ( 1-\varrho ) ^{s+2}}{ ( s+1 ) ( s+2 ) } ) ^{\frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac { ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \frac{1-2\varrho +2\varrho ^{2}}{2} ) ^{1-\frac{1}{q}} ( \frac{s+1- ( s+2 ) \varrho +2\varrho ^{s+2}}{ ( s+1 ) ( s+2 ) } ) ^{\frac{1}{q}}}. \end{aligned}$$
Corollary 3.37
By setting \(\delta =1\) in Theorem 3.33we get, for \(\varrho >1\),
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ & \quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{2 ( b-a ) } ( \frac{2}{ ( s+1 ) ( s+2 ) } ) ^{\frac{1}{q}}} \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{2 ( b-a ) } ( \frac{2}{s+2} ) ^{\frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \varrho -\frac{1}{2} ) ^{1-\frac{1}{q}} ( \frac{\varrho ( s+2 ) -1}{ ( s+1 ) ( s+2 ) } ) ^{ \frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac { ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \varrho - \frac{1}{2} ) ^{1-\frac{1}{q}} ( \frac{ ( s+2 ) ( \varrho -1 ) +1}{ ( s+1 ) ( s+2 ) } ) ^{\frac{1}{q}}}. \end{aligned}$$
Corollary 3.38
Taking \(\delta =s=1\) and \(\varrho \in [ 0,1 ] \) in Theorem 3.33, we get
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ & \quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{2 ( b-a ) } ( \frac{1}{3} ) ^{\frac{1}{q}}} \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{2 ( b-a ) } ( \frac{2}{3} ) ^{\frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \frac{1-2\varrho +2\varrho ^{2}}{2} ) ^{1-\frac{1}{q}} ( \frac{1-3\varrho +6\varrho ^{2}-2\varrho ^{3}}{6} ) ^{ \frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac { ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \frac{1-2\varrho +2\varrho ^{2}}{2} ) ^{1-\frac{1}{q}} ( \frac{2-3\varrho +2\varrho ^{3}}{6} ) ^{\frac{1}{q}}}. \end{aligned}$$
Corollary 3.39
Taking \(\delta =s=1\) and \(\varrho >1\) in Theorem 3.33, we get
$$\begin{aligned} & \biggl\vert \bigl( \mathcal{R} ( x ) \mathcal{R} ( a+b-x ) \bigr) ^{ \frac{2 ( x-a ) +\varrho ( a+b-2x ) }{2 ( b-a ) }} \biggl( \mathcal{R} \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac{ ( 1-\varrho ) ( a+b-2x ) }{b-a}} \biggl( \int^{b}_{a} \bigl( \mathcal{R} ( u ) \bigr) ^{du} \biggr) ^{ \frac{1}{a-b}} \biggr\vert \\ &\quad \leq \bigl( \mathcal{R}^{\ast } ( a ) \mathcal{R}^{\ast } ( b ) \bigr) ^{\frac{ ( x-a ) ^{2}}{2 ( b-a ) } ( \frac{1}{3} ) ^{\frac{1}{q}}} \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{\ast } ( a+b-x ) \bigr) ^{ \frac{ ( x-a ) ^{2}}{2 ( b-a ) } ( \frac{2}{3} ) ^{\frac{1}{q}}} \\ &\qquad{} \times \bigl( \mathcal{R}^{\ast } ( x ) \mathcal{R}^{ \ast } ( a+b-x ) \bigr) ^{ \frac{ ( a+b-2x ) ^{2}}{4 ( b-a ) } ( \varrho -\frac{1}{2} ) ^{1-\frac{1}{q}} ( \frac{3\varrho -1}{6} ) ^{\frac{1}{q}}} \\ &\qquad{} \times \biggl( \mathcal{R}^{\ast } \biggl( \frac {a+b}{2} \biggr) \biggr) ^{ \frac { ( a+b-2x ) ^{2}}{2 ( b-a ) } ( \varrho - \frac{1}{2} ) ^{1-\frac{1}{q}} ( \frac{3\varrho -2}{6} ) ^{ \frac{1}{q}}}. \end{aligned}$$