For establishing our generalized fractional Bullen-type inequalities, the following lemma plays a significant role.
Lemma 1
Let \(f:I\subseteq \mathbf{R}^{+}\rightarrow \mathbf{R}\) be a differentiable function on \(I^{\circ}\), the interior of I, \(\varrho _{1},\varrho _{2}\in I^{\circ}\) with \(\varrho _{1}<\varrho _{2}\); \(p,\rho ,\beta >0\); let \(g(x)=\sqrt[p]{x}\), \(x>0\); \(u,w,\gamma \in \mathbf{R}\) and \(\lambda \in [0,1]\), then
$$\begin{aligned}& \Omega (f,\varrho _{1},\varrho _{2};u) \\& \quad := \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \biggl\{ (1-\lambda )^{1+\beta}(1- \gamma ) \int _{0}^{1} \bigl\{ t^{\beta} \mathfrak{F}_{\rho ,\beta +1}^{ \sigma} \bigl[w \bigl((1-\lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr]-u \bigr\} \\& \qquad {} \times \bigl[(1-t) \bigl(\lambda \varrho _{1}^{p}+(1- \lambda ) \varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{\frac{1-p}{p}} f' \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t\varrho _{1}^{p}} \bigr)\,dt \\& \qquad {} +\gamma \lambda ^{1+\beta} \int _{0}^{1} \bigl\{ t^{\beta} \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p} - \varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr] - u \bigr\} \\& \qquad {} \times \bigl[t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr) + (1 - t) \varrho _{2}^{p} \bigr]^{ \frac{1-p}{p}} \\& \qquad {} \times f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr)\,dt \biggr\} \\& \quad =-p \bigl[(1-\lambda )^{\beta}(1-\gamma ) \bigl[ \bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1-\lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} f( \varrho _{1}) \\& \qquad {} +u f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr]+\lambda ^{\beta}\gamma \bigl[ \bigl\{ \mathfrak{F}_{ \rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} \\& \qquad {} \times f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) +\mathit{uf} (\varrho _{2} ) \bigr] \bigr]+ \frac{p}{(\varrho _{2}^{p}-\varrho _{1}^{p})^{\beta}} \bigl[(1- \gamma ) \\& \qquad {} \times \bigl(\mathfrak{J}_{\rho ,\beta ,\varrho _{1}^{p}+;w}^{ \sigma}f\circ g \bigr) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}\bigr)+ \gamma \bigl(\mathfrak{J}_{\rho ,\beta ,[\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}]+;w}^{\sigma}f \circ g \bigr) \bigl(\varrho _{2}^{p}\bigr) \bigr] . \end{aligned}$$
(7)
Proof
Integrating by parts
$$\begin{aligned} I_{1} :=& \int _{0}^{1} \bigl\{ t^{\beta} \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1 - \lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr] - u \bigr\} \bigl[(1-t) \bigl(\lambda \varrho _{1}^{p}+(1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{ \frac{1-p}{p}} \\ &{} \times f' \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p}} \bigr)\,dt \\ =& \biggl\vert \frac{p t^{\beta}\mathfrak{F}_{\rho ,\beta +1}^{\sigma} [w ((1 - \lambda )( \varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho}t^{\rho} ] - \mathit{pu}}{(\lambda - 1)(\varrho _{2}^{p} - \varrho _{1}^{p})} f \bigl( \sqrt[p]{(1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p}} \bigr) \biggr\vert _{0}^{1} \\ & {}- \int _{0}^{1} \frac{p t^{\beta -1}\mathfrak{F}_{\rho ,\beta}^{\sigma} [w ((1 - \lambda )( \varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho}t^{\rho} ]}{(\lambda - 1)(\varrho _{2}^{p} - \varrho _{1}^{p})} f \bigl( \sqrt[p]{(1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p}} \bigr)\,dt \\ =& \frac{p}{(\lambda -1)(\varrho _{2}^{p}-\varrho _{1}^{p})} \bigl[\bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1-\lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u\bigr\} f( \varrho _{1}) \\ & {}+u f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr] - \int _{0}^{1} \frac{p t^{\beta -1}\mathfrak{F}_{\rho ,\beta}^{\sigma} [w ((1-\lambda )( \varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho}t^{\rho} ]}{(\lambda -1)(\varrho _{2}^{p}-\varrho _{1}^{p})} \\ &{} \times f \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p}} \bigr)\,dt, \end{aligned}$$
setting \(x\rightarrow (1-t) (\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} )+t \varrho _{1}^{p}\), so that \(dx\rightarrow (\lambda -1)(\varrho _{2}^{p}-\varrho _{1}^{p})\,dt\) and \(0\leq t\leq 1\Leftrightarrow \lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p}\leq x\leq \varrho _{1}^{p}\), we have
$$\begin{aligned} I_{1} =& - \frac{p}{(1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p})} \bigl[\bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1-\lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u\bigr\} f( \varrho _{1}) \\ & {}+\mathit{uf} \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr] - \frac{p}{[(1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p})]^{1+\beta}} \\ &{} \times \int _{\varrho _{1}^{p}}^{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigl[\lambda \varrho _{1}^{p}+(1 - \lambda ) \varrho _{2}^{p} - x\bigr]^{\beta -1} \\ &{}\times \mathfrak{F}_{\rho ,\beta}^{\sigma} \bigl[w\bigl( \lambda \varrho _{1}^{p} +(1 - \lambda )\varrho _{2}^{p} - x\bigr)^{\rho} \bigr] (f \circ g ) (x ) \,dx \\ \Rightarrow& \frac{[(1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p})]^{1+\beta}}{p}I_{1} \\ =& -\bigl[(1-\lambda ) \bigl(\varrho _{2}^{p}-\varrho _{1}^{p}\bigr)\bigr]^{ \beta} \bigl[\bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1- \lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} f(\varrho _{1}) \\ & {}+u f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr] + \bigl( \mathfrak{J}_{\rho ,\beta ,\varrho _{1}^{p}+;w}^{ \sigma}f\circ g \bigr) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr). \end{aligned}$$
(8)
Again, integrating by parts
$$\begin{aligned} I_{2} :=& \int _{0}^{1}\bigl\{ t^{\beta} \mathfrak{F}_{ \rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr]-u\bigr\} \bigl[(t \bigl( \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}} \\ &{} \times f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr)\,dt \\ =& \biggl\vert \frac{p t^{\beta}\mathfrak{F}_{\rho ,\beta +1}^{\sigma} [w (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho}t^{\rho} ] - \mathit{pu}}{ \lambda (\varrho _{1}^{p} - \varrho _{2}^{p})}f \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1 - \lambda )\varrho _{2}^{p} \bigr)+(1 - t) \varrho _{2}^{p}} \bigr) \biggr\vert _{0}^{1} \\ & {}- \int _{0}^{1} \frac{p t^{\beta -1}\mathfrak{F}_{\rho ,\beta}^{\sigma} [w (\lambda ( \varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho}t^{\rho} ]}{\lambda (\varrho _{1}^{p} - \varrho _{2}^{p})} f \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1 - \lambda )\varrho _{2}^{p} \bigr)+(1 - t) \varrho _{2}^{p}} \bigr)\,dt \\ & =\frac{p}{\lambda (\varrho _{1}^{p}-\varrho _{2}^{p})} \bigl[\bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl( \lambda \bigl( \varrho _{2}^{p} - \varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr] - u\bigr\} f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1 - \lambda )\varrho _{2}^{p}} \bigr) + u f (\varrho _{2} ) \bigr] \\ & {}- \int _{0}^{1} \frac{p t^{\beta -1}\mathfrak{F}_{\rho ,\beta}^{\sigma} [w (\lambda ( \varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho}t^{\rho} ]}{\lambda (\varrho _{1}^{p} - \varrho _{2}^{p})} f \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1 - \lambda )\varrho _{2}^{p} \bigr)+(1 - t) \varrho _{2}^{p}} \bigr) \,dt, \end{aligned}$$
setting \(y\rightarrow t (\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} )+(1-t) \varrho _{2}^{p}\), so that \(dy\rightarrow \lambda (\varrho _{1}^{p}-\varrho _{2}^{p})\,dt\) and \(0\leq t\leq 1\Leftrightarrow \lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p}\geq y\geq \varrho _{2}^{p}\), we have
$$\begin{aligned} I_{2} =& \frac{p}{\lambda (\varrho _{1}^{p}-\varrho _{2}^{p})} \bigl[\bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u\bigr\} f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}} \bigr)+u f (\varrho _{2} ) \bigr] \\ & {}+ \frac{p}{[\lambda (\varrho _{2}^{p}-\varrho _{1}^{p})]^{1+\beta}} \int _{\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}}^{ \varrho _{2}^{p}}\bigl[\varrho _{2}^{p}-y \bigr]^{\beta -1}\mathfrak{F}_{\rho , \beta}^{\sigma} \bigl[w\bigl( \varrho _{2}^{p}-y\bigr)^{\rho} \bigr](f\circ g) (y) \,dy. \\ \Rightarrow& \frac{[\lambda (\varrho _{2}^{p}-\varrho _{1}^{p})]^{1+\beta}}{p}I_{2}=-\bigl[ \lambda \bigl(\varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr]^{\beta} \bigl[\bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u\bigr\} \\ & {}\times f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) +u f (\varrho _{2} ) \bigr]+ \bigl(\mathfrak{J}_{ \rho ,\beta ,[\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}]+;w}^{ \sigma}f\circ g \bigr) \bigl(\varrho _{2}^{p}\bigr). \end{aligned}$$
(9)
Multiplying (8) by \(1-\gamma \) and (9) by γ and then adding the resulting equalities yields the desired identity (7). □
Proposition 1
Let \(f:I\subseteq \mathbf{R}^{+}\rightarrow \mathbf{R}\) be a differentiable function on \(I^{\circ}\), the interior of I, \(\varrho _{1},\varrho _{2}\in I^{\circ}\) with \(\varrho _{1}<\varrho _{2}\); \(p,\rho ,\beta >0\); let \(g(x)=\sqrt[p]{x}\), \(x>0\); \(u,w\in \mathbf{R}\) and \(\lambda \in [0,1]\), then
$$\begin{aligned}& \lambda ^{1+\beta}\bigl(\varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \int _{0}^{1} \bigl\{ t^{\beta} \mathfrak{F}_{\rho , \beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr] - u\bigr\} \\& \qquad {}\times \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t) \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}} \\& \qquad {} \times f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr)\,dt \\& \quad =-p\lambda ^{\beta} \bigl[\bigl\{ \mathfrak{F}_{ \rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u\bigr\} f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}} \bigr) +u f (\varrho _{2} ) \bigr] \\& \qquad {} +\frac{p}{(\varrho _{2}^{p}-\varrho _{1}^{p})^{\beta}} \bigl( \mathfrak{J}_{\rho ,\beta ,[\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p}]+;w}^{\sigma}f\circ g \bigr) \bigl(\varrho _{2}^{p}\bigr). \end{aligned}$$
Remark 1
-
Lemma 1 is the generalization of [19, Lemma 1.2]. For \(\gamma ,\lambda ,u\rightarrow \frac{1}{2}\); \(p,\beta ,\sigma (0)\rightarrow 1\); \(w\rightarrow 0\), Lemma 1 reduces to [19, Lemma 1.2].
-
Proposition 1 is the generalization of [19, Lemma 1.1]. For \(p,\beta ,\lambda ,\sigma (0)\rightarrow 1\); \(u\rightarrow \frac{1}{2}\); \(w\rightarrow 0\), it reduces to [19, Lemma 1.1].
We make the following assumptions before going to our main results:
$$\begin{aligned}& A_{k,s,j} := \sqrt[s]{\frac{ (\beta +\rho k+1)^{j-1} (\lambda \vert f'(\varrho _{1}) \vert ^{s}+ (1 - \lambda ) \vert f'(\varrho _{2}) \vert ^{s} )+(\beta +\rho k+1)^{2-j} \vert f'(\varrho _{j}) \vert ^{s}}{\beta +\rho k+2}}, \\& B_{s},_{j} := (2-j+\lambda ) \bigl\vert f'( \varrho _{1}) \bigr\vert ^{s}+(j- \lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s}. \end{aligned}$$
Theorem 1
Let \(f:I\subseteq \mathbf{R}^{+}\rightarrow \mathbf{R}\) be a differentiable function on \(I^{\circ}\), the interior of I. If \(f'\in L^{1}[ \varrho _{1},\varrho _{2}]\) and \(\vert f^{\prime}\vert \) is p-convex on \([ \varrho _{1},\varrho _{2} ] \), where \(\varrho _{1}\), \(\varrho _{2}\in I^{\circ}\) with \(\varrho _{1}<\varrho _{2}\); \(p,\rho ,\beta >0\); let \(g(x)=\sqrt[p]{x}\), \(x>0\); \(u,w,\gamma \in \mathbf{R}\) and \(\lambda \in [0,1]\), then
$$\begin{aligned}& \bigl\vert \Omega (f,\varrho _{1},\varrho _{2};u) \bigr\vert \\& \quad \leq\textstyle\begin{cases} \varrho _{1}^{1-p}(\varrho _{2}^{p}-\varrho _{1}^{p}) [(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \\ \quad {}\times \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)}A_{k,1,1}+ \frac{ \vert u \vert }{2}B_{1,1} \} \\ \quad {} + \vert \gamma \vert \lambda ^{1+\beta} \{\sum_{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)} A_{k,1,2}+\frac{ \vert u \vert }{2}B_{1,2} \} ], & p \in [1,\infty ); \\ \varrho _{2}^{1-p}(\varrho _{2}^{p}-\varrho _{1}^{p}) [(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \\ \quad {}\times \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)}A_{k,1,1}+ \frac{ \vert u \vert }{2}B_{1,1} \} \\ \quad {} + \vert \gamma \vert \lambda ^{1+\beta} \{\sum_{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)} A_{k,1,2}+\frac{ \vert u \vert }{2}B_{1,2} \} ], & p \in (0,1). \end{cases}\displaystyle \end{aligned}$$
(10)
Proof
By the properties of modulus and repeated application of p-convexity to Lemma 1 for the case \(p\in [1,\infty )\), we have the following:
$$\begin{aligned}& \bigl\vert \Omega (f,\varrho _{1},\varrho _{2};u) \bigr\vert \\& \quad = \biggl\vert -p \bigl[(1-\lambda )^{\beta}(1-\gamma ) \bigl[ \bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1- \lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} f(\varrho _{1}) \\& \qquad {} +u f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr]+\lambda ^{\beta}\gamma \bigl[ \bigl\{ \mathfrak{F}_{ \rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} \\& \qquad {} \times f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) +u f (\varrho _{2} ) \bigr] \bigr]+ \frac{p}{(\varrho _{2}^{p}-\varrho _{1}^{p})^{\beta}} \bigl[(1- \gamma ) \\& \qquad {} \times \bigl(\mathfrak{J}_{\rho ,\beta ,\varrho _{1}^{p}+;w}^{ \sigma}f\circ g \bigr) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+\gamma \bigl(\mathfrak{J}_{\rho ,\beta ,[ \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}]+;w}^{\sigma}f \circ g \bigr) \bigl(\varrho _{2}^{p}\bigr) \bigr] \biggr\vert \\& \quad = \biggl\vert \bigl(\varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \biggl[(1-\lambda )^{1+\beta}(1-\gamma ) \int _{0}^{1} \bigl\{ t^{ \beta} \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1- \lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr]-u \bigr\} \\& \qquad {} \times \bigl[(1-t) \bigl(\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{ \frac{1-p}{p}} \\& \qquad {} \times f' \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t \varrho _{1}^{p}} \bigr)\,dt \\& \qquad {}+\gamma \lambda ^{1+\beta} \int _{0}^{1} \bigl\{ t^{ \beta} \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p} - \varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr] - u \bigr\} \\& \qquad {} \times \bigl[t \bigl(\lambda \varrho _{1}^{p}+(1 - \lambda )\varrho _{2}^{p} \bigr)+(1 - t)\varrho _{2}^{p} \bigr]^{ \frac{1-p}{p}} \\& \qquad {} \times f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr)\,dt \biggr] \biggr\vert \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Biggl[ (1 - \lambda )^{1+\beta} \vert 1 - \gamma \vert \int _{0}^{1} \Biggl\vert \sum _{k=0}^{\infty} \frac{\sigma (k)w^{k} ((1 - \lambda ) (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} t^{\beta +\rho k} - u \Biggr\vert \\& \qquad {} \times \varrho _{1}^{1-p} \bigl\{ (1-t) \bigl\vert f' \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr\vert +t \bigl\vert f'( \varrho _{1}) \bigr\vert \bigr\} \,dt \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \int _{0}^{1} \Biggl\vert \sum _{k=0}^{\infty} \frac{\sigma (k)w^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} t^{\beta +\rho k}-u \Biggr\vert \\& \qquad {} \times \varrho _{1}^{1-p} \bigl\{ t \bigl\vert f' \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr\vert +(1-t) \bigl\vert f'(\varrho _{2}) \bigr\vert \bigr\} \,dt \Biggr] \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr)\varrho _{1}^{1-p} \Biggl[(1-\lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \int _{0}^{1} t^{\beta +\rho k} \bigl\{ (1-t) \bigl( \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert +(1- \lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert \bigr) +t \bigl\vert f'(\varrho _{1}) \bigr\vert \bigr\} \,dt \\& \qquad {}+ \vert u \vert \int _{0}^{1} \bigl\{ (1-t) \bigl( \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert +(1-\lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert \bigr) +t \bigl\vert f'( \varrho _{1}) \bigr\vert \bigr\} \,dt \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \int _{0}^{1} t^{\beta +\rho k} \bigl\{ t \bigl( \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert +(1- \lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert \bigr) +(1-t) \bigl\vert f'(\varrho _{2}) \bigr\vert \bigr\} \,dt \\& \qquad {} + \vert u \vert \int _{0}^{1} \bigl\{ t \bigl( \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert +(1-\lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert \bigr) +(1-t) \bigl\vert f'(\varrho _{2}) \bigr\vert \bigr\} \,dt \Biggr\} \Biggr] \\& \quad = \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr)\varrho _{1}^{1-p} \Bigg[(1-\lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \frac{\lambda \vert f'(\varrho _{1}) \vert +(1 - \lambda ) \vert f'(\varrho _{2}) \vert }{(\beta + \rho k + 1)(\beta + \rho k + 2)} + \frac{ \vert f'(\varrho _{1}) \vert }{\beta + \rho k + 2} \biggr) \\& \qquad {}+ \vert u \vert \biggl( \frac{\lambda \vert f'(\varrho _{1}) \vert +(1-\lambda ) \vert f'(\varrho _{2}) \vert }{2} +\frac{ \vert f'( \varrho _{1}) \vert }{2} \biggr) \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \frac{\lambda \vert f'(\varrho _{1}) \vert + (1 - \lambda ) \vert f'(\varrho _{2}) \vert }{\beta + \rho k + 2} + \frac{ \vert f'(\varrho _{2}) \vert }{(\beta + \rho k + 1)(\beta + \rho k + 2)} \biggr) \\& \qquad {} + \vert u \vert \biggl( \frac{\lambda \vert f'(\varrho _{1}) \vert +(1-\lambda ) \vert f'(\varrho _{2}) \vert }{2} + \frac{ \vert f'(\varrho _{2}) \vert }{2} \biggr) \Biggr\} \Bigg] \\& \quad = \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr)\varrho _{1}^{1-p} \Biggl[(1-\lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \frac{\lambda \vert f'(\varrho _{1}) \vert +(1-\lambda ) \vert f'(\varrho _{2}) \vert +(\beta +\rho k+1) \vert f'(\varrho _{1}) \vert }{(\beta +\rho k+1)(\beta +\rho k+2)} \biggr) \\& \qquad {}+\frac{ \vert u \vert }{2} \bigl((1+\lambda ) \bigl\vert f'( \varrho _{1}) \bigr\vert +(1-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert \bigr) \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \frac{(\beta +\rho k+1) (\lambda \vert f'(\varrho _{1}) \vert +(1-\lambda ) \vert f'(\varrho _{2}) \vert )+ \vert f'(\varrho _{2}) \vert }{(\beta +\rho k+1)(\beta +\rho k+2)} \biggr) \\& \qquad {} +\frac{ \vert u \vert }{2} \bigl(\lambda \bigl\vert f'( \varrho _{1}) \bigr\vert +(2-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert \bigr) \Biggr\} \Biggr]. \end{aligned}$$
Now, for the case \(p\in (0,1)\), using the fact
$$\begin{aligned}& \bigl[(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{\frac{1-p}{p}} \\& \quad \leq \bigl[(1-t) \bigl(\lambda \varrho _{2}^{p}+(1- \lambda ) \varrho _{2}^{p} \bigr)+t \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}} \leq \varrho _{2}^{1-p} \end{aligned}$$
(11)
and
$$\begin{aligned}& \bigl[t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}} \\& \quad \leq \bigl[t \bigl(\lambda \varrho _{2}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}}\leq \varrho _{2}^{1-p} \end{aligned}$$
(12)
and applying the same procedure like \(p\in [1,\infty )\), we obtain the desired result. □
Corollary 1
Under the conditions of Theorem 1, for \(\sigma (0),p,\beta \rightarrow 1\), \(\lambda ,\gamma ,u\rightarrow \frac{1}{2}\), and \(w\rightarrow 0\), we have
$$\begin{aligned}& \biggl\vert \frac{1}{2} \biggl\{ \frac{f(\varrho _{1})+f(\varrho _{2})}{2}+f \biggl( \frac{\varrho _{1}+\varrho _{2}}{2} \biggr) \biggr\} - \frac{1}{\varrho _{2}-\varrho _{1}} \int _{\varrho _{1}}^{\varrho _{2}}f(x)\,dx \biggr\vert \\& \quad \leq \frac{\varrho _{2}-\varrho _{1}}{48} \bigl[13 \bigl\vert {f'(\varrho _{1})} \bigr\vert +11 \bigl\vert {f'(\varrho _{2})} \bigr\vert \bigr] \\& \quad \leq \frac{(\varrho _{2}-\varrho _{1}) ( \vert {f'(\varrho _{1})} \vert + \vert {f'(\varrho _{2})} \vert )}{2}. \end{aligned}$$
Theorem 2
Let \(f:I\subseteq \mathbf{R}^{+}\rightarrow \mathbf{R}\) be a differentiable function on \(I^{\circ}\), the interior of I. If \(f'\in L^{1}[ \varrho _{1},\varrho _{2}]\) and \(\vert f^{\prime}\vert ^{s}\) is p-convex on \([ \varrho _{1},\varrho _{2} ] \), where \(\varrho _{1}\), \(\varrho _{2}\in I^{\circ}\) with \(\varrho _{1}<\varrho _{2}\); \(p,\rho ,\beta >0\); \(s\geq 1\); let \(g(x)=\sqrt[p]{x}\), \(x>0\); \(u,w,\gamma \in \mathbf{R}\) and \(\lambda \in [0,1]\), then
$$\begin{aligned}& \bigl\vert \Omega (f,\varrho _{1},\varrho _{2};u) \bigr\vert \\& \quad \leq \textstyle\begin{cases} \varrho _{1}^{1-p}(\varrho _{2}^{p}-\varrho _{1}^{p}) [(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \\ \quad {}\times \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)} A_{k,s,1}+ \vert u \vert (\frac{B_{s,1}}{2} )^{ \frac{1}{s}} \} \\ \quad {} + \lambda ^{1+\beta} \vert \gamma \vert \{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)} A_{k,s,2}+ \vert u \vert (\frac{B_{s,2}}{2} )^{\frac{1}{s}} \} ], & p\in [1,\infty ); \\ \varrho _{2}^{1-p}(\varrho _{2}^{p}-\varrho _{1}^{p}) [(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \\ \quad {}\times \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p}- \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)} A_{k,s,1}+ \vert u \vert ( \frac{B_{s,1}}{2} )^{ \frac{1}{s}} \} \\ \quad {} + \lambda ^{1+\beta} \vert \gamma \vert \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)} A_{k,s,2}+ \vert u \vert (\frac{B_{s,2}}{2} )^{\frac{1}{s}} \} ], & p\in (0,1). \end{cases}\displaystyle \end{aligned}$$
Proof
By the properties of modulus, power-mean inequality, and repeated application of p-convexity to Lemma 1 for the case \(p\in [1,\infty )\), we obtain the following:
$$\begin{aligned}& \bigl\vert \Omega (f,\varrho _{1},\varrho _{2};u) \bigr\vert \\& \quad = \biggl\vert -p \bigl[(1-\lambda )^{\beta}(1-\gamma ) \bigl[ \bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1- \lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} f(\varrho _{1}) \\& \qquad {} +u f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr]+\lambda ^{\beta}\gamma \bigl[ \bigl\{ \mathfrak{F}_{ \rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} \\& \qquad {} \times f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) +u f (\varrho _{2} ) \bigr] \bigr]+ \frac{p}{(\varrho _{2}^{p}-\varrho _{1}^{p})^{\beta}} \bigl[(1- \gamma ) \\& \qquad {} \times \bigl(\mathfrak{J}_{\rho ,\beta ,\varrho _{1}^{p}+;w}^{ \sigma}f\circ g \bigr) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+\gamma \bigl(\mathfrak{J}_{\rho ,\beta ,[ \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}]+;w}^{\sigma}f \circ g \bigr) \bigl(\varrho _{2}^{p}\bigr) \bigr] \biggr\vert \\& \quad = \biggl\vert \bigl(\varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \biggl[(1-\lambda )^{1+\beta}(1-\gamma ) \\& \qquad {}\times \int _{0}^{1} \bigl\{ t^{ \beta} \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1- \lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr]-u \bigr\} \\& \qquad {} \times \bigl[(1-t) \bigl(\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{ \frac{1-p}{p}} \\& \qquad {}\times f' \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t \varrho _{1}^{p}} \bigr)\,dt \\& \qquad {}+\gamma \lambda ^{1+\beta} \int _{0}^{1} \bigl\{ t^{ \beta} \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p} - \varrho _{1}^{p}\bigr) \bigr)^{\rho}t^{\rho} \bigr] - u \bigr\} \\& \qquad {}\times \bigl[t \bigl(\lambda \varrho _{1}^{p}+(1 - \lambda )\varrho _{2}^{p} \bigr)+(1 - t)\varrho _{2}^{p} \bigr]^{ \frac{1-p}{p}} \\& \qquad {} \times f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr)\,dt \biggr] \biggr\vert \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Biggl[(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \int _{0}^{1} t^{\beta +\rho k} \bigl[(1 - t) \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {}\times\bigl\vert f' \bigl( \sqrt[p]{ (1 - t) \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t\varrho _{1}^{p}} \bigr) \bigr\vert \,dt \\& \qquad {}+ \vert u \vert \int _{0}^{1} \bigl[(1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {}\times \bigl\vert f' \bigl( \sqrt[p]{ (1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t\varrho _{1}^{p}} \bigr) \bigr\vert \,dt \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \int _{0}^{1} t^{\beta +\rho k} \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) +(1 - t) \varrho _{2}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {}\times \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t)\varrho _{2}^{p}} \bigr) \bigr\vert \,dt \\& \qquad {} + \vert u \vert \int _{0}^{1} \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) +(1 - t) \varrho _{2}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {}\times \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t)\varrho _{2}^{p}} \bigr) \bigr\vert \,dt \Biggr\} \Biggr] \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Biggl[(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \int _{0}^{1}t^{\beta +\rho k} \bigl[(1-t) \bigl( \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{\frac{1-p}{p}}\,dt \biggr)^{1-\frac{1}{s}} \\& \qquad {} \times \biggl[ \int _{0}^{1} t^{\beta +\rho k} \bigl[(1 - t) \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{ \frac{1-p}{p}} \\& \qquad {}\times \bigl\vert f' \bigl( \sqrt[p]{ (1 - t) \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t\varrho _{1}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr]^{\frac{1}{s}} \\& \qquad {}+ \vert u \vert \biggl( \int _{0}^{1} \bigl[(1-t) \bigl( \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{\frac{1-p}{p}}\,dt \biggr)^{1-\frac{1}{s}} \\& \qquad {} \times \biggl[ \int _{0}^{1} \bigl[(1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{\frac{1-p}{p}} \\& \qquad {}\times \bigl\vert f' \bigl( \sqrt[p]{ (1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t\varrho _{1}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr]^{\frac{1}{s}} \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \int _{0}^{1}t^{\beta +\rho k} \bigl[t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}}\,dt \biggr)^{1-\frac{1}{s}} \\& \qquad {} \times \biggl[ \int _{0}^{1} t^{\beta +\rho k} \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda ) \varrho _{2}^{p} \bigr) + (1 - t) \varrho _{2}^{p} \bigr]^{ \frac{1-p}{p}} \\& \qquad {}\times \bigl\vert f' \bigl( \sqrt[p]{ t \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t)\varrho _{2}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr]^{\frac{1}{s}} \\& \qquad {}+ \vert u \vert \biggl( \int _{0}^{1} \bigl[t \bigl( \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}}\,dt \biggr)^{1-\frac{1}{s}} \\& \qquad {} \times \Bigg( \int _{0}^{1} \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t) \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}} \\& \qquad {}\times \bigl\vert f' \bigl( \sqrt[p]{ t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t)\varrho _{2}^{p}} \bigr) \bigr\vert ^{s} \,dt \Biggr]^{\frac{1}{s}} \Biggr\} \Bigg] \\& \quad \leq \varrho _{1}^{1-p}\bigl(\varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \Biggl[(1-\lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl[ \int _{0}^{1} t^{\beta +\rho k}\,dt \biggr]^{1-\frac{1}{s}} \\& \qquad {}\times \biggl[ \int _{0}^{1} t^{\beta +\rho k} \bigl\{ (1 - t) \bigl(\lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} + (1 - \lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr) + t \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \bigr\} \,dt \biggr]^{\frac{1}{s}} \\& \qquad {}+ \vert u \vert \biggl( \int _{0}^{1} \bigl\{ (1-t) \bigl(\lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} +(1- \lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr)+t \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \bigr\} \,dt \biggr)^{\frac{1}{s}} \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl[ \int _{0}^{1} t^{\beta +\rho k}\,dt \biggr]^{1-\frac{1}{s}} \\& \qquad {}\times \biggl[ \int _{0}^{1} t^{\beta +\rho k} \bigl\{ t \bigl( \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} + (1 - \lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr) + (1 - t) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr\} \,dt \biggr]^{\frac{1}{s}} \\& \qquad {}+ \vert u \vert \biggl( \int _{0}^{1} \bigl\{ t \bigl(\lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} +(1-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr)+(1-t) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr\} \,dt \biggr)^{\frac{1}{s}} \Biggr\} \Biggr] \\& \quad = \varrho _{1}^{1-p}\bigl(\varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \Biggl[(1-\lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)(\beta +\rho k+1)^{1-\frac{1}{s}}} \\& \qquad {} \times \biggl( \frac{\lambda \vert f'(\varrho _{1}) \vert ^{s} +(1-\lambda ) \vert f'(\varrho _{2}) \vert ^{s}}{(\beta +\rho k+1)(\beta +\rho k+2)}+ \frac{ \vert f'(\varrho _{1}) \vert ^{s}}{\beta +\rho k+2} \biggr)^{ \frac{1}{s}} \\& \qquad {}+\frac{ \vert u \vert }{2^{\frac{1}{s}}} \bigl\{ \lambda \bigl\vert f'( \varrho _{1}) \bigr\vert ^{s} +(1-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s}+ \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \bigr\} ^{ \frac{1}{s}} \Biggr\} \\& \qquad {} + \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1){(\beta + \rho k + 1)^{1-\frac{1}{s}}}} \\& \qquad {} \times \biggl\{ \frac{\lambda \vert f'(\varrho _{1}) \vert ^{s} + (1 - \lambda ) \vert f'(\varrho _{2}) \vert ^{s}}{\beta +\rho k+2} + \frac{ \vert f'(\varrho _{2}) \vert ^{s}}{(\beta + \rho k + 1)(\beta + \rho k + 2)} \biggr\} ^{\frac{1}{s}} \\& \qquad {}+\frac{ \vert u \vert }{2^{\frac{1}{s}}} \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} +(1-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s}+ \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr\} ^{ \frac{1}{s}} \Biggr\} \Biggr] \\& \quad = \varrho _{1}^{1-p}\bigl(\varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \Biggl[(1-\lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)} \\& \qquad {} \times \biggl( \frac{\lambda \vert f'(\varrho _{1}) \vert ^{s} +(1-\lambda ) \vert f'(\varrho _{2}) \vert ^{s}+ (\beta +\rho k+1) \vert f'(\varrho _{1}) \vert ^{s}}{\beta +\rho k+2} \biggr)^{\frac{1}{s}} \\& \qquad {}+\frac{ \vert u \vert }{2^{\frac{1}{s}}} \bigl\{ (1+ \lambda ) \bigl\vert f'( \varrho _{1}) \bigr\vert ^{s} +(1-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr\} ^{\frac{1}{s}} \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +2)} \\& \qquad {} \times \biggl( \frac{(\beta +\rho k+1)(\lambda \vert f'(\varrho _{1}) \vert ^{s} +(1-\lambda ) \vert f'(\varrho _{2}) \vert ^{s})+ \vert f'(\varrho _{2}) \vert ^{s}}{\beta +\rho k+2} \biggr)^{\frac{1}{s}} \\& \qquad {}+\frac{ \vert u \vert }{2^{\frac{1}{s}}} \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} +(2-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr\} ^{\frac{1}{s}} \Biggr\} \Biggr]. \end{aligned}$$
Now, for the case \(p\in (0,1)\), using inequalities (11) and (12), and repeating the same procedure like \(p\in [1,\infty )\), we obtain the desired result. □
Corollary 2
Under the conditions of Theorem 2for \(\beta ,u,s,p,\sigma (0)\rightarrow 1\), \(\lambda \rightarrow \frac{1}{2}\), and \(w,\gamma \rightarrow 0\), we have
$$\begin{aligned}& \biggl\vert {f \biggl(\frac{\varrho _{1}+\varrho _{2}}{2} \biggr)}- \frac{2}{\varrho _{2}-\varrho _{1}} \int _{\varrho _{1}}^{ \frac{\varrho _{1}+\varrho _{2}}{2}}f(x)\,dx \biggr\vert \leq \frac{\varrho _{2}-\varrho _{1}}{12} \bigl(7{ \bigl\vert {f'(\varrho _{1}) \bigr\vert }}+2{ \bigl\vert {f'(\varrho _{2}) \bigr\vert }} \bigr). \end{aligned}$$
Theorem 3
Let \(f:I\subseteq \mathbf{R}^{+}\rightarrow \mathbf{R}\) be a differentiable function on \(I^{\circ}\), the interior of I. If \(f'\in L^{1}[ \varrho _{1},\varrho _{2}]\) and \(\vert f^{\prime}\vert ^{s}\) is p-convex on \([ \varrho _{1},\varrho _{2} ] \), where \(\varrho _{1}\), \(\varrho _{2}\in I^{\circ}\) with \(\varrho _{1}<\varrho _{2}\); \(p,\rho ,\beta >0\); \(r,s>1\) such that \(\frac{1}{r}+\frac{1}{s}=1\); let \(g(x)=\sqrt[p]{x}\), \(x>0\); \(u,w,\gamma \in \mathbf{R}\) and \(\lambda \in [0,1]\), then
$$\begin{aligned}& \bigl\vert \Omega (f,\varrho _{1},\varrho _{2};u) \bigr\vert \\& \quad \leq \textstyle\begin{cases} \frac{\varrho _{1}^{1-p}(\varrho _{2}^{p}-\varrho _{1}^{p})}{\sqrt[s]{2}} [(1-\lambda )^{1+\beta} \vert 1-\gamma \vert B_{s,1}^{ \frac{1}{s}} \\ \quad {}\times \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{ (r(\beta +\rho k)+1 )^{\frac{1}{r}}\Gamma (\rho k+\beta +1)}+ \vert u \vert \} \\ \quad {} + \vert \gamma \vert \lambda ^{1+\beta} B_{s,2}^{\frac{1}{s}} \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{ (r(\beta +\rho k)+1 )^{\frac{1}{r}}\Gamma (\rho k+\beta +1)}+ \vert u \vert \} ], & p\in [1,\infty ); \\ \frac{\varrho _{2}^{1-p}(\varrho _{2}^{p}-\varrho _{1}^{p})}{\sqrt[s]{2}} [(1-\lambda )^{1+\beta} \vert 1-\gamma \vert B_{s,1}^{ \frac{1}{s}} \\ \quad {}\times \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{ (r(\beta +\rho k)+1 )^{\frac{1}{r}}\Gamma (\rho k+\beta +1)}+ \vert u \vert \} \\ \quad {}+ \vert \gamma \vert \lambda ^{1+\beta} B_{s,2}^{\frac{1}{s}} \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{ (r(\beta +\rho k)+1 )^{\frac{1}{r}}\Gamma (\rho k+\beta +1)}+ \vert u \vert \} ], & p\in (0,1). \end{cases}\displaystyle \end{aligned}$$
Proof
By the properties of modulus, Holder’s inequality, and repeated application of p-convexity to Lemma 1 for the case \(p\in [1,\infty )\), we have the following:
$$\begin{aligned}& \bigl\vert \Omega (f,\varrho _{1},\varrho _{2};u) \bigr\vert \\& \quad = \biggl\vert -p \bigl[(1-\lambda )^{\beta}(1-\gamma ) \bigl[ \bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1- \lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} f(\varrho _{1}) \\& \qquad {}+u f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr]+\lambda ^{\beta}\gamma \bigl[ \bigl\{ \mathfrak{F}_{ \rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} \\& \qquad {} \times f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) +u f (\varrho _{2} ) \bigr] \bigr]+ \frac{p}{(\varrho _{2}^{p}-\varrho _{1}^{p})^{\beta}} \bigl[(1- \gamma ) \\& \qquad {} \times \bigl(\mathfrak{J}_{\rho ,\beta ,\varrho _{1}^{p}+;w}^{ \sigma}f\circ g \bigr) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+\gamma \bigl(\mathfrak{J}_{\rho ,\beta ,[ \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}]+;w}^{\sigma}f \circ g \bigr) \bigl(\varrho _{2}^{p}\bigr) \bigr] \biggr\vert \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Biggl[(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \int _{0}^{1} t^{\beta +\rho k} \bigl[(1 - t) \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {} \times \bigl\vert f' \bigl( \sqrt[p]{ (1 - t) \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t\varrho _{1}^{p}} \bigr) \bigr\vert \,dt \\& \qquad {}+ \vert u \vert \int _{0}^{1} \bigl[(1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {} \times \bigl\vert f' \bigl( \sqrt[p]{ (1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t\varrho _{1}^{p}} \bigr) \bigr\vert \,dt \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \int _{0}^{1} t^{\beta +\rho k} \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) +(1 - t) \varrho _{2}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {} \times \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t)\varrho _{2}^{p}} \bigr) \bigr\vert \,dt \\& \qquad {} + \vert u \vert \int _{0}^{1} \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) +(1 - t) \varrho _{2}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {} \times \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t)\varrho _{2}^{p}} \bigr) \bigr\vert \,dt \Biggr\} \Biggr] \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Bigg[(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \int _{0}^{1} \bigl(t^{\beta +\rho k} \bigl[(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{\frac{1-p}{p}} \bigr)^{r}\,dt \biggr)^{\frac{1}{r}} \\& \qquad {} \times \biggl( \int _{0}^{1} \bigl\vert f' \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t\varrho _{1}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr)^{\frac{1}{s}} \\& \qquad {}+ \vert u \vert \biggl( \int _{0}^{1} \bigl[(1-t) \bigl( \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{\frac{r(1-p)}{p}}\,dt \biggr)^{\frac{1}{r}} \\& \qquad {} \times \biggl( \int _{0}^{1} \bigl\vert f' \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t\varrho _{1}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr)^{\frac{1}{s}} \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Bigg\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \int _{0}^{1}t^{\beta +\rho k} \bigl[t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{r(1-p)}{p}}\,dt \biggr)^{\frac{1}{r}} \\& \qquad {} \times \biggl( \int _{0}^{1} \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr) \bigr\vert ^{s}\,dt \biggr)^{\frac{1}{s}} \\& \qquad {}+ \vert u \vert \biggl( \int _{0}^{1} \bigl[t \bigl( \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{r(1-p)}{p}}\,dt \biggr)^{\frac{1}{r}} \\& \qquad {} \times \biggl( \int _{0}^{1} \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr)^{\frac{1}{s}} \Bigg\} \Bigg] \\& \quad \leq \varrho _{1}^{1-p}\bigl(\varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \Biggl[(1-\lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl[ \int _{0}^{1} t^{r(\beta +\rho k)}\,dt \biggr]^{\frac{1}{r}} \biggl[ \int _{0}^{1} \bigl[(1 - t) \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} + (1 - \lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr\} + t \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \bigr]\,dt \biggr]^{\frac{1}{s}} \\& \qquad {}+ \vert u \vert \biggl( \int _{0}^{1} \bigl[(1-t) \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} +(1- \lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr\} +t \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \bigr]\,dt \biggr)^{\frac{1}{s}} \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \biggl[ \int _{0}^{1}t^{r(\beta +\rho k)}\,dt \biggr]^{ \frac{1}{r}} \\& \qquad {} \times \biggl[ \int _{0}^{1} \bigl[t \bigl\{ \lambda \bigl\vert f'( \varrho _{1}) \bigr\vert ^{s} +(1-\lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr\} +(1-t) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr]\,dt \biggr]^{ \frac{1}{s}} \\& \qquad {}+ \vert u \vert \biggl( \int _{0}^{1} \bigl[t \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} +(1-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr\} +(1-t) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr]\,dt \biggr)^{\frac{1}{s}} \Biggr\} \Biggr] \\& \quad = \frac{1}{2^{\frac{1}{s}}} \varrho _{1}^{1-p}\bigl( \varrho _{2}^{p} - \varrho _{1}^{p}\bigr) \Biggl[ (1 - \lambda )^{1+ \beta} \vert 1 - \gamma \vert \bigl((1 + \lambda ) \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} + (1 - \lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr)^{\frac{1}{s}} \\& \qquad {} \times \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{ (r(\beta + \rho k) + 1 )^{\frac{1}{r}}\Gamma (\rho k + \beta + 1)} + \vert u \vert \Biggr\} \\& \qquad {} \times + \vert \gamma \vert \lambda ^{1+ \beta} \bigl(\lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} + (2 - \lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr)^{\frac{1}{s}} \\& \qquad {} \times \Biggl\{ \sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{ (r(\beta +\rho k)+1 )^{\frac{1}{r}}\Gamma (\rho k+\beta +1)}+ \vert u \vert \Biggr\} \Biggr]. \end{aligned}$$
Now, for the case \(p\in (0,1)\), using inequalities (11) and (12), and repeating the same procedure like \(p\in [1,\infty )\), we obtain the desired result. □
Corollary 3
Under the conditions of Theorem 3for \(p,\beta ,\lambda ,\gamma ,\sigma (0)\rightarrow 1\), \(s\rightarrow 2\), \(u\rightarrow \frac{1}{2}\), and \(w\rightarrow 0\), we have
$$\begin{aligned}& \biggl\vert \frac{f(\varrho _{1}) + f(\varrho _{2})}{2} - \frac{1}{\varrho _{2} - \varrho _{1}} \int _{\varrho _{1}}^{\varrho _{2}} f(x)\,dx \biggr\vert \leq \frac{\varrho _{2} - \varrho _{1}}{2\sqrt{6}} (2 + \sqrt{3} ) \sqrt{{ \bigl\vert {f'(\varrho _{1})} \bigr\vert }^{2} + { \bigl\vert {f'( \varrho _{2})} \bigr\vert }^{2}}. \end{aligned}$$
Theorem 4
Let \(f:I\subseteq \mathbf{R}^{+}\rightarrow \mathbf{R}\) be a differentiable function on \(I^{\circ}\), the interior of I. If \(f'\in L^{1}[ \varrho _{1},\varrho _{2}]\) and \(\vert f^{\prime}\vert ^{s}\) is p-convex on \([ \varrho _{1},\varrho _{2} ] \), where \(\varrho _{1}\), \(\varrho _{2}\in I^{\circ}\) with \(\varrho _{1}<\varrho _{2}\); \(p,\rho ,\beta >0\); \(r,s>1\) such that \(\frac{1}{r}+\frac{1}{s}=1\); let \(g(x)=\sqrt[p]{x}\), \(x>0\); \(u,w,\gamma \in \mathbf{R}\) and \(\lambda \in [0,1]\), then
$$\begin{aligned}& \bigl\vert \Omega (f,\varrho _{1},\varrho _{2};u) \bigr\vert \\& \quad \leq \textstyle\begin{cases} \frac{\varrho _{2}^{p}-\varrho _{1}^{p}}{2\mathit{rs}} [(1-\lambda )^{1+ \beta} \vert 1-\gamma \vert \{\sum_{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{ (r(\beta +\rho k)+1 )\Gamma (\rho k+\beta +1)} \\ \quad {} \times (2s\varrho _{1}^{r(1-p)}+r(r(\beta +\rho k)+1)B_{s,1} )+ \vert u \vert (2s\varrho _{1}^{r(1-p)}+\mathit{rB}_{s,1} ) \} \\ \quad {}+ \vert \gamma \vert \lambda ^{1+\beta} \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k} ( 2s\varrho _{1}^{r(1-p)} + r(r(\beta + \rho k) + 1)B_{s,2} )}{ (r(\beta +\rho k)+1 )\Gamma (\rho k+\beta +1)} \\ \quad {} + \vert u \vert (2s\varrho _{1}^{r(1-p)}+\mathit{rB}_{s,2} ) \} ], & p\in [1, \infty ); \\ \frac{\varrho _{2}^{p}-\varrho _{1}^{p}}{2\mathit{rs}} [(1-\lambda )^{1+ \beta} \vert 1-\gamma \vert \{\sum_{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{ (r(\beta +\rho k)+1 )\Gamma (\rho k+\beta +1)} \\ \quad {} \times (2s\varrho _{2}^{r(1-p)}+r(r(\beta +\rho k)+1)B_{s,1} )+ \vert u \vert (2s\varrho _{2}^{r(1-p)}+\mathit{rB}_{s,1} ) \} \\ \quad {}+ \vert \gamma \vert \lambda ^{1+\beta} \{\sum_{k=0}^{\infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k} ( 2s\varrho _{2}^{r(1-p)} + r(r(\beta + \rho k) + 1)B_{s,2} )}{ (r(\beta +\rho k)+1 )\Gamma (\rho k+\beta +1)} \\ \quad {} + \vert u \vert (2s\varrho _{2}^{r(1-p)}+\mathit{rB}_{s,2} ) \} ], & p\in (0,1). \end{cases}\displaystyle \end{aligned}$$
(13)
Proof
By the properties of modulus, Young’s inequality, and repeated application of p-convexity to Lemma 1 for the case \(p\in [1,\infty )\), we have the following:
$$\begin{aligned}& \bigl\vert \Omega (f,\varrho _{1},\varrho _{2};u) \bigr\vert \\& \quad = \biggl\vert -p \bigl[(1-\lambda )^{\beta}(1-\gamma ) \bigl[ \bigl\{ \mathfrak{F}_{\rho ,\beta +1}^{\sigma} \bigl[w \bigl((1- \lambda ) \bigl( \varrho _{2}^{p}-\varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} f(\varrho _{1}) \\& \qquad {} +u f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) \bigr]+\lambda ^{\beta}\gamma \bigl[ \bigl\{ \mathfrak{F}_{ \rho ,\beta +1}^{\sigma} \bigl[w \bigl(\lambda \bigl( \varrho _{2}^{p}- \varrho _{1}^{p}\bigr) \bigr)^{\rho} \bigr]-u \bigr\} \\& \qquad {} \times f \bigl( \sqrt[p]{\lambda \varrho _{1}^{p}+(1- \lambda )\varrho _{2}^{p}} \bigr) +u f (\varrho _{2} ) \bigr] \bigr]+ \frac{p}{(\varrho _{2}^{p}-\varrho _{1}^{p})^{\beta}} \bigl[(1- \gamma ) \\& \qquad {} \times \bigl(\mathfrak{J}_{\rho ,\beta ,\varrho _{1}^{p}+;w}^{ \sigma}f\circ g \bigr) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+\gamma \bigl(\mathfrak{J}_{\rho ,\beta ,[ \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p}]+;w}^{\sigma}f \circ g \bigr) \bigl(\varrho _{2}^{p}\bigr) \bigr] \biggr\vert \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Biggl[(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \int _{0}^{1} t^{\beta +\rho k} \bigl[(1 - t) \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {} \times \bigl\vert f' \bigl( \sqrt[p]{ (1 - t) \bigl( \lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t\varrho _{1}^{p}} \bigr) \bigr\vert \,dt \\& \qquad {}+ \vert u \vert \int _{0}^{1} \bigl[(1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t \varrho _{1}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {} \times \bigl\vert f' \bigl( \sqrt[p]{ (1 - t) \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + t\varrho _{1}^{p}} \bigr) \bigr\vert \,dt \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \int _{0}^{1} t^{\beta +\rho k} \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) +(1 - t) \varrho _{2}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {} \times \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t)\varrho _{2}^{p}} \bigr) \bigr\vert \,dt \\& \qquad {} + \vert u \vert \int _{0}^{1} \bigl[t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) +(1 - t) \varrho _{2}^{p} \bigr]^{\frac{1 - p}{p}} \\& \qquad {} \times \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p} + (1 - \lambda )\varrho _{2}^{p} \bigr) + (1 - t)\varrho _{2}^{p}} \bigr) \bigr\vert \,dt \Biggr\} \Biggr] \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Biggl[(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl\{ {\frac{1}{r}} \int _{0}^{1} \bigl(t^{ \beta +\rho k} \bigl[(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{\frac{1-p}{p}} \bigr)^{r}\,dt \\& \qquad {}+{\frac{1}{s}} \int _{0}^{1} \bigl\vert f' \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t\varrho _{1}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr\} \\& \qquad {}+ \vert u \vert \biggl\{ {\frac{1}{r}} \int _{0}^{1} \bigl[(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+t \varrho _{1}^{p} \bigr]^{\frac{r(1-p)}{p}}\,dt \\& \qquad {} +{\frac{1}{s}} \int _{0}^{1} \bigl\vert f' \bigl( \sqrt[p]{(1-t) \bigl(\lambda \varrho _{1}^{p}+(1-\lambda ) \varrho _{2}^{p} \bigr)+t\varrho _{1}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr\} \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl\{ {\frac{1}{r}} \int _{0}^{1} \bigl(t^{\beta + \rho k} \bigl[t \bigl( \lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{1-p}{p}} \bigr)^{r}\,dt \\& \qquad {}+{\frac{1}{s}} \int _{0}^{1} \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr) \bigr\vert ^{s}\,dt \biggr\} \\& \qquad {}+ \vert u \vert \biggl\{ {\frac{1}{r}} \int _{0}^{1} \bigl[t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p} \bigr]^{\frac{r(1-p)}{p}}\,dt \\& \qquad {} +{\frac{1}{s}} \int _{0}^{1} \bigl\vert f' \bigl( \sqrt[p]{t \bigl(\lambda \varrho _{1}^{p}+(1-\lambda )\varrho _{2}^{p} \bigr)+(1-t) \varrho _{2}^{p}} \bigr) \bigr\vert ^{s} \,dt \biggr\} \Biggr\} \Biggr] \\& \quad \leq \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Biggl[(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1 - \lambda )(\varrho _{2}^{p} - \varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl\{ {\frac{\varrho _{1}^{r(1-p)}}{r}} \int _{0}^{1} t^{r(\beta +\rho k)}\,dt \\& \qquad {} + { \frac{1}{s}} \int _{0}^{1} \bigl[(1 - t) \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} + (1 - \lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr\} + t \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \bigr]\,dt \biggr\} \\& \qquad {}+ \vert u \vert \biggl\{ { \frac{\varrho _{1}^{r(1-p)}}{r}}+{\frac{1}{s}} \int _{0}^{1} \bigl[(1-t) \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \\& \qquad {} +(1- \lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr\} +t \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \bigr]\,dt \biggr\} \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \biggl\{ { \frac{\varrho _{1}^{r(1-p)}}{r}} \int _{0}^{1}t^{r( \beta +\rho k)}\,dt \\& \qquad {}+{\frac{1}{s}} \int _{0}^{1} \bigl[t \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} +(1-\lambda ) \bigl\vert f'( \varrho _{2}) \bigr\vert ^{s} \bigr\} +(1-t) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr]\,dt \biggr\} \\& \qquad {}+ \vert u \vert \biggl\{ { \frac{\varrho _{1}^{r(1-p)}}{r}}+{ \frac{1}{s}} \int _{0}^{1} \bigl[t \bigl\{ \lambda \bigl\vert f'(\varrho _{1}) \bigr\vert ^{s} \\& \qquad {} +(1 - \lambda ) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr\} +(1-t) \bigl\vert f'(\varrho _{2}) \bigr\vert ^{s} \bigr]\,dt \biggr\} \Biggr\} \Biggr] \\& \quad =\bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) [(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl\{ \frac{\varrho _{1}^{r(1-p)}}{r(r(\beta +\rho k)+1)} +{ \frac{\lambda \vert f'(\varrho _{1}) \vert ^{s} +(1-\lambda ) \vert f'(\varrho _{2}) \vert ^{s}+ \vert f'(\varrho _{1}) \vert ^{s}}{2s}} \biggr\} \\& \qquad {}+ \vert u \vert \biggl\{ { \frac{\varrho _{1}^{r(1-p)}}{r}}+{ \frac{\lambda \vert f'(\varrho _{1}) \vert ^{s} +(1-\lambda ) \vert f'(\varrho _{2}) \vert ^{s}+ \vert f'(\varrho _{1}) \vert ^{s}}{2s}} \biggr\} \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \{\sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \{ \frac{\varrho _{1}^{r(1-p)}}{r(r(\beta +\rho k)+1)} \\& \qquad {}+{ \frac{\lambda \vert f'(\varrho _{1}) \vert ^{s} +(1-\lambda ) \vert f'(\varrho _{2}) \vert ^{s}+ \vert f'(\varrho _{2}) \vert ^{s}}{2s}} \} \\& \qquad {} + \vert u \vert \biggl\{ { \frac{\varrho _{1}^{r(1-p)}}{r}}+{ \frac{\lambda \vert f'(\varrho _{1}) \vert ^{s} +(1-\lambda ) \vert f'(\varrho _{2}) \vert ^{s}+ \vert f'(\varrho _{2}) \vert ^{s}}{2s}} \biggr\} \} ] \\& \quad = \bigl(\varrho _{2}^{p}-\varrho _{1}^{p} \bigr) \Biggl[(1- \lambda )^{1+\beta} \vert 1-\gamma \vert \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} ((1-\lambda )(\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \\& \qquad {} \times \biggl( \frac{2s\varrho _{1}^{r(1-p)}+r(r(\beta +\rho k)+1)B_{s,1}}{2\mathit{rs}(r(\beta +\rho k)+1)} \biggr)+ \vert u \vert \biggl({ \frac{2s\varrho _{1}^{r(1-p)}+\mathit{rB}_{s,1}}{2\mathit{rs}}} \biggr) \Biggr\} \\& \qquad {}+ \vert \gamma \vert \lambda ^{1+\beta} \Biggl\{ \sum _{k=0}^{ \infty} \frac{\sigma (k) \vert w \vert ^{k} (\lambda (\varrho _{2}^{p}-\varrho _{1}^{p}) )^{\rho k}}{\Gamma (\rho k+\beta +1)} \biggl( \frac{2s\varrho _{1}^{r(1-p)}+r(r(\beta +\rho k)+1)B_{s,2}}{2\mathit{rs}(r(\beta +\rho k)+1)} \biggr) \\& \qquad {} + \vert u \vert \biggl({ \frac{2s\varrho _{1}^{r(1-p)}+\mathit{rB}_{s,2}}{2\mathit{rs}}} \biggr) \Biggr\} \Biggr]. \end{aligned}$$
Now, for the case \(p\in (0,1)\), using inequalities (11) and (12), and repeating the same procedure like \(p\in [1,\infty )\), we obtain the desired result. □
Corollary 4
Under the conditions of Theorem 4for \(\lambda ,w\rightarrow 0\), \(\gamma ,p,u\rightarrow \frac{1}{2}\), \(s\rightarrow 2\) and \(\beta ,\sigma (0)\rightarrow 1\), we have
$$\begin{aligned}& \biggl\vert \frac{f(\varrho _{1})+f(\varrho _{2})}{2}- \frac{1}{\sqrt{\varrho _{2}}-\sqrt{\varrho _{1}}} \int _{\sqrt{ \varrho _{1}}}^{\sqrt{\varrho _{2}}}f(\sqrt{x})\,dx \biggr\vert \\& \quad \leq \frac{\sqrt{\varrho _{2}}-\sqrt{\varrho _{1}}}{12} \bigl\{ 10 \varrho _{2}+ 9 \bigl({ \bigl\vert {f'(\varrho _{1})} \bigr\vert }^{2}+{ \bigl\vert {f'( \varrho _{2})} \bigr\vert }^{2} \bigr) \bigr\} . \end{aligned}$$