In this section the Ostrowski inequality is proved for an n-polynomial \(\mathscr{P} \)-convex function via a generalized k-fractional Hilfer–Katugampola derivative.
Lemma 3.1
(See [18])
For a differentiable function \(\mu >0\), \(\mathscr{P}\in \mathbb{R}\setminus \{0\}\) and \(\psi ^{( \mu )} :\Omega \rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\) and \(\psi ^{( \mu +1)}\in {M}([{a_{1}},{a_{2}}])\), the following inequality holds:
$$\begin{aligned} \begin{aligned} & \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} -\frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu }\psi \bigr) (z) \bigr] \\ & \quad =- \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{ \zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{ \zeta {a_{2}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta . \end{aligned} \end{aligned}$$
(3.1)
Theorem 3.1
For a differentiable function \(n\in \mathbb{N}\), \(\mu >0\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert \) a n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \\ &\quad\quad{} \times \frac{1}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\mu +2\theta +1}{(\mu +1)(\mu +\theta +1)}-\mathbb{B}(\mu -1, \theta -1) \biggr], \end{aligned} $$
(3.2)
and the following inequality holds with \(\forall z\in ({a_{1}},{a_{2}}) \) and \(\mathscr{P}\in (- \infty , 0)\cup (0, 1)\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{2}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }} \biggl[ \frac{(z^{\mathscr{P}} -{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \\ &\quad\quad{} \times \frac{1}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\mu +2\theta +1}{(\mu +1)(\mu +\theta +1)} -\mathbb{B}(\mu -1, \theta -1) \biggr]. \end{aligned} $$
(3.3)
Proof
We use Lemma 3.1 to prove the inequality (3.2) for the n-polynomial \(\mathscr{P} \)-convexity of \(\vert \psi ^{( \mu +1)} \vert \) to yield
$$\begin{aligned} \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{ \zeta {a_{2}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \\ & \qquad {}\times \Biggl[\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-(1- \zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-\zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert \Biggr] \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \\ & \qquad {}\times \Biggl[ \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-(1- \zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{2}}) \bigr\vert + \frac{1}{n}\sum_{ \theta =1}^{n} \bigl[1-\zeta ^{ \theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert \Biggr] \,d\zeta . \end{aligned} \end{aligned}$$
(3.4)
As \(\mathscr{P}\in (1,\infty )\), we can infer that
$$ \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}}\leq \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}}\leq {a_{1}}^{1- \mathscr{P}}. $$
(3.5)
We proceed by simplifying
$$ \begin{aligned} & \int _{0}^{1}\zeta ^{\mu } \Biggl[ \frac{1}{n}\sum_{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] +\frac{1}{n}\sum _{ \theta =1}^{n} \bigl[1-\zeta ^{\theta } \bigr] \Biggr] \,d\zeta \\ & \quad =\frac{1}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\mu +2\theta +1}{(\mu +1)(\mu +\theta +1)}-\mathbb{B}(\mu -1, \theta -1) \biggr]. \end{aligned} $$
(3.6)
The first inequality of Theorem 3.1 is proved. Now for the second part, we let \(\mathscr{P}\in (-\infty , 0)\cup (0,1)\) to yield
$$ \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}}\leq \bigl(\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}}\leq {a_{2}}^{1- \mathscr{P}}. $$
(3.7)
The above inequality completes the proof of the second part of Theorem 3.1. □
Theorem 3.2
For a differentiable function \(n \in \mathbb{N}\), \(\lambda , \vartheta >1 \) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \(a_{1}, a_{2}\in \Omega \) with \(a_{1}< a_{2}\), and \(\psi ^{( \mu )} :\Omega \subset (0,\infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([a_{1},a_{2}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) an n-polynomial convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [a_{1},a_{2}]\), the following inequality holds for all \(z\in (a_{1},a_{2})\) and \(\mathscr{P}\in (1, \infty )\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}} \mathscr{Q}}{{\mathscr{P}}^{1+\mu }(1+\lambda \mu )^{1/\lambda }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1} +({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \Biggl( \frac{1}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{1/\vartheta }, \end{aligned} $$
(3.8)
and the following inequality holds with \(\forall z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (-\infty , 0)\cup (0, 1)\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{2}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }(1+\lambda \mu )^{1/\lambda }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \Biggl( \frac{1}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{1/\vartheta }. \end{aligned} $$
(3.9)
Proof
We use Lemma 3.1, (3.5) and the Hölder inequality to prove the first inequality of Theorem 3.3,
$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} -\frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {} + \frac{{a_{2}}^{1-\mathscr{P}}({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \biggl( \int _{0}^{1}\zeta ^{\lambda \mu } \,d\zeta \biggr)^{1/ \lambda } \biggl( \int _{0}^{1} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/ \vartheta } \\ & \qquad {}+ \frac{{a_{2}}^{1-\mathscr{P}}({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \biggl( \int _{0}^{1} \zeta ^{\lambda \mu } \,d\zeta \biggr)^{1/ \lambda } \biggl( \int _{0}^{1} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/ \vartheta }. \end{aligned}$$
(3.10)
As \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) is n-polynomial \(\mathscr{P} \)-convex and \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q} \forall z \in [{a_{1}}, {a_{2}}]\), we have
$$ \begin{aligned} & \int _{0}^{1} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1} \Biggl[\frac{1}{n}\sum _{\theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta }+\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr] \,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl[2-(1-\zeta )^{\theta }- \zeta ^{\theta } \bigr] d \zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \end{aligned} $$
(3.11)
and
$$ \int _{0}^{1} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \leq \frac{\mathscr{Q}^{\vartheta }}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1}. $$
(3.12)
Since \(\int _{0}^{1}\zeta ^{\lambda \mu } \,d\zeta = \frac{1}{\lambda \mu +1}\). We get the first inequality of Theorem 3.3 by combining all above inequalities. Continuing in the same way the second inequality of Theorem 3.3 can be proved. □
Theorem 3.3
For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0,\infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) an n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }(1+\lambda \mu )^{1/\lambda }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \\ & \qquad {}\times \Biggl(\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{ \theta =1}^{n} \biggl[ \frac{\vartheta \mu +2\theta +1}{(\mu \vartheta +1)(\mu \vartheta +\theta +1)} - \mathbb{B}(\theta +1,\vartheta \mu +1) \biggr] \Biggr)^{1/ \vartheta } \end{aligned} $$
(3.13)
and the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (-\infty , 0) \cup (0,1)\):
$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} -\frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{{a_{2}}^{1-\mathscr{P}}\mathscr{Q}}{{\mathscr{P}}^{1+\mu }(1+\lambda \mu )^{1/\lambda }} \biggl[ \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{({a_{2}}-{a_{1}})} \biggr] \\ & \qquad {} \times \Biggl(\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{ \theta =1}^{n} \biggl[ \frac{\vartheta \mu +2\theta +1}{(\mu \vartheta +1)(\mu \vartheta +\theta +1)} - \mathbb{B}(\theta +1,\vartheta \mu +1) \biggr] \Biggr)^{1/ \vartheta } . \end{aligned}$$
(3.14)
Proof
We use the Lemma 3.1, to prove Theorem 3.3 and the power-mean inequality to yield
$$\begin{aligned} \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {} + \frac{{a_{2}}^{1-\mathscr{P}}({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{{a_{1}}^{1-\mathscr{P}}(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl( \int _{0}^{1}\zeta ^{ \vartheta \mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {} + \frac{{a_{2}}^{1-\mathscr{P}}({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl( \int _{0}^{1}\zeta ^{\vartheta \mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta }. \end{aligned} \end{aligned}$$
(3.15)
As \(\vert \psi ^{( \mu )} \vert ^{\vartheta }\) is n-polynomial \(\mathscr{P} \)-convex and \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z \in [{a_{1}}, {a_{2}}]\), we obtain
$$\begin{aligned} \begin{aligned} & \int _{0}^{1}\zeta ^{\vartheta \mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}\zeta ^{\vartheta \mu } \Biggl[ \frac{1}{n} \sum_{\theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1 )}({a_{1}}) \bigr\vert ^{\vartheta }+ \frac{1}{n}\sum _{\theta =1}^{n} \bigl[1-\zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr]\,d \zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl[2\zeta ^{\mu \vartheta } -\zeta ^{\mu \vartheta }(1- \zeta )^{ \theta }+\zeta ^{\vartheta \mu } \bigl(1-\zeta ^{\theta } \bigr) \bigr]\,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\vartheta \mu +2\theta +1}{(\mu \vartheta +1) (\mu \vartheta +\theta +1)}- \mathbb{B}(\theta +1,\vartheta \mu +1) \biggr]. \end{aligned} \end{aligned}$$
(3.16)
Similarly,
$$ \begin{aligned} & \int _{0}^{1}\zeta ^{\vartheta \mu } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \biggl[ \frac{\vartheta \mu +2\theta +1}{(\mu \vartheta +1)(\mu \vartheta +\theta +1)}- \mathbb{B}(\theta +1,\vartheta \mu +1) \biggr]. \end{aligned} $$
(3.17)
We arrive at the first inequality of Theorem 3.3 by combining all above inequalities. For the second part continuing in the same fashion, we find the required result. □
Theorem 3.4
For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) be an n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1,\infty )\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \frac{({a_{1}}^{\lambda (1-\mathscr{P})})}{\lambda (\lambda \mu +1)}+ \frac{1}{\vartheta } \Biggl(\frac{\mathscr{Q}}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{\vartheta } \Biggr] \end{aligned} $$
(3.18)
and the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (-\infty , 0) \cup (0,1)\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \frac{({a_{2}}^{\lambda (1-\mathscr{P})})}{\lambda (\lambda \mu +1)} + \frac{1}{\vartheta } \Biggl(\frac{\mathscr{Q}}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{\vartheta } \Biggr]. \end{aligned} $$
(3.19)
Proof
The Young inequality is \(cd\leq \frac{1}{\lambda }c^{\lambda }+\frac{1}{\vartheta }d^{ \lambda }\), \(c,d\geq 0\), \(\lambda ,\vartheta >1\), \(\lambda ^{-1}+ \vartheta ^{-1}=1\). We use Lemma 3.1 to prove the first part of Theorem 3.3, and using the n-polynomial \(\mathscr{P}\)-convexity of \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) we find
$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \int _{0}^{1} \biggl(\frac{1}{\lambda } \bigl\vert \zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert ^{ \lambda }+ \frac{1}{\vartheta } \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \biggr) \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \int _{0}^{1} \biggl(\frac{1}{\lambda } \bigl\vert \zeta ^{\mu } \bigl( \zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert ^{ \lambda }+ \frac{1}{\vartheta } \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \biggr) \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl(\frac{\zeta ^{\lambda \mu }}{\lambda } \bigl\vert \bigl( \zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert ^{\lambda } \\ & \qquad {} +\frac{1}{\vartheta } \Biggl\vert \frac{1}{n}\sum _{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1- \zeta ^{ \theta } \bigr] \Biggr\vert \psi ^{( \mu +1)}(z) \vert \vert ^{\vartheta } \Biggr) \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1} \Biggl(\frac{\zeta ^{\lambda \mu }}{\lambda } \bigl\vert \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert ^{ \lambda } \\ & \qquad {} +\frac{1}{\vartheta } \Biggl\vert \frac{1}{n}\sum _{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{2}}) \bigr\vert + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1- \zeta ^{ \theta } \bigr] \Biggr\vert \psi ^{( \mu +1)}(z) \vert \vert ^{\vartheta } \Biggr) \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \frac{({a_{1}}^{\lambda (1-\mathscr{P})})}{\lambda (\lambda \mu +1)}+ \frac{1}{\vartheta } \Biggl(\frac{\mathscr{Q}}{n}\sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr)^{\vartheta } \Biggr]. \end{aligned}$$
(3.20)
Continuing in the same fashion, we can prove the second part. □
Theorem 3.5
For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) a n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \frac{\lambda {a_{1}}^{1-\mathscr{P}}}{(\mu +1)}+ \frac{\vartheta \mathscr{Q}}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr] \end{aligned} $$
(3.21)
and the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (-\infty , 0) \cup (0,1)\),
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[\frac{\lambda {a_{2}}^{1-\mathscr{P}}}{(\mu +1)}+ \frac{\vartheta \mathscr{Q}}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr] . \end{aligned} $$
(3.22)
Proof
Use the weighted \(\mathscr{AM}-\mathscr{GM}\) inequality
$$ \begin{aligned} c^{\lambda }d^{\vartheta }\leq {\lambda }c+{ \vartheta }d,\quad c,d \geq 0, \lambda , \vartheta >0, \lambda +\vartheta =1. \end{aligned} $$
(3.23)
Using Lemma 3.1 and by using the n-polynomial \(\mathscr{P}\)-convexity of \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) we find
$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}}) +({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1} \bigl[\zeta ^{\mu } \bigl( \zeta {a_{1}}^{ \mathscr{P}} +(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr]^{ \lambda } \bigl[ \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \bigr]^{ \vartheta } \,d\zeta \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} \times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} ]^{ \lambda } \bigl[ \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \bigr]^{\vartheta } \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \biggl[ \int _{0}^{1}\lambda \zeta ^{\mu } \bigl\vert \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert \,d\zeta \\ &\quad\quad{} + \int _{0}^{1} \vartheta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \biggr] \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \biggl[ \int _{0}^{1}\lambda \zeta ^{\mu } \bigl\vert \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigr\vert \,d\zeta \\ &\quad\quad{} + \int _{0}^{1} \vartheta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \biggr] \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \Biggl[ \int _{0}^{1}{a_{1}}^{1-\mathscr{P}} \lambda \zeta ^{\mu } \,d\zeta + \int _{0}^{1}\vartheta \Biggl\vert \frac{1}{n} \sum_{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert \\ &\quad\quad{} + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-\zeta ^{ \theta } \bigr] \big\vert \psi ^{( \mu +1)}(z) \big\vert \Biggr\vert \,d\zeta \Biggr] \\ & \qquad {} + \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ & \qquad {}\times \Biggl[ \int _{0}^{1}{a_{1}}^{1-\mathscr{P}} \lambda \zeta ^{\mu } \,d\zeta + \int _{0}^{1}\vartheta \Biggl\vert \frac{1}{n} \sum_{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{2}}) \bigr\vert \\ &\quad\quad{} + \frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-\zeta ^{ \theta } \bigr] \big\vert \psi ^{( \mu +1)}(z) \big\vert \Biggr\vert \,d\zeta \Biggr] \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[\frac{\lambda {a_{1}}^{1-\mathscr{P}}}{(\mu +1)}+ \frac{\vartheta \mathscr{Q}}{n} \sum_{\theta =1}^{n} \frac{2\theta }{\theta +1} \Biggr]. \end{aligned}$$
(3.24)
Continuing in the same fashion, we can prove the second part. □
Theorem 3.6
For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ }\) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) a n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):
$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \bigl(\Lambda _{1}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{\theta =1}^{n} \frac{\theta }{\theta +1} \Biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{2}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{\theta ^{2}+2\theta -1}{(\theta +2)(\theta +1)} \Biggr)^{1/ \vartheta } \Biggr] \end{aligned}$$
(3.25)
and the following inequality holds for all \(z\in ({a_{1}}, {a_{2}})\) and \(\mathscr{P}\in (-\infty , 0) \cup (0,1)\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}- \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z) + \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \bigl(\Lambda _{3}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{\theta =1}^{n} \frac{\theta }{\theta +1} \Biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{4}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{\theta ^{2}+2\theta -1}{(\theta +2)(\theta +1)} \Biggr)^{1/ \vartheta } \Biggr]. \end{aligned} , $$
(3.26)
Here
$$ \begin{aligned} & \Lambda _{1}({a_{1}},z; \mathscr{P})= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ]}{{a_{1}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \\ & \Lambda _{2}({a_{1}},z;\mathscr{P})= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2, \lambda \mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (- \infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2, \lambda \mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ]}{{a_{1}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (1, \infty ), \end{cases}\displaystyle \\ & \Lambda _{3}({a_{2}},z;\mathscr{P})= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-({a_{2}}/z)^{\mathscr{P}} ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-(z/{a_{2}})^{\mathscr{P}} ]}{{a_{2}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \end{aligned} $$
(3.27)
and
$$ \Lambda _{4}({a_{2}},z;\mathscr{P}) = \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-({a_{2}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (- \infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-(z/{a_{2}})^{\mathscr{P}} ) ]}{{a_{2}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (1, \infty ). \end{cases} $$
(3.28)
Proof
We use Lemma 3.1 to prove the first part of the inequality and we use the Hölder–İşcan inequality to find
$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \biggl( \int _{0}^{1}\zeta ^{\lambda \mu }(1-\zeta ) \bigl( \zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{0}^{1}\zeta ^{\lambda \mu +1} \bigl( \zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta ) z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{ \zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \biggl( \int _{0}^{1} \zeta ^{\lambda \mu }(1-\zeta ) \bigl( \zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \lambda (\frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1/ \lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{0}^{1}\zeta ^{\lambda \mu +1} \bigl( \zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{ \lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})}\,d\zeta \biggr)^{1/ \lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{ \zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}\times \biggl[ \bigl(\Lambda _{1}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \biggl( \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{2}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \biggl( \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}\times \biggl[ \bigl(\Lambda _{3}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \biggl( \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{4}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \biggl( \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}}\sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr]. \end{aligned}$$
(3.29)
As \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) is n-polynomial \(\mathscr{P} \)-convex and \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z \in [{a_{1}}, {a_{2}}]\), we find
$$\begin{aligned} & \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}(1-\zeta ) \Biggl[ \frac{1}{n} \sum_{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta }+\frac{1}{n}\sum_{ \theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{\vartheta } \Biggr]\,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n} \sum_{\theta =1}^{n} \int _{0}^{1} \bigl[2(1-\zeta )-(1-\zeta )^{\theta +1}-\zeta ^{ \theta }(1-\zeta ) \bigr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{\theta }{\theta +1}, \\ & \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}\zeta \Biggl[\frac{1}{n} \sum _{ \theta =1}^{n} \bigl[1-(1-\zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta }+\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl[2\zeta -\zeta (1-\zeta )^{\theta }-\zeta ^{ \theta +1} \bigr]\,d\zeta \\ & \quad \leq \frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \frac{\theta ^{2}+2\theta -1}{(\theta +2)(\theta +1)}. \end{aligned}$$
(3.30)
Similarly, we obtain
$$ \begin{aligned} & \int _{0}^{1}(1-\zeta ) \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \leq \frac{\mathscr{Q}^{\vartheta }}{n} \sum_{\theta =1}^{n} \frac{\theta }{\theta +1}, \\ & \int _{0}^{1}\zeta \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}}\sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \leq \frac{\mathscr{Q}^{\vartheta }}{n} \sum_{\theta =1}^{n} \frac{\theta ^{2}+2\theta -1}{(\theta +2)(\theta +1)}. \end{aligned} $$
(3.31)
We have the result
$$\begin{aligned}& \begin{aligned} \Lambda _{1}({a_{1}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \lambda \mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \\ &= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-({a_{1}}/z)^{\mathscr{P}} ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-(z/{a_{1}})^{\mathscr{P}} ]}{{a_{1}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.32)
$$\begin{aligned}& \begin{aligned} \Lambda _{2}({a_{1}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \lambda \mu +1} \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{\lambda (\frac{1-\mathscr{P}}{\mathscr{P}})} \,d \zeta \\ &= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (- \infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ]}{{a_{1}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (1, \infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.33)
$$\begin{aligned}& \begin{aligned} \Lambda _{3}({a_{2}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \lambda \mu }(1-\zeta ) \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \\ &= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-({a_{2}}/z)^{\mathscr{P}} ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +1,\lambda \mu +3,1-(z/{a_{2}})^{\mathscr{P}} ]}{{a_{2}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +1)(\lambda \mu +2)}, &\mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.34)
$$\begin{aligned}& \begin{aligned} \Lambda _{4}({a_{2}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \lambda \mu +1} \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{\lambda (\frac{1-\mathscr{P}}{\mathscr{P}})} \,d \zeta \\ &= \textstyle\begin{cases} \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-({a_{2}}/z)^{\mathscr{P}} ) ]}{z^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (- \infty , 0)\cup (0,1), \\ \frac{ [{}_{2}\mathscr{F}_{1} (\lambda (1-1/\mathscr{P}),\lambda \mu +2,\lambda \mu +3,1-(z/{a_{2}})^{\mathscr{P}} ) ]}{{a_{2}}^{\lambda (\mathscr{P}-1)}(\lambda \mu +2)}, &\mathscr{P}\in (1, \infty ). \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.35)
□
Theorem 3.7
For a differentiable function \(n\in \mathbb{N}\), \(\lambda , \vartheta >1\) with \(\lambda ^{-1}+\vartheta ^{-1}=1\), \({a_{1}}, {a_{2}}\in \Omega \) with \({a_{1}}< {a_{2}}\), and \(\psi ^{( \mu )} :\Omega \subset (0, \infty )\rightarrow \mathbb{R}\) on \(\Omega ^{\circ } \) such that \(\psi ^{( \mu +1)}\in M ([{a_{1}},{a_{2}}])\) and \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) an n-polynomial \(\mathscr{P}\)-convex function satisfying \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}},{a_{2}}]\), the following inequality holds for all \(z\in ({a_{1}},{a_{2}})\) and \(\mathscr{P}\in (1, \infty )\):
$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}- \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \bigl(\Lambda _{1}^{*}({a_{1}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{\theta =1}^{n} \mathrm{T} _{1}({a_{1}},z; \mathscr{P}) \Biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{2}^{*}({a_{1}},z; \mathscr{P}) \bigr)^{1/\lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{ \theta =1}^{n}\mathrm{T} _{2}({a_{1}},z; \mathscr{P}) \Biggr)^{1/ \vartheta } \Biggr] \end{aligned}$$
(3.36)
and the following inequality holds for all \(z\in ({a_{1}}, {a_{2}})\) and \(\mathscr{P}\in (-\infty , 0)\cup (0,1)\):
$$ \begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}} -z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}- \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu } \psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \Biggl[ \bigl(\Lambda _{3}^{*}({a_{2}},z; \mathscr{P}) \bigr)^{1/ \lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{\theta =1}^{n} \mathrm{T} _{3}({a_{2}},z; \mathscr{P}) \Biggr)^{1/\vartheta } \\ & \qquad {} + \bigl(\Lambda _{4}^{*}({a_{2}},z; \mathscr{P}) \bigr)^{1/\lambda } \Biggl( \frac{\mathscr{Q}^{\vartheta }}{n}\sum _{ \theta =1}^{n}\mathrm{T} _{4}({a_{2}},z; \mathscr{P}) \Biggr)^{1/ \vartheta } \Biggr] .\end{aligned} $$
(3.37)
Here
$$\begin{aligned} & \mathrm{T} _{1}({a_{1}}z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{(\mathscr{P}-1)}} [ \frac{2}{(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +1,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\theta +\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ] \\ \quad {}-\mathbb{B}(\mu +\theta +1,2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +\theta +1,\theta +\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{(\mathscr{P}-1)}} [ \frac{2}{(\mu +1)(\mu +2)} \mathscr{F}_{1} (1-1/\mathscr{P},\mu +1, \mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +1,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\theta +\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ] \\ \quad {}-\mathbb{B}(\mu +\theta +1,2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +\theta +1,\theta +\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \\ & \mathrm{T} _{2}({a_{1}}z;\mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{(\mathscr{P}-1)}} [\frac{2}{(\mu +2)} \mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +2,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\theta +\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ] \\ \quad {}-\frac{1}{\mu +\theta +2}{}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P}, \mu +\theta +2,\theta +\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) ], \\ \quad \mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{(\mathscr{P}-1)}} [\frac{2}{(\mu +2)} \mathscr{F}_{1} (1-1/\mathscr{P},\mu +2,\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +2,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\theta +\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ] \\ \quad {}-\frac{1}{\mu +\theta +2}{}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P}, \mu +\theta +2,\theta +\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) ], \\ \quad \mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \\ & \Lambda _{1}^{*}({a_{1}},z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +1)(\mu +2)} \mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +1,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{\mathscr{P}-1}(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +1,\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \\ & \Lambda _{2}^{*}({a_{1}},z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +2,a\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \\ & \Lambda _{3}^{*}({a_{2}},z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +1)(\mu +2)} \mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +1,\mu +3,1-({a_{2}}/z)^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{2}}^{\mathscr{P}-1}(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +1,\mu +3,1-(z/{a_{2}})^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \end{aligned}$$
(3.38)
and
$$ \Lambda _{4}^{*}({a_{2}},z; \mathscr{P})= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-({a_{2}}/z)^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{2}}^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-(z/{a_{2}})^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ). \end{cases}\displaystyle . $$
(3.39)
Proof
By using Lemma 3.1 and the improved power-mean inequality
$$\begin{aligned} & \biggl\vert \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu } \psi ^{( \mu )} ({a_{1}})+({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu }\psi ^{( \mu )} ({a_{2}})}{{\mathscr{P}}^{\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{} - \frac{\Gamma _{k} (\mu +k)}{{a_{2}}-{a_{1}}} \bigl[ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{1}}^{+}}^{\mu }\psi \bigr) (z)+ \bigl({}_{k}^{\mathscr{P}} \mathscr{D}_{{a_{2}}^{-}}^{ \mu } \psi \bigr) (z) \bigr] \biggr\vert \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1 )} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{ \mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \\ &\quad\quad{}\times \int _{0}^{1}\zeta ^{\mu } \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{\frac{1-\mathscr{P}}{\mathscr{P}}} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert \,d\zeta \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl( \zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad\quad{}\times \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl(\zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl( \zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta ) z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad\quad{}\times \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta ) z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \biggr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{2}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \quad \leq \frac{(z^{\mathscr{P}}-{a_{1}}^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \bigl(\Lambda _{1}^{*}({a_{1}},z; \mathscr{P}) \bigr)^{1-1/ \lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu }(1- \zeta ) \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad\quad{}\times \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{ \vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{2}^{*}({a_{1}},z; \mathscr{P}) \bigr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr] \\ & \qquad {}+ \frac{({a_{2}}^{\mathscr{P}}-z^{\mathscr{P}})^{\mu +1}}{{\mathscr{P}}^{1+\mu }({a_{2}}-{a_{1}})} \biggl[ \bigl(\Lambda _{3}^{*}({a_{2}},z; \mathscr{P}) \bigr)^{1-1/ \lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl( \zeta {a_{2}}^{ \mathscr{P}}+(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad\quad{}\times \bigl\vert \psi ^{( \mu +1)} \bigl({}^{ \mathscr{P}} \sqrt{\zeta {a_{2}}^{\mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \\ & \qquad {}+ \bigl(\Lambda _{4}^{*}({a_{2}},z; \mathscr{P}) \bigr)^{1-1/\lambda } \\ & \qquad {}\times \biggl( \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{2}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{\lambda ( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{2}}^{ \mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \biggr)^{1/\vartheta } \biggr]. \end{aligned}$$
(3.40)
As \(\vert \psi ^{( \mu +1)} \vert ^{\vartheta }\) is n-polynomial \(\mathscr{P} \)-convex and \(\vert \psi ^{( \mu +1)}(z) \vert \leq \mathscr{Q}\), \(\forall z\in [{a_{1}}, {a_{2}}]\), we have
$$\begin{aligned} \begin{aligned} & \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}} +(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}\zeta ^{\mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ & \qquad {} \times \Biggl[\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-(1- \zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta } +\frac{1}{n}\sum _{\theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{(\frac{1-\mathscr{P}}{\mathscr{P}})} \bigl[2 \zeta ^{\mu }(1- \zeta )-\zeta ^{\mu }(1-\zeta )^{ \theta +1}-\zeta ^{ \mu +\theta }(1- \zeta ) \bigr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \mathrm{T} _{1}({a_{1}}z;\mathscr{P}), \end{aligned} \end{aligned}$$
(3.41)
where
$$\begin{aligned} \mathrm{T} _{1}({a_{1}}z; \mathscr{P})&:= \int _{0}^{1} \bigl( \zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ &\quad {}\times \bigl[2\zeta ^{\mu }(1-\zeta )-\zeta ^{ \mu }(1- \zeta )^{\theta +1}-\zeta ^{\mu +\theta }(1-\zeta ) \bigr]\,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{(\mathscr{P}-1)}} [ \frac{2}{(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} (1-1/\mathscr{P}, \mu +1,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +1,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\theta +\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ] \\ \quad {}-\mathbb{B}(\mu +\theta +1,2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +\theta +1,\theta +\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{(\mathscr{P}-1)}} [ \frac{2}{(\mu +1)(\mu +2)} \mathscr{F}_{1} (1-1/\mathscr{P},\mu +1, \mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +1,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +1,\theta +\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ] \\ \quad {}-\mathbb{B}(\mu +\theta +1,2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +\theta +1,\theta +\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ). \end{cases}\displaystyle \end{aligned}$$
(3.42)
Similarly, we obtain
$$ \begin{aligned} & \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl\vert \psi ^{( \mu +1)} \bigl({}^{\mathscr{P}} \sqrt{\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}}} \bigr) \bigr\vert ^{\vartheta } \,d\zeta \\ & \quad \leq \int _{0}^{1}\zeta ^{\mu +1} \bigl(\zeta {a_{1}}^{ \mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \\ & \qquad {} \times \Biggl[\frac{1}{n}\sum_{\theta =1}^{n} \bigl[1-(1- \zeta )^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}({a_{1}}) \bigr\vert ^{\vartheta }+ \frac{1}{n}\sum _{\theta =1}^{n} \bigl[1- \zeta ^{\theta } \bigr] \bigl\vert \psi ^{( \mu +1)}(z) \bigr\vert ^{ \vartheta } \Biggr]\,d\zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \int _{0}^{1} \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{ \mathscr{P}} \bigr)^{(\frac{1-\mathscr{P}}{\mathscr{P}})} \bigl[2 \zeta ^{\mu +1} - \zeta ^{\mu +1}(1-\zeta )^{\theta }-\zeta ^{\mu + \theta +1} \bigr]\,d \zeta \\ & \quad =\frac{\mathscr{Q}^{\vartheta }}{n}\sum_{\theta =1}^{n} \mathrm{T} _{2}({a_{1}}z;\mathscr{P}), \end{aligned} $$
(3.43)
where
$$\begin{aligned} \begin{aligned} \mathrm{T} _{2}({a_{1}}z; \mathscr{P}) :={}& \int _{0}^{1} \bigl( \zeta {a_{1}}^{ \mathscr{P}} +(1-\zeta )z^{\mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})} \bigl[2\zeta ^{\mu +1}- \zeta ^{\mu +1}(1- \zeta )^{\theta }-\zeta ^{\mu +\theta +1} \bigr]\,d \zeta \\ ={}& \textstyle\begin{cases} \frac{1}{z^{(\mathscr{P}-1)}} [\frac{2}{(\mu +2)} \mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +2,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\theta +\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ] \\ \quad {}- \frac{1}{\mu +\theta +2}{}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P}, \mu +\theta +2,\theta +\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ) ], \\ \quad \mathscr{P} \in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{(\mathscr{P}-1)}} [\frac{2}{(\mu +2)} \mathscr{F}_{1} (1-1/\mathscr{P},\mu +2,\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) \\ \quad {}-\mathbb{B}(\mu +2,\theta +2){}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P},\mu +2,\theta +\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ] \\ \quad {}-\frac{1}{\mu +\theta +2}{}_{2}\mathscr{F}_{1} (1-1/ \mathscr{P}, \mu +\theta +2,\theta +\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ) ], \\ \quad \mathscr{P} \in (1,\infty ). \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.44)
We obtain \(\mathrm{T} _{3}({a_{2}},z;\mathscr{P})\) and \(\mathrm{T} _{4}({a_{2}},z;\mathscr{P})\) by replacing \({a_{1}}\) into \({a_{2}}\) in the above results and using the facts
$$\begin{aligned}& \begin{aligned} \Lambda _{1}^{*}({a_{1}},z; \mathscr{P})&:= \int _{0}^{1} \zeta ^{ \mu }(1-\zeta ) \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1- \zeta )z^{\mathscr{P}} \bigr)^{(\frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +1)(\mu +2)} \mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +1,\mu +3,1-({a_{1}}/z)^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{\mathscr{P}-1}(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +1,\mu +3,1-(z/{a_{1}})^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.45)
$$\begin{aligned}& \begin{aligned} \Lambda _{2}^{*}({a_{1}},z; \mathscr{P})&:= \int _{0}^{1} \zeta ^{ \mu +1} \bigl(\zeta {a_{1}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{( \frac{1-\mathscr{P}}{\mathscr{P}})}\,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-({a_{1}}/z)^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{1}}^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +2,a\mu +3,1-(z/{a_{1}})^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.46)
$$\begin{aligned}& \begin{aligned} \Lambda _{3}^{*}({a_{2}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \mu }(1-\zeta ) \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{(\frac{1-\mathscr{P}}{\mathscr{P}})} \,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +1)(\mu +2)} \mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +1,\mu +3,1-({a_{2}}/z)^{ \mathscr{P}} ), \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{2}}^{\mathscr{P}-1}(\mu +1)(\mu +2)}{}_{2} \mathscr{F}_{1} ((1-1/\mathscr{P}),\mu +1,\mu +3,1-(z/{a_{2}})^{ \mathscr{P}} ), \\ \quad \mathscr{P} \in (1,\infty ), \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.47)
and
$$ \begin{aligned} \Lambda _{4}^{*}({a_{2}},z; \mathscr{P})&:= \int _{0}^{1}\zeta ^{ \mu +1} \bigl(\zeta {a_{2}}^{\mathscr{P}}+(1-\zeta )z^{ \mathscr{P}} \bigr)^{ \frac{1-\mathscr{P}}{\mathscr{P}}} \,d\zeta \\ &= \textstyle\begin{cases} \frac{1}{z^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-({a_{2}}/z)^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (-\infty , 0)\cup (0,1), \\ \frac{1}{{a_{2}}^{\mathscr{P}-1}(\mu +2)} [{}_{2}\mathscr{F}_{1} ((1-1/ \mathscr{P}),\mu +2,a\mu +3,1-(z/{a_{2}})^{\mathscr{P}} ) ], \\ \quad \mathscr{P}\in (1,\infty ), \end{cases}\displaystyle \end{aligned} $$
(3.48)
which completes the proof. □