The resulting new Hermite–Hadamard-type inequalities are presented in this section. Conformable fractional integrals are used for doubly differentiable functions when obtaining these inequalities. These inequalities are examined under two separate subheadings. In the first subheading, midpoint-type inequalities, which are the left side of Hermite–Hadamard inequalities, and trapezoid-type inequalities, which are the right side of Hermite–Hadamard inequalities, will be discussed in the other.
3.1 Inequalities of midpoint type involving conformable fractional integrals
In this subsection, midpoint-type inequalities are created for twice-differentiable functions with the help of conformable fractional integrals. To obtain these inequalities, let us first set up the following identity.
Lemma 1
Let \(\mathscr{F}:[\sigma ,\delta ]\rightarrow \mathbb{R}\) be a twice-differentiable mapping on \((\sigma ,\delta )\) such that \(\mathscr{F}^{\prime \prime }\in L_{1} [ \sigma ,\delta ] \). Then, the following equality holds:
$$\begin{aligned} & \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta } \mathcal{J}_{\delta -}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta } \mathcal{J}_{\sigma +}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \\ &\quad =\frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl[ \int _{0}^{1} \biggl( \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{ \beta } \biggr] \,ds \biggr) \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2}\delta \biggr) \,d\mu \\ &\quad \quad{}+ \int _{0}^{1} \biggl( \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr) \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma + \frac{1-\mu }{2}\delta \biggr) \,d\mu \biggr] . \end{aligned}$$
(3.1)
Proof
With the help of integrating by parts, we obtain
$$\begin{aligned} A_{1}& = \int _{0}^{1} \biggl( \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr) \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma + \frac{1+\mu }{2}\delta \biggr) \,d\mu \\ & = \frac{2}{\delta -\sigma } \biggl( \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{1-\mu }{2}\sigma + \frac{1+\mu }{2}\delta \biggr) \bigg\vert _{0}^{1} \\ & \quad{}+\frac{2}{\delta -\sigma } \int _{0}^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-\mu ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \mathscr{F}^{\prime } \biggl( \frac{1-\mu }{2}\sigma + \frac{1+\mu }{2}\delta \biggr) \,d\mu \\ & =-\frac{2}{\delta -\sigma } \biggl( \int _{0}^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) \\ & \quad{}+\frac{2}{\delta -\sigma } \biggl\{ \frac{2}{\delta -\sigma } \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-\mu ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \mathscr{F} \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2} \delta \biggr) \bigg\vert _{0}^{1} \\ & \quad{}+ \frac{2\beta }{\delta -\sigma } \int _{0}^{1} \biggl( \frac{1- ( 1-\mu ) ^{\alpha }}{\alpha } \biggr) ^{ \beta -1} ( 1-\mu ) ^{\alpha -1}\mathscr{F} \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2}\delta \biggr) \,d\mu \biggr\} . \end{aligned}$$
If we use the change of variables \(x=\frac{1-\mu }{2}\sigma +\frac{1+\mu }{2}\delta \), then we have
$$\begin{aligned} A_{1}& =-\frac{2}{\delta -\sigma } \biggl( \int _{0}^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) \\ &\quad {}- \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2}\frac{1}{\alpha ^{\beta }} \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \\ & \quad{}+ \biggl( \frac{2}{\delta -\sigma } \biggr) ^{\alpha \beta +2} \frac{\Gamma ( \beta +1 ) }{\Gamma ( \beta ) } \int _{\frac{\sigma +\delta }{2}}^{\delta } \biggl( \frac{ ( \frac{\delta -\sigma }{2} ) ^{\alpha }- ( \delta -x ) ^{\alpha }}{\alpha } \biggr) ^{\beta -1} \frac{\mathscr{F} ( x ) }{ ( \delta -x ) ^{1-\alpha }}\mathscr{F} ( x ) \,dx \\ & =-\frac{2}{\delta -\sigma } \biggl( \int _{0}^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) \\ &\quad {}- \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2}\frac{1}{\alpha ^{\beta }} \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \\ & \quad{}+ \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2+\alpha \beta } \Gamma ( \beta +1 ) {}^{\beta }\mathcal{J}_{\delta -}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) . \end{aligned}$$
(3.2)
Then, similar to the foregoing process, we can easily obtain
$$\begin{aligned} A_{2}& = \int _{0}^{1} \biggl( \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr) \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma + \frac{1-\mu }{2}\delta \biggr) \,d\mu \\ & =\frac{2}{\delta -\sigma } \biggl( \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr) \\ &\quad {}\times\mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) - \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2}\frac{1}{\alpha ^{\beta }} \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \\ & \quad{}+ \biggl( \frac{2}{\delta -\sigma } \biggr) ^{\alpha \beta +2} \Gamma ( \beta +1 ) {}^{\beta }\mathcal{J}_{\sigma +}^{ \alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) . \end{aligned}$$
(3.3)
If (3.2) and (3.3) are added together and then multiplied by \(\frac{(\delta -\sigma )^{2}\alpha ^{\beta }}{8}\), the proof of Lemma 1 is completed. □
Theorem 1
Note that \(\mathscr{F}:[\sigma ,\delta ]\rightarrow \mathbb{R}\) is a twice-differentiable function on \((\sigma ,\delta )\) so that \(\mathscr{F}^{\prime \prime }\in L_{1} [ \sigma ,\delta ] \). Note also that \(\vert \mathscr{F}^{\prime \prime } \vert \) is convex on \([ \sigma ,\delta ] \). Then, the following inequality holds:
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ & \quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \Upsilon _{1} ( \alpha ,\beta ) \bigl( \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert + \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert \bigr) . \end{aligned}$$
Here,
$$\begin{aligned} \Upsilon _{1} ( \alpha ,\beta ) & = \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \,d\mu \\ & =\frac{1}{\alpha ^{\beta }} \int _{0}^{1} \biggl\vert 1-\mu - \frac{1}{\alpha } \biggl( \mathcal{B} \biggl( \beta +1,\frac{1}{\alpha } \biggr) -\mathscr{B} \biggl( \beta +1,\frac{1}{\alpha },1- ( 1-\mu ) ^{\alpha } \biggr) \biggr) \biggr\vert \,d\mu , \end{aligned}$$
(3.4)
where \(\mathcal{B}\) and \(\mathscr{B}\) denote the beta function and incomplete beta function, respectively.
Proof
Let us start by taking the absolute values of both sides of (3.1), we have
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl[ \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \biggl\vert \mathscr{F}^{ \prime \prime } \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2} \delta \biggr) \biggr\vert \,d\mu \\ &\quad \quad{}+ \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma +\frac{1-\mu }{2} \delta \biggr) \biggr\vert \,d\mu \biggr] . \end{aligned}$$
(3.5)
It is known that \(\vert \mathscr{F}^{\prime \prime } \vert \) is convex on \([\sigma ,\delta ]\). Then, we have
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ & \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl[ \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \\ &\quad {}\times\biggl( \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert + \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert \biggr) \,d\mu \\ & \quad{}+ \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \biggl( \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert +\frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert \biggr) \,d\mu \biggr] \\ & =\frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \,d\mu \biggr) \bigl( \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert + \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert \bigr) . \end{aligned}$$
Hence, the proof of Theorem 1 is completed. □
Remark 1
If we set \(\alpha =1\) in Theorem 1, then we have [5, Corollary 4.6].
Remark 2
Consider \(\alpha =1\) and \(\beta =1\) in Theorem 1. Then, Theorem 1 equals [20, Theorem 3].
Theorem 2
Let \(\mathscr{F}:[\sigma ,\delta ]\rightarrow \mathbb{R}\) be a twice-differentiable function on \((\sigma ,\delta )\) such that \(\mathscr{F}^{\prime \prime }\in L_{1} ( [ \sigma ,\delta ] ) \) and \(\vert \mathscr{F}^{\prime \prime } \vert ^{q}\) is convex on \([ \sigma ,\delta ] \) with \(q>1\). Then, the following inequalities hold:
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \bigl( \Upsilon _{\alpha }^{\beta } ( p ) \bigr) ^{ \frac{1}{p}} \biggl[ \biggl( \frac{ \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{1/q} \\ &\quad \quad {}+ \biggl( \frac{3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+ \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{1/q} \biggr] \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \bigl( 4\Upsilon _{\alpha }^{\beta } ( p ) \bigr) ^{ \frac{1}{p}} \bigl[ \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert + \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert \bigr] , \end{aligned}$$
where \(\frac{1}{p}+\frac{1}{q}=1\) and
$$ \Upsilon _{\alpha }^{\beta } ( p ) = \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert ^{p}\,d\mu . $$
Proof
Let us start by using the Hölder inequality in (3.5). Then, we have
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl[ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert ^{p}\,d\mu \biggr) ^{ \frac{1}{p}} \\ &\quad \quad {}\times\biggl( \int _{0}^{1} \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma + \frac{1+\mu }{2}\delta \biggr) \biggr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \\ &\quad \quad{}+ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert ^{p}\,d\mu \biggr) ^{\frac{1}{p}} \\ &\quad \quad {}\times \biggl( \int _{0}^{1} \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma + \frac{1-\mu }{2}\delta \biggr) \biggr\vert ^{q}\,d\mu \biggr) ^{ \frac{1}{q}} \biggr] . \end{aligned}$$
Since \(\vert \mathscr{F}^{\prime \prime } \vert ^{q}\) is convex on \([\sigma ,\delta ]\), we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert ^{p}\,d\mu \biggr) ^{ \frac{1}{p}} \\ &\quad \quad{}\times \biggl[ \biggl( \int _{0}^{1} \biggl( \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+\frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) \,d\mu \biggr) ^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl( \int _{0}^{1} \biggl( \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+\frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) \,d\mu \biggr) ^{\frac{1}{q}} \biggr] \\ &\quad =\frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert ^{p}\,d\mu \biggr) ^{\frac{1}{p}} \\ &\quad \quad{} \times \biggl[ \biggl( \frac{ \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}}+ \biggl( \frac{3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+ \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
Let us consider \(\varpi _{1}= \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}\), \(\varrho _{1}=3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}\), \(\varpi _{2}=3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}\), and \(\varrho _{2}= \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}\) and applying the inequality:
$$ \sum_{k=1}^{n} ( \varpi _{k}+\varrho _{k} ) ^{s} \leq \sum _{k=1}^{n}\varpi _{k}^{s}+ \sum_{k=1}^{n} \varrho _{k}^{s},\quad 0\leq s< 1. $$
(3.6)
This finishes the proof of Theorem 2. □
Corollary 1
If we assign \(\alpha =1\) in Theorem 2, then we derive
$$\begin{aligned} & \biggl\vert \frac{2^{\beta -1}\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\beta }} \biggl[ J_{\delta -}^{\beta } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) + J_{\sigma +}^{\beta } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}}{8} \bigl( \psi ^{ \beta } ( p ) \bigr) ^{\frac{1}{p}} \biggl[ \biggl( \frac{ \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{1/q}+ \biggl( \frac{3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+ \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{1/q} \biggr] \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}}{8} \bigl( 4\psi ^{ \beta } ( p ) \bigr) ^{\frac{1}{p}} \bigl[ \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert + \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert \bigr] , \end{aligned}$$
where \(\frac{1}{p}+\frac{1}{q}=1\) and
$$ \psi ^{\beta } ( p ) = \int _{0}^{1} \biggl\vert 1- \mu + \frac{\mu ^{\beta +1}-1}{\beta +1} \biggr\vert ^{p}\,d\mu . $$
Remark 3
If we choose \(\alpha =1\) and \(\beta =1\) in Theorem 2, then Theorem 2 reduces to [5, Corollary 4.8].
Theorem 3
Suppose that \(\mathscr{F}:[\sigma ,\delta ]\rightarrow \mathbb{R}\) is a twice-differentiable function on \((\sigma ,\delta )\) such that \(\mathscr{F}^{\prime \prime }\in L_{1} ( [ \sigma ,\delta ] ) \). If \(\vert \mathscr{F}^{\prime \prime } \vert ^{q}\) is convex on \([ \sigma ,\delta ] \) with \(q\geq 1\), then the following inequality holds:
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \bigl( \Upsilon _{1} ( \alpha ,\beta ) \bigr) ^{1- \frac{1}{q}} \biggl[ \biggl( \frac{\Upsilon _{1} ( \alpha ,\beta ) -\Upsilon _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q} \\ &\quad \quad {}+ \frac{\Upsilon _{1} ( \alpha ,\beta ) +\Upsilon _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\ &\quad \quad{}+ \biggl( \frac{\Upsilon _{1} ( \alpha ,\beta ) +\Upsilon _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{\Upsilon _{1} ( \alpha ,\beta ) -\Upsilon _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
Here, \(\Upsilon _{1} ( \alpha ,\beta ) \) is defined as in (3.4) and
$$\begin{aligned} \Upsilon _{2} ( \alpha ,\beta ) & = \int _{0}^{1} \mu \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \,d\mu \\ & =\frac{1}{\alpha ^{\beta }} \int _{0}^{1}\mu \biggl\vert 1- \mu - \frac{1}{\alpha } \biggl( \mathcal{B} \biggl( \beta +1,\frac{1}{\alpha } \biggr) -\mathscr{B} \biggl( \beta +1,\frac{1}{\alpha },1- ( 1-\mu ) ^{\alpha } \biggr) \biggr) \biggr\vert \,d\mu , \end{aligned}$$
where \(\mathcal{B}\) and \(\mathscr{B}\) denote the beta function and incomplete beta function, respectively.
Proof
Let us start by applying the power-mean inequality in (3.5). Then, we have
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl[ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \,d\mu \biggr) ^{1- \frac{1}{q}} \\ &\quad \quad{}\times \biggl( \int _{0}^{1} \biggl\vert \int _{ \mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2} \delta \biggr) \biggr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \\ &\quad \quad{}+ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \,d\mu \biggr) ^{1- \frac{1}{q}} \\ &\quad \quad{}\times \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma + \frac{1-\mu }{2}\delta \biggr) \biggr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
Since \(\vert \mathscr{F}^{\prime \prime } \vert ^{q}\) is convex on \([\sigma ,\delta ]\), we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \,d\mu \biggr) ^{1-\frac{1}{q}} \\ &\quad \quad{}\times \biggl[ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+\frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q}\,d\mu \biggr) ^{ \frac{1}{q}} \\ &\quad \quad{}+ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1}{\alpha ^{\beta }}- \biggl( \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr) ^{\beta } \biggr] \,ds \biggr\vert \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q} + \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
It is clearly seen that
$$\begin{aligned} & \biggl\vert \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \bigl( \Upsilon _{1} ( \alpha ,\beta ) \bigr) ^{1- \frac{1}{q}} \\ &\quad \quad{}\times \biggl[ \biggl( \frac{\Upsilon _{1} ( \alpha ,\beta ) -\Upsilon _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{\Upsilon _{1} ( \alpha ,\beta ) +\Upsilon _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{\frac{1}{q}} \\ &\quad \quad{}+ \biggl( \frac{\Upsilon _{1} ( \alpha ,\beta ) +\Upsilon _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{\Upsilon _{1} ( \alpha ,\beta ) -\Upsilon _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
□
Corollary 2
Let us consider \(\alpha =1\) in Theorem 3. Then, the following inequality holds:
$$\begin{aligned} & \biggl\vert \frac{2^{\beta -1}\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\beta }} \biggl[ J_{\delta -}^{\beta } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +J_{\sigma +}^{\beta } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] -\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}}{8} \biggl( \frac{1}{2}- \frac{1}{\beta +2} \biggr) ^{1-\frac{1}{q}} \\ &\quad \quad{} \times \biggl[ \biggl( \biggl( \frac{1}{6}- \frac{\beta +4}{4 ( \beta +2 ) ( \beta +3 ) } \biggr) \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \biggl( \frac{1}{3}- \frac{3\beta +8}{4 ( \beta +2 ) ( \beta +3 ) } \biggr) \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \\ &\quad \quad{}+ \biggl( \biggl( \frac{1}{3}- \frac{3\beta +8}{4 ( \beta +2 ) ( \beta +3 ) } \biggr) \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \biggl( \frac{1}{6}- \frac{\beta +4}{4 ( \beta +2 ) ( \beta +3 ) } \biggr) \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
Remark 4
If we take \(\alpha =1\) and \(\beta =1\), then Theorem 3 becomes [19, Proposition 5].
3.2 Inequalities of trapezoid type involving conformable fractional integrals
In this subsection, inequalities of trapezoid-type are obtained for twice-differentiable functions. We use the conformable fractional integral operators to obtain these inequalities.
Lemma 2
If \(\mathscr{F}:[\sigma ,\delta ]\rightarrow \mathbb{R}\) is a twice-differentiable mapping on \((\sigma ,\delta )\) such that \(\mathscr{F}^{\prime \prime }\in L_{1} [ \sigma ,\delta ] \), then the following equality holds:
$$\begin{aligned} & \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta } \mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \\ &\quad =\frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl\{ \int _{0}^{1} \biggl( \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr) \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2} \delta \biggr) \,d\mu \\ &\quad \quad{}+ \int _{0}^{1} \biggl( \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{ \beta }\,ds \biggr) \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma +\frac{1-\mu }{2}\delta \biggr) \,d\mu \biggr\} . \end{aligned}$$
(3.7)
Proof
By employing integration by parts, we have
$$\begin{aligned} A_{3}& = \int _{0}^{1} \biggl( \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr) \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma + \frac{1+\mu }{2}\delta \biggr) \,d\mu \\ & = \frac{2}{\delta -\sigma } \biggl( \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{ \beta }\,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{1-\mu }{2} \sigma +\frac{1+\mu }{2}\delta \biggr) \bigg\vert _{0}^{1} \\ & \quad{}+\frac{2}{\delta -\sigma } \int _{0}^{1} \biggl[ \frac{1- ( 1-\mu ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\mathscr{F}^{ \prime } \biggl( \frac{1-\mu }{2} \sigma +\frac{1+\mu }{2}\delta \biggr) \,d\mu \\ & =-\frac{2}{\delta -\sigma } \biggl( \int _{0}^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) \\ & \quad{}+\frac{2}{\delta -\sigma } \biggl\{ \frac{2}{\delta -\sigma } \biggl[ \frac{1- ( 1-\mu ) ^{\alpha }}{\alpha } \biggr] ^{ \beta }\mathscr{F} \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2}\delta \biggr) \bigg\vert _{0}^{1} \\ & \quad{}-\frac{2\beta }{\delta -\sigma } \int _{0}^{1} \biggl[ \frac{1- ( 1-\mu ) ^{\alpha }}{\alpha } \biggr] ^{ \beta -1} ( 1-\mu ) ^{\alpha -1}\mathscr{F} \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2}\delta \biggr) \,d\mu \biggr\} \\ & =-\frac{2}{\delta -\sigma } \biggl( \int _{0}^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) + \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2}\frac{1}{\alpha ^{\beta }}\mathscr{F} ( \delta ) \\ & \quad{}- \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2} \frac{\Gamma ( \beta +1 ) }{\Gamma ( \beta ) } \int _{ \frac{\sigma +\delta }{2}}^{\delta } \biggl( \frac{1- ( \frac{2}{\delta -\sigma } ( \delta -x ) ) ^{\alpha }}{\alpha } \biggr) ^{\beta -1} \\ &\quad {}\times\biggl( \frac{2}{\delta -\sigma } ( \delta -x ) \biggr) ^{\alpha -1} \frac{2}{\delta -\sigma }\mathscr{F} ( x ) \,dx \\ & =-\frac{2}{\delta -\sigma } \biggl( \int _{0}^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) + \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2}\frac{1}{\alpha ^{\beta }}\mathscr{F} ( \delta ) \\ & \quad{}- \biggl( \frac{2}{\delta -\sigma } \biggr) ^{\alpha \beta +2} \frac{\Gamma ( \beta +1 ) }{\Gamma ( \beta ) } \int _{\frac{\sigma +\delta }{2}}^{\delta } \biggl( \frac{ ( \frac{\delta -\sigma }{2} ) ^{\alpha }- ( \delta -x ) ^{\alpha }}{\alpha } \biggr) ^{\beta -1} \frac{\mathscr{F} ( x ) }{ ( \delta -x ) ^{1-\alpha }}\mathscr{F} ( x ) \,dx \\ & =-\frac{2}{\delta -\sigma } \biggl( \int _{0}^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) + \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2}\frac{1}{\alpha ^{\beta }}\mathscr{F} ( \delta ) \\ & \quad{}- \biggl( \frac{2}{\delta -\sigma } \biggr) ^{\alpha \beta +2} \Gamma ( \beta +1 ) {}^{\beta }\mathcal{J}_{\delta -}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) . \end{aligned}$$
In a similar manner,
$$\begin{aligned} A_{4}& = \int _{0}^{1} \biggl( \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr) \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma + \frac{1-\mu }{2}\delta \biggr) \,d\mu \\ & =\frac{2}{\delta -\sigma } \biggl( \int _{0}^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr) \mathscr{F}^{\prime } \biggl( \frac{\sigma +\delta }{2} \biggr) + \biggl( \frac{2}{\delta -\sigma } \biggr) ^{2}\frac{1}{\alpha ^{\beta }} \mathscr{F} ( \delta ) \\ & \quad{}- \biggl( \frac{2}{\delta -\sigma } \biggr) ^{\alpha \beta +2} \Gamma ( \beta +1 ) {} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) . \end{aligned}$$
Then, it follows that
$$\begin{aligned} & \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \{ A_{3}+A_{4} \} \\ &\quad = \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ ^{\beta } \mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{\alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] . \end{aligned}$$
Thus, the proof of Lemma 2 is accomplished. □
Theorem 4
Let \(\mathscr{F}:[\sigma ,\delta ]\rightarrow \mathbb{R}\) be a twice-differentiable mapping on \((\sigma ,\delta )\) such that \(\mathscr{F}^{\prime \prime }\in L_{1} ( [ \sigma ,\delta ] ) \). If \(\vert \mathscr{F}^{\prime \prime } \vert \) is convex on \([ \sigma ,\delta ] \), then the following expression holds:
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \Phi _{1}(\alpha ,\beta ) \bigl( \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert + \bigl\vert \mathscr{F}^{ \prime \prime } ( \delta ) \bigr\vert \bigr) . \end{aligned}$$
(3.8)
Here,
$$\begin{aligned} \Phi _{1}(\alpha ,\beta )& = \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \,d\mu \\ & =\frac{1}{\alpha ^{\beta +1}} \int _{0}^{1} \biggl\vert \mathcal{B} \biggl( \beta +1,\frac{1}{\alpha } \biggr) -\mathscr{B} \biggl( \beta +1, \frac{1}{\alpha },1- ( 1-\mu ) ^{\alpha } \biggr) \biggr\vert \,d\mu , \end{aligned}$$
where \(\mathcal{B}\) and \(\mathscr{B}\) denote the beta function and incomplete beta function, respectively.
Proof
Taking the absolute values of both sides of (3.7), we derive
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl\{ \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma + \frac{1+\mu }{2}\delta \biggr) \biggr\vert \,d\mu \\ &\quad \quad{}+ \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{ \beta }\,ds \biggr\vert \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma +\frac{1-\mu }{2}\delta \biggr) \biggr\vert \,d\mu \biggr\} . \end{aligned}$$
(3.9)
If we use the convexity of the \(\vert \mathscr{F}^{\prime \prime } \vert \) on \([\sigma ,\delta ]\), then we establish
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl\{ \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \biggl[ \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert + \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{ \prime \prime } ( \delta ) \bigr\vert \biggr] \,d\mu \\ &\quad \quad{}+ \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{ \beta }\,ds \biggr\vert \biggl[ \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{ \prime \prime } ( \sigma ) \bigr\vert + \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert \biggr] \,d\mu \biggr\} \\ & \quad =\frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \,d\mu \biggr) \bigl( \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert + \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert \bigr) . \end{aligned}$$
□
Remark 5
If \(\alpha =1\) is chosen in (3.8), then Theorem 4 reduces to [5, Corolalry 3.6].
Remark 6
Let us consider \(\alpha =1\) and \(\beta =1\) in (3.8). Then, Theorem 4 is equal to [19, Proposition 2].
Theorem 5
Let \(\mathscr{F}:[\sigma ,\delta ]\rightarrow \mathbb{R}\) be a twice-differentiable function on \((\sigma ,\delta )\) such that \(\mathscr{F}^{\prime \prime }\in L_{1} [ \sigma ,\delta ] \) with \(\sigma <\delta \). If \(\vert \mathscr{F}^{\prime \prime } \vert ^{q}\) is convex on \([ \sigma ,\delta ] \) with \(q>1\), then the following double inequality holds:
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \bigl( \Theta _{\alpha }^{\beta } ( p ) \bigr) ^{ \frac{1}{p}} \biggl[ \biggl( \frac{ \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}}+ \biggl( \frac{3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+ \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}} \biggr] \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \bigl( 4\Theta _{\alpha }^{\beta } ( p ) \bigr) ^{ \frac{1}{p}} \bigl[ \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \bigr] , \end{aligned}$$
(3.10)
where \(\frac{1}{p}+\frac{1}{q}=1\) and
$$ \Theta _{\alpha }^{\beta } ( p ) = \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert ^{p}\,d\mu . $$
Proof
Using the Hölder inequality in (3.9), we have
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl\{ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert ^{p}\,d\mu \biggr) ^{\frac{1}{p}} \\ &\quad \quad {}\times \biggl( \int _{0}^{1} \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma + \frac{1+\mu }{2}\delta \biggr) \biggr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \\ &\quad \quad{}+ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert ^{p}\,d\mu \biggr) ^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma + \frac{1-\mu }{2}\delta \biggr) \biggr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Since \(\vert \mathscr{F}^{\prime \prime } \vert ^{q}\) is convex on \([\sigma ,\delta ]\), we obtain
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert ^{p}\,d\mu \biggr) ^{\frac{1}{p}} \\ &\quad \quad{}\times \biggl[ \biggl( \int _{0}^{1} \biggl( \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+\frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) \,d\mu \biggr) ^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl( \int _{0}^{1} \biggl( \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+\frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) \,d\mu \biggr) ^{\frac{1}{q}} \biggr] \\ &\quad =\frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert ^{p}\,d\mu \biggr) ^{\frac{1}{p}} \\ &\quad \quad{}\times \biggl[ \biggl( \frac{ \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}}+ \biggl( \frac{3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+ \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
Let us consider \(\varpi _{1}= \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}\), \(\varrho _{1}=3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}\), \(\varpi _{2}=3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}\) and \(\varrho _{2}= \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}\) and with the help of the inequality (3.6). Finally, the proof of Theorem 5 is completed. □
Remark 7
If we choose \(\alpha =1\) in Theorem 5, then we derive
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\beta -1}\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\beta }} \biggl[ J_{\delta -}^{\beta } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{}J_{\sigma +}^{\beta } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}}{8 ( \beta +1 ) } \biggl( \frac{p ( \beta +1 ) }{p ( \beta +1 ) +1} \biggr) ^{\frac{1}{p}} \biggl[ \biggl( \frac{ \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl( \frac{3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q} + \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}} \biggr] \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}}{8 ( \beta +1 ) } \biggl( \frac{4p ( \beta +1 ) }{p ( \beta +1 ) +1} \biggr) ^{\frac{1}{p}} \bigl[ \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \bigr] , \end{aligned}$$
which is given in [5, Corollary 3.9].
Proof
It will be sufficient to write down the solution of the integral below,
$$ \Theta _{1}^{\beta } ( p ) = \int _{0}^{1} \biggl\vert \frac{1}{\beta +1}-\frac{\mu ^{\beta +1}}{\beta +1} \biggr\vert ^{p}\,d\mu . $$
Under conditions \(A>B>0\) and \(p>1\), the following inequality is satisfied
$$ \vert A-B \vert ^{p}\leq A^{p}-B^{p}. $$
(3.11)
From the inequality (3.11), we have
$$ \begin{aligned} \Theta _{1}^{\beta } ( p ) &= \frac{1}{ ( \beta +1 ) ^{p}}\int _{0}^{1} \bigl\vert 1-\mu ^{\beta +1} \bigr\vert ^{p}\,d\mu \leq \frac{1}{ ( \beta +1 ) ^{p}} \biggl( \int _{0}^{1} \bigl( 1-\mu ^{p ( \beta +1 ) } \bigr) \,d\mu \biggr) \\ &= \frac{1}{ ( \beta +1 ) ^{p}} \biggl( \frac{p ( \beta +1 ) }{p ( \beta +1 ) +1} \biggr) . \end{aligned} $$
When the solution of \(\Theta _{\alpha }^{\beta } ( p ) \) is substituted for (3.10), the proof is finished. □
Corollary 3
If we take \(\alpha =1\) and \(\beta =1\) in Theorem 5, then we obtain
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}-\frac{1}{\delta -\sigma } \int _{\sigma }^{ \delta }\mathscr{F} ( x ) \,dx \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}}{16} \biggl( \frac{2p}{2p+1} \biggr) ^{\frac{1}{p}} \biggl[ \biggl( \frac{ \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+3 \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}}+ \biggl( \frac{3 \vert \mathscr{F}^{\prime \prime } ( \sigma ) \vert ^{q}+ \vert \mathscr{F}^{\prime \prime } ( \delta ) \vert ^{q}}{4} \biggr) ^{\frac{1}{q}} \biggr] \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}}{16} \biggl( \frac{8p}{2p+1} \biggr) ^{\frac{1}{p}} \bigl[ \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \bigr] . \end{aligned}$$
Theorem 6
Assume that \(\mathscr{F}:[\sigma ,\delta ]\rightarrow \mathbb{R} \) is a twice-differentiable mapping on \((\sigma ,\delta )\) such that \(\mathscr{F}^{\prime \prime }\in L_{1} [ \sigma ,\delta ] \) and \(\vert \mathscr{F}^{\prime \prime } \vert ^{q}\) is convex on \([ \sigma ,\delta ] \) with \(q\geq 1\). Then, the following inequality holds:
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \bigl( \Phi _{1} ( \alpha ,\beta ) \bigr) ^{1- \frac{1}{q}} \biggl[ \biggl( \frac{\Phi _{1} ( \alpha ,\beta ) -\Phi _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q} \\ &\quad\quad {}+ \frac{\Phi _{1} ( \alpha ,\beta ) +\Phi _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{\frac{1}{q}} \\ &\quad \quad{}+ \biggl( \frac{\Phi _{1} ( \alpha ,\beta ) +\Phi _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{\Phi _{1} ( \alpha ,\beta ) -\Phi _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{ \frac{1}{q}} \biggr] . \end{aligned}$$
Here,
$$\begin{aligned} \Phi _{2}(\alpha ,\beta )& = \int _{0}^{1}\mu \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \,d\mu \\ & =\frac{1}{\alpha ^{\beta +1}} \int _{0}^{1}\mu \biggl\vert \mathcal{B} \biggl( \beta +1,\frac{1}{\alpha } \biggr) -\mathscr{B} \biggl( \beta +1,\frac{1}{\alpha },1- ( 1-\mu ) ^{\alpha } \biggr) \biggr\vert \,d\mu , \end{aligned}$$
where \(\mathcal{B}\) and \(\mathscr{B}\) denote the beta function and incomplete beta function, respectively.
Proof
With the help of the power-mean inequality in (3.9), we have
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl[ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \,d\mu \biggr) ^{1-\frac{1}{q}} \\ &\quad \quad{}\times \biggl( \int _{0}^{1} \biggl\vert \int _{ \mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1-\mu }{2}\sigma +\frac{1+\mu }{2}\delta \biggr) \biggr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \\ &\quad \quad{}+ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{ \beta }\,ds \biggr\vert \,d\mu \biggr) ^{1-\frac{1}{q}} \\ &\quad \quad{}\times \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \biggl\vert \mathscr{F}^{\prime \prime } \biggl( \frac{1+\mu }{2}\sigma +\frac{1-\mu }{2}\delta \biggr) \biggr\vert ^{q}\,d\mu \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
Since \(\vert \mathscr{F}^{\prime \prime } \vert ^{q}\) is convex on \([\sigma ,\delta ]\), we obtain
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\alpha \beta -1}\alpha ^{\beta }\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\alpha \beta }} \biggl[ {}^{\beta }\mathcal{J}_{\delta -}^{\alpha }\mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} ^{\beta }\mathcal{J}_{\sigma +}^{ \alpha } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \biggl[ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \,d\mu \biggr) ^{1-\frac{1}{q}} \\ & \quad \quad{}\times \biggl[ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \biggl( \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) \,d\mu \biggr) ^{\frac{1}{q}} \\ & \quad \quad{}+ \biggl( \int _{0}^{1} \biggl\vert \int _{\mu }^{1} \biggl[ \frac{1- ( 1-s ) ^{\alpha }}{\alpha } \biggr] ^{\beta }\,ds \biggr\vert \biggl( \frac{1+\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{1-\mu }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) \,d\mu \biggr) ^{\frac{1}{q}} \biggr] \\ & \quad =\frac{ ( \delta -\sigma ) ^{2}\alpha ^{\beta }}{8} \bigl( \Phi _{1} ( \alpha ,\beta ) \bigr) ^{1- \frac{1}{q}} \biggl[ \biggl( \frac{\Phi _{1} ( \alpha ,\beta ) -\Phi _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q} \\ &\quad \quad {}+ \frac{\Phi _{1} ( \alpha ,\beta ) +\Phi _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{\frac{1}{q}} \\ & \quad \quad{}+ \biggl( \frac{\Phi _{1} ( \alpha ,\beta ) +\Phi _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{\Phi _{1} ( \alpha ,\beta ) -\Phi _{2} ( \alpha ,\beta ) }{2} \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{ \frac{1}{q}} \biggr] . \end{aligned}$$
Finally, we obtain the required result. □
Corollary 4
Let us consider \(\alpha =1\) in Theorem 6. Then, we derive
$$\begin{aligned} & \biggl\vert \frac{\mathscr{F} ( \sigma ) +\mathscr{F} ( \delta ) }{2}- \frac{2^{\beta -1}\Gamma ( \beta +1 ) }{ ( \delta -\sigma ) ^{\beta }} \biggl[ J_{\delta -}^{\beta } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) +{} J_{\sigma +}^{\beta } \mathscr{F} \biggl( \frac{\sigma +\delta }{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{ ( \delta -\sigma ) ^{2}}{8} \biggl( \frac{1}{\beta +2} \biggr) ^{1-\frac{1}{q}} \biggl[ \biggl( \frac{\beta +4}{4 ( \beta +2 ) ( \beta +3 ) } \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{3\beta +8}{2 ( \beta +2 ) ( \beta +3 ) } \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{\frac{1}{q}} \\ &\quad \quad{}+ \biggl( \frac{3\beta +8}{2 ( \beta +2 ) ( \beta +3 ) } \bigl\vert \mathscr{F}^{\prime \prime } ( \sigma ) \bigr\vert ^{q}+ \frac{\beta +4}{4 ( \beta +2 ) ( \beta +3 ) } \bigl\vert \mathscr{F}^{\prime \prime } ( \delta ) \bigr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$
Remark 8
If we take \(\alpha =1\) and \(\beta =1\) in Theorem 6, then Theorem 6 becomes [19, Proposition 6].