In this section, we establish some new Simpson’s type inequalities for differentiable convex functions via generalized fractional integrals.
Theorem 3
We assume that the conditions of Lemma 1hold. If the mapping \(\vert \mathcal{F}^{\prime } \vert \) is convex on \([ \kappa _{1},\kappa _{2} ] \), then we have the following inequality for generalized fractional integrals:
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \bigl[ \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert \bigl\{ \Pi _{1}^{ \varphi } ( \lambda ) +\Pi _{3}^{\varphi } ( \mu ) \bigr\} + \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert \bigl\{ \Pi _{2}^{\varphi } ( \lambda ) +\Pi _{4}^{\varphi } ( \mu ) \bigr\} \bigr], \end{aligned}$$
(4.1)
where
$$\begin{aligned}& \Pi _{1}^{\varphi } ( \lambda ) = \int _{0}^{\frac{1}{2}} \tau \bigl\vert \Delta ( \tau ) -\Delta ( 1 ) \lambda \bigr\vert \,d\tau ,\qquad \Pi _{2}^{\varphi } ( \lambda ) = \int _{0}^{\frac{1}{2}} ( 1-\tau ) \bigl\vert \Delta ( \tau ) -\Delta ( 1 ) \lambda \bigr\vert \,d\tau , \\& \Pi _{3}^{\varphi } ( \mu ) = \int _{\frac{1}{2}}^{1} \tau \bigl\vert \Delta ( \tau ) -\Delta ( 1 ) \mu \bigr\vert \,d\tau ,\qquad \Pi _{4}^{\varphi } ( \mu ) = \int _{\frac{1}{2}}^{1} ( 1-\tau ) \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \,d\tau . \end{aligned}$$
Proof
By taking the modulus in Lemma 1 and using the properties of the modulus, we obtain that
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert \bigl\vert \mathcal{F}^{\prime } \bigl( \tau \kappa _{2}+ ( 1-\tau ) \kappa _{1} \bigr) \bigr\vert \,d\tau \\& \qquad {} + \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \bigl\vert \mathcal{F}^{\prime } \bigl( \tau \kappa _{2}+ ( 1-\tau ) \kappa _{1} \bigr) \bigr\vert \,d\tau \biggr] . \end{aligned}$$
(4.2)
Since the mapping \(\vert \mathcal{F}^{\prime } \vert \) is convex on \([ \kappa _{1},\kappa _{2} ] \), we have
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert \biggl( \int _{0}^{ \frac{1}{2}}\tau \bigl\vert \Delta ( \tau ) -\Delta ( 1 ) \lambda \bigr\vert \,d\tau + \int _{\frac{1}{2}}^{1}\tau \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \,d\tau \biggr) \\& \qquad {} + \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert \biggl( \int _{0}^{\frac{1}{2}} ( 1-\tau ) \bigl\vert \Delta ( \tau ) -\Delta ( 1 ) \lambda \bigr\vert \,d\tau + \int _{\frac{1}{2}}^{1} ( 1- \tau ) \bigl\vert \Delta ( \tau ) -\Delta ( 1 ) \mu \bigr\vert \,d\tau \biggr) \biggr] \\& \quad {}= ( \kappa _{2}-\kappa _{1} ) \bigl[ \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert \bigl\{ \Pi _{1}^{ \varphi } ( \lambda ) +\Pi _{3}^{\varphi } ( \mu ) \bigr\} + \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert \bigl\{ \Pi _{2}^{\varphi } ( \lambda ) +\Pi _{4}^{\varphi } ( \mu ) \bigr\} \bigr], \end{aligned}$$
which ends the proof. □
Remark 2
In Theorem 3, if we take \(\varphi ( \tau ) =\tau \), then Theorem 3 reduces to [10, Theorem 2.1 for \(s=m=1\)].
Corollary 3
In Theorem 3, if we use \(\varphi ( \tau ) =\frac{\tau ^{\alpha }}{\Gamma ( \kappa _{1} ) }\), then we obtain the following parameterized Simpson’s type inequality for Riemann–Liouville fractional integrals:
$$\begin{aligned}& \biggl\vert \lambda \mathcal{F} ( \kappa _{1} ) + ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \frac{\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \bigl[ J_{\frac{\kappa _{1}+\kappa _{2}}{2}+}^{\alpha } \mathcal{F} ( \kappa _{2} ) +J_{\frac{\kappa _{1}+\kappa _{2}}{2}-}^{\alpha } \mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \bigl[ \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert \bigl\{ \Pi _{1}^{ \alpha } ( \lambda ) +\Pi _{3}^{\alpha } ( \mu ) \bigr\} + \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert \bigl\{ \Pi _{2}^{\alpha } ( \lambda ) +\Pi _{4}^{\alpha } ( \mu ) \bigr\} \bigr], \end{aligned}$$
(4.3)
where
$$\begin{aligned}& \Pi _{1}^{\alpha } ( \lambda ) = \frac{\alpha }{\alpha +2} \lambda ^{\frac{\alpha +2}{\alpha }}- \frac{\lambda }{8}+\frac{1}{2^{\alpha +2} ( \alpha +2 ) }, \\& \Pi _{2}^{\alpha } ( \lambda ) = \frac{2\alpha }{\alpha +1}\lambda ^{\frac{\alpha +1}{\alpha }}-\frac{\lambda }{2}+ \frac{1}{2^{\alpha +1} ( \alpha +1 ) }-\Pi _{1}^{\alpha } ( \lambda ) , \\& \Pi _{3}^{\alpha } ( \mu ) =\frac{\alpha }{\alpha +2} \mu ^{\frac{\alpha +2}{\alpha }}-\frac{5}{8}\mu + \frac{2^{\alpha +2}+1}{2^{\alpha +2} ( \alpha +2 ) }, \end{aligned}$$
and
$$ \Pi _{4}^{\alpha } ( \mu ) =\frac{2\alpha }{\alpha +1} \mu ^{\frac{\alpha +1}{\alpha }}-\frac{3}{2}\mu + \frac{2^{\alpha +1}+1}{2^{\alpha +1} ( \alpha +1 ) }-\Pi _{3}^{ \alpha } ( \mu ) . $$
Corollary 4
In Theorem 3, if we use \(\varphi ( \tau ) =\frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \kappa _{1} ) }\), then we obtain the following parameterized Simpson’s type inequality for k-Riemann–Liouville fractional integrals:
$$\begin{aligned}& \biggl\vert \lambda \mathcal{F} ( \kappa _{1} ) + ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \frac{\Gamma _{k} ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \bigl[ J_{\frac{\kappa _{1}+\kappa _{2}}{2}+,k}^{\alpha } \mathcal{F} ( \kappa _{2} ) +J_{\frac{\kappa _{1}+\kappa _{2}}{2}-,k}^{\alpha } \mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \bigl[ \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert \bigl\{ \Pi _{1}^{ \frac{\alpha }{k}} ( \lambda ) +\Pi _{3}^{\frac{\alpha }{k}} ( \mu ) \bigr\} + \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert \bigl\{ \Pi _{2}^{ \frac{\alpha }{k}} ( \lambda ) +\Pi _{4}^{ \frac{\alpha }{k}} ( \mu ) \bigr\} \bigr], \end{aligned}$$
(4.4)
where
$$\begin{aligned} \Pi _{1}^{\frac{\alpha }{k}} ( \lambda ) =& \frac{\alpha }{\alpha +2k} \lambda ^{\frac{\alpha +2k}{\alpha }}- \frac{\lambda }{8}+ \frac{1}{2^{\frac{\alpha +2k}{k}} ( \frac{\alpha +2k}{k} ) }, \\ \Pi _{2}^{\frac{\alpha }{k}} ( \lambda ) =& \frac{2\alpha }{\alpha +k} \lambda ^{\frac{\alpha +k}{\alpha }}- \frac{\lambda }{2}+ \frac{1}{2^{\frac{\alpha +k}{k}} ( \frac{\alpha +k}{k} ) }-\Pi _{1}^{ \frac{\alpha }{k}}, \\ \Pi _{3}^{\frac{\alpha }{k}} ( \mu ) =& \frac{\alpha }{\alpha +2k}\mu ^{\frac{\alpha +2k}{\alpha }}-\frac{5}{8}\mu + \frac{2^{\frac{\alpha +2k}{k}}+1}{2^{\frac{\alpha +2k}{k}} ( \frac{\alpha +2k}{k} )}, \end{aligned}$$
and
$$ \Pi _{4}^{\frac{\alpha }{k}} ( \mu ) = \frac{2\alpha }{\alpha +k}\mu ^{\frac{\alpha +1}{\alpha }}-\frac{3}{2}\mu + \frac{2^{\frac{\alpha +k}{k}}+1}{2^{\frac{\alpha +k}{k}} ( \frac{\alpha +k}{k} ) }- \Pi _{3}^{\frac{\alpha }{k}}. $$
Theorem 4
We assume that the conditions of Lemma 1hold. If the mapping \(\vert \mathcal{F} \vert ^{p_{1}}\), \(p_{1}\geq 1\), is convex on \([ \kappa _{1},\kappa _{2} ] \), then we have the following inequality of Simpson’s type for generalized fractional integrals:
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert \,d\tau \biggr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1}^{\varphi } ( \lambda ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+ \Pi _{2}^{\varphi } ( \lambda ) \bigl\vert \mathcal{F}^{ \prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \,d\tau \biggr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{3}^{\varphi } ( \mu ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{4}^{ \varphi } ( \mu ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$
(4.5)
where \(\Pi _{1}^{\varphi } ( \lambda ) \), \(\Pi _{2}^{\varphi } ( \lambda ) \), \(\Pi _{3}^{\varphi } ( \mu ) \), and \(\Pi _{4}^{\varphi } ( \mu ) \) are defined in Theorem 3.
Proof
Reusing inequality (4.2), by the power mean inequality we have
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert \,d\tau \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{0}^{\frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert \bigl\vert \mathcal{F}^{\prime } \bigl( \tau \kappa _{2}+ ( 1-\tau ) \kappa _{1} \bigr) \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \,d\tau \biggr) ^{1-\frac{1}{p_{1}}}\\& \qquad {}\times \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \bigl\vert \mathcal{F}^{\prime } \bigl( \tau \kappa _{2}+ ( 1-\tau ) \kappa _{1} \bigr) \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{ \frac{1}{p_{1}}} \biggr] . \end{aligned}$$
Using the convexity of \(\vert \mathcal{F}^{\prime } \vert ^{p_{1}}\), we have
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert \,d\tau \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \int _{0}^{\frac{1}{2}}\tau \bigl\vert \Delta ( \tau ) -\Delta ( 1 ) \lambda \bigr\vert \,d\tau \\& \qquad {}+ \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \int _{0}^{\frac{1}{2}} ( 1-\tau ) \bigl\vert \Delta ( \tau ) -\Delta ( 1 ) \lambda \bigr\vert \,d\tau \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \,d\tau \biggr) ^{1- \frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \int _{\frac{1}{2}}^{1}\tau \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \,d\tau \\& \qquad {}+ \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \int _{\frac{1}{2}}^{1} ( 1-\tau ) \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggr] \\& \quad = ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert \,d\tau \biggr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1}^{ \varphi } ( \lambda ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{\varphi } ( \lambda ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert \,d\tau \biggr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{3}^{\varphi } ( \mu ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{4}^{ \varphi } ( \mu ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$
which finishes the proof. □
Remark 3
In Theorem 4, if we assume that \(\varphi ( \tau ) =\tau \), then Theorem 4 reduces to [10, Theorem 2.3 for \(s=m=1\)].
Corollary 5
If we assume that \(\varphi ( \tau ) = \frac{\tau ^{\alpha }}{\Gamma ( \alpha ) }\) in Theorem 4, then we have the following parameterized Simpson’s type inequality for Riemann–Liouville fractional integrals:
$$\begin{aligned}& \biggl\vert \lambda \mathcal{F} ( \kappa _{1} ) + ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \frac{\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \bigl[ J_{\frac{\kappa _{1}+\kappa _{2}}{2}+}^{\alpha } \mathcal{F} ( \kappa _{2} ) +J_{\frac{\kappa _{1}+\kappa _{2}}{2}-}^{\alpha } \mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \bigl[ \bigl( \Pi _{1}^{ \alpha } ( \lambda ) +\Pi _{2}^{\alpha } ( \lambda ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1}^{\alpha } ( \lambda ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{\alpha } ( \lambda ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \\& \qquad {}+ \bigl( \Pi _{3}^{\alpha } ( \mu ) +\Pi _{4}^{ \alpha } ( \mu ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{3}^{\alpha } ( \mu ) \bigl\vert \mathcal{F}^{ \prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{4}^{ \alpha } ( \mu ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \bigr], \end{aligned}$$
(4.6)
where \(\Pi _{1}^{\alpha } ( \lambda ) \), \(\Pi _{2}^{\alpha } ( \lambda ) \), \(\Pi _{3}^{\alpha } ( \mu ) \), and \(\Pi _{4}^{\alpha } ( \mu ) \) are defined in Corollary 3.
Corollary 6
If we assume that \(\varphi ( \tau ) = \frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\) in Theorem 4, then we have the following parameterized Simpson’s type inequality for k-Riemann–Liouville fractional integrals:
$$\begin{aligned}& \biggl\vert \lambda \mathcal{F} ( \kappa _{1} ) + ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \frac{\Gamma _{k} ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \bigl[ J_{\frac{\kappa _{1}+\kappa _{2}}{2}+,k}^{\alpha } \mathcal{F} ( \kappa _{2} ) +J_{\frac{\kappa _{1}+\kappa _{2}}{2}-,k}^{\alpha } \mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \bigl[ \bigl( \Pi _{1}^{ \frac{\alpha }{k}} ( \lambda ) +\Pi _{2}^{\frac{\alpha }{k}} ( \lambda ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1}^{ \frac{\alpha }{k}} ( \lambda ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{\frac{\alpha }{k}} ( \lambda ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \bigl( \Pi _{3}^{\frac{\alpha }{k}} ( \mu ) + \Pi _{4}^{\frac{\alpha }{k}} ( \mu ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{3}^{\frac{\alpha }{k}} ( \mu ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+ \Pi _{4}^{\frac{\alpha }{k}} ( \mu ) \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \bigr], \end{aligned}$$
(4.7)
where \(\Pi _{1}^{\frac{\alpha }{k}} ( \lambda ) \), \(\Pi _{2}^{ \frac{\alpha }{k}} ( \lambda ) \), \(\Pi _{3}^{\frac{\alpha }{k}} ( \mu ) \), and \(\Pi _{4}^{\frac{\alpha }{k}} ( \mu ) \) are described in Corollary 4.
Theorem 5
We assume that the conditions of Lemma 1hold. If the mapping \(\vert \mathcal{F}^{\prime } \vert ^{r_{1}}\), \(r_{1}>1\) is convex on \([ \kappa _{1},\kappa _{2} ] \), then we have the following inequality of Simpson’s type for generalized fractional integrals:
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{ \frac{1}{p_{1}}} \biggl( \frac{ \vert \mathcal{F}^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+3 \vert \mathcal{F}^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{8} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \mathcal{F}^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \mathcal{F}^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{8} \biggr) ^{ \frac{1}{r_{1}}} \biggr], \end{aligned}$$
(4.8)
where \(\frac{1}{p_{1}}+\frac{1}{r_{1}}=1\).
Proof
Reusing inequality (4.2), by the well-known Hölder inequality we have
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{ \frac{1}{p_{1}}} \biggl( \int _{0}^{\frac{1}{2}} \bigl\vert \mathcal{F}^{\prime } \bigl( \tau \kappa _{2}+ ( 1-\tau ) \kappa _{1} \bigr) \bigr\vert ^{r_{1}}\,d\tau \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \mathcal{F}^{\prime } \bigl( \tau \kappa _{2}+ ( 1-\tau ) \kappa _{1} \bigr) \bigr\vert ^{r_{1}}\,d\tau \biggr) ^{ \frac{1}{r_{1}}} \biggr] . \end{aligned}$$
Since \(\vert \mathcal{F}^{\prime } \vert ^{r_{1}}\) is convex, we have
$$\begin{aligned}& \biggl\vert \Delta ( 1 ) \lambda \mathcal{F} ( \kappa _{1} ) + \Delta ( 1 ) ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\Delta ( 1 ) ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \bigl[ _{\frac{\kappa _{1}+\kappa _{2}}{2}+}I_{\varphi } \mathcal{F} ( \kappa _{2} ) +_{\frac{\kappa _{1}+\kappa _{2}}{2}-}I_{ \varphi }\mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{ \frac{1}{p_{1}}}\\& \qquad {} \times \biggl( \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{r_{1}} \int _{0}^{\frac{1}{2}}\tau \,d\tau + \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{r_{1}} \int _{0}^{\frac{1}{2}} ( 1-\tau ) \,d\tau \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}}\\& \qquad {} \times \biggl( \bigl\vert \mathcal{F}^{\prime } ( \kappa _{2} ) \bigr\vert ^{r_{1}} \int _{\frac{1}{2}}^{1}\tau \,d\tau + \bigl\vert \mathcal{F}^{\prime } ( \kappa _{1} ) \bigr\vert ^{r_{1}} \int _{\frac{1}{2}}^{1} ( 1-\tau ) \,d\tau \biggr) ^{\frac{1}{r_{1}}} \biggr] \\& \quad = ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \lambda \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{ \vert \mathcal{F}^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+3 \vert \mathcal{F}^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{8} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \Delta ( \tau ) - \Delta ( 1 ) \mu \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \mathcal{F}^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \mathcal{F}^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{8} \biggr) ^{ \frac{1}{r_{1}}} \biggr], \end{aligned}$$
which completes the proof. □
Remark 4
In Theorem 5, if we set \(\varphi ( \tau ) =\tau \), then Theorem 5 reduces to [10, Theorem 2.2 for \(s=m=1\)].
Corollary 7
In Theorem 5, if we set \(\varphi ( \tau ) =\frac{\tau ^{\alpha }}{\Gamma ( \alpha ) }\), then we obtain the following parameterized Simpson’s type inequality for Riemann-Liouville fractional integrals:
$$\begin{aligned}& \biggl\vert \lambda \mathcal{F} ( \kappa _{1} ) + ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \frac{\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \bigl[ J_{\frac{\kappa _{1}+\kappa _{2}}{2}+}^{\alpha } \mathcal{F} ( \kappa _{2} ) +J_{\frac{\kappa _{1}+\kappa _{2}}{2}-}^{\alpha } \mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \tau ^{\alpha }-\lambda \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{ \vert \mathcal{F}^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+3 \vert \mathcal{F}^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{8} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \tau ^{\alpha }- \mu \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \mathcal{F}^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \mathcal{F}^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{8} \biggr) ^{\frac{1}{r_{1}}} \biggr] . \end{aligned}$$
Corollary 8
In Theorem 5, if we set \(\varphi ( \tau ) =\frac{\tau ^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we obtain the following parameterized Simpson’s type inequality for k-Riemann–Liouville fractional integrals:
$$\begin{aligned}& \biggl\vert \lambda \mathcal{F} ( \kappa _{1} ) + ( \mu -\lambda ) \mathcal{F} \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\mu ) \mathcal{F} ( \kappa _{2} ) \\& \qquad {} - \frac{\Gamma _{k} ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \bigl[ J_{\frac{\kappa _{1}+\kappa _{2}}{2}+,k}^{\alpha } \mathcal{F} ( \kappa _{2} ) +J_{\frac{\kappa _{1}+\kappa _{2}}{2}-,k}^{\alpha } \mathcal{F} ( \kappa _{1} ) \bigr] \biggr\vert \\& \quad \leq ( \kappa _{2}-\kappa _{1} ) \biggl[ \biggl( \int _{0}^{ \frac{1}{2}} \bigl\vert \tau ^{\frac{\alpha }{k}}-\lambda \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{ \vert \mathcal{F}^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+3 \vert \mathcal{F}^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{8} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \tau ^{ \frac{\alpha }{k}}-\mu \bigr\vert ^{p_{1}}\,d\tau \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \mathcal{F}^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \mathcal{F}^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{8} \biggr) ^{\frac{1}{r_{1}}} \biggr]. \end{aligned}$$