In this section, we present some Simpson-type inequalities for generalized fractional integrals.
Theorem 3
We assume that the conditions of Lemma 2hold. If the mapping \(\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s} \vert \) is convex on △, then we have the following inequality for generalized fractional integrals:
$$\begin{aligned} & \bigl\vert \Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4\Lambda (1)\triangledown (1)} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda (t)}{2}- \frac{\Lambda (1)}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown (s)}{2}- \frac{\bigtriangledown (1)}{3} \biggr\vert \,ds\,dt \biggr) \\ &\qquad{}\times \biggl[ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr], \end{aligned}$$
where \(\Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Lemma 2.
Proof
By taking modulus in Lemma 2, we obtain
$$\begin{aligned} & \bigl\vert \Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4\Lambda (1)\bigtriangledown (1)} \\ &\qquad{}\times \biggl\{ \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}-\frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\qquad{}+ \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \biggr\} . \end{aligned}$$
(4.1)
Since the mapping \(|\frac{\partial \digamma }{\partial t\,\partial s} |\) is co-ordinated convex on △, we obtain
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad\leq \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr)\,ds\,dt. \end{aligned}$$
(4.2)
Similarly, we obtain
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \end{aligned}$$
(4.3)
$$\begin{aligned} &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad\leq \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr)\,ds\,dt, \\ & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad\leq \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr)\,ds\,dt \end{aligned}$$
(4.4)
and
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad\leq \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert \\ &\qquad{}\times \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert \\ &\qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr)\,ds\,dt. \end{aligned}$$
(4.5)
Using the inequalities (4.2)–(4.5) in (4.1), the proof is completed. □
Corollary 4
If we take \(\phi (t)=t\) and \(\psi (s)=s\) in Theorem 3, then Theorem 3reduces to [15, Theorem 3].
Corollary 5
In Theorem 3, if we use \(\phi (t)=\frac{t^{\alpha }}{\Gamma {(\alpha )}}\) and \(\psi (s)=\frac{s^{\beta }}{\Gamma (\beta )}\), then we obtain the following inequality for Riemann–Loiuville fractional integrals:
$$\begin{aligned} & \bigl\vert \Omega (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad \leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4} \\ &\qquad{}\times \biggl( \frac{\alpha }{\alpha +1} \biggl( \frac{2}{3} \biggr) ^{ \frac{1}{\alpha }+1}+\frac{1}{2(\alpha +1)}- \frac{1}{3} \biggr) \biggl( \frac{\beta }{\beta +1} \biggl( \frac{2}{3} \biggr) ^{\frac{1}{\beta }+1}+ \frac{1}{2(\beta +1)}- \frac{1}{3} \biggr) \\ & \qquad{}\times \biggl[ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr], \end{aligned}$$
where \(|\Omega (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})|\) is defined as in Corollary 2.
Corollary 6
If we take \(\phi (t)=\frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k}{(\alpha )}}\) and \(\psi (s)=\frac{s^{\frac{\beta }{k}}}{k\Gamma _{k}{(\beta )}}\) in Theorem 3, we obtain the following inequality for k-Riemann–Louville fractional integrals:
$$\begin{aligned} & \bigl\vert \$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4} \biggl( \frac{\alpha }{\alpha +k} \biggl( \frac{2}{3} \biggr) ^{ \frac{k}{\alpha }+1}+\frac{k}{2(\alpha +k)}- \frac{1}{3} \biggr) \\ &\qquad{}\times \biggl( \frac{\beta }{\beta +k} \biggl( \frac{2}{3} \biggr) ^{\frac{k}{\beta }+1}+ \frac{k}{2(\beta +k)}- \frac{1}{3} \biggr) \\ &\qquad{}\times \biggl[ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert + \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert \biggr], \end{aligned}$$
where \(\$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Corollary 3.
Theorem 4
Suppose that the assumptions of Lemma 2hold. If the mapping \(\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s} \vert ^{q}\) is co-ordinated convex on Δ, then we have the following inequality for generalized fractional integrals:
$$\begin{aligned} & \bigl\vert \Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{4\Lambda (1)\bigtriangledown (1)} \\ &\qquad{}\times \biggl[ \biggl( \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac{ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac{3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac{3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}} \\ &\qquad{}+ \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \frac{9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Lemma 2and \(\frac{1}{p}+\frac{1}{q}=1\).
Proof
From Hölder’s inequality and co-ordinated convexity of \(|\frac{\partial ^{2}\digamma }{\partial t\,\partial s} |^{q}\), we have
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert \biggl\vert \biggl(\frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ & \qquad{}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}}{\partial {t}\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2} \kappa _{2},\frac{1-s}{2}\kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert ^{q}\,dtds \biggr)^{\frac{1}{q}} \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ & \qquad{}\times \int _{0}^{1} \int _{0}^{1} \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}}\,ds\,dt \\ & \quad= \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl( \frac{ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}}. \end{aligned}$$
(4.6)
Similarly, we obtain
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \end{aligned}$$
(4.7)
$$\begin{aligned} &\qquad{} \times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad \leq \biggl( \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr) \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \biggl( \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ & \qquad{}\times \biggl( \frac{3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}}, \\ & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad \leq \biggl( \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \biggl( \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr) \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl( \frac{3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}}, \end{aligned}$$
(4.8)
and
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \biggl( \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr) \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert \biggl(\frac{\bigtriangledown ({1})}{3}-\frac{\bigtriangledown ({s})}{2} \biggr) \biggr\vert ^{p} \,ds \biggr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl( \frac{9 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \vert ^{q}+3 \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \vert ^{q}+ \vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{4}) \vert ^{q}}{16} \biggr) ^{\frac{1}{q}}. \end{aligned}$$
(4.9)
Using the inequalities (4.6)–(4.9) in (4.1), we obtain the required result. □
Corollary 7
If we take \(\phi (t)=t\) and \(\psi (s)=s\) in Theorem 3, we have the following Simpson inequality for Riemann integrals:
$$\begin{aligned} & \bigl\vert \Upsilon (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \frac{(\kappa _{2}-\kappa _{1})(\kappa _{4}-\kappa _{3})}{144} \biggl(\frac{1+2^{p+1}}{3(p+1)} \biggr)^{\frac{2}{p}} \biggl(\frac{1}{16} \biggr)^{ \frac{1}{q}} \\ &\qquad{}\times \biggl[ \biggl( \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{2},\kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\Upsilon (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Corollary 1.
Corollary 8
If we take \(\phi (t)=\frac{t^{\alpha }}{\Gamma {(\alpha )}}\) and \(\psi (s)=\frac{t^{\beta }}{\Gamma {(\beta )}}\) in Theorem 3, we obtain the following Simpson inequality for Riemann–Liouville fractional integrals:
$$\begin{aligned} & \bigl\vert \Omega (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{t^{\alpha }}{2}- \frac{1}{3} \biggr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{1} \biggl\vert \frac{s^{\alpha }}{2}- \frac{1}{3} \biggr\vert ^{p}\,ds \biggr)^{\frac{1}{p}} \biggl(\frac{1}{16} \biggr)^{\frac{1}{q}} \\ &\qquad{}\times \biggl[ \biggl( \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{2},\kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\Omega (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Corollary 2.
Corollary 9
If we take \(\phi (t)=\frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k}{(\alpha )}}\) and \(\psi (s)=\frac{t^{\frac{\beta }{k}}}{k\Gamma _{k}{(\beta )}}\) in Theorem 3, we obtain the following Simpson inequality for k-Riemann–Liouville fractional integrals:
$$\begin{aligned} & \bigl\vert \$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad\leq \biggl( \biggl\vert \frac{t^{\frac{\alpha }{k}}}{2}-\frac{1}{3} \biggr\vert \,dt \biggr)^{\frac{1}{p}} \biggl( \biggl\vert \frac{s^{\frac{\beta }{k}}}{2}-\frac{1}{3} \biggr\vert \,ds \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl[ \biggl( \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{2},\kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{}+ \biggl(9 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}( \kappa _{1},\kappa _{3}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{4}) \biggr\vert ^{q}+3 \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\$(\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Corollary 3.
Theorem 5
Suppose that the assumptions of Lemma 2hold. If the mapping \(\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s} \vert ^{q}\), \(q>1\), is co-ordinated convex on Δ, then we have the following inequality for generalized fractional integrals:
$$\begin{aligned} & \bigl\vert \Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4}) \bigr\vert \\ &\quad \leq \frac{(\kappa _{4}-\kappa _{3})(\kappa _{2}-\kappa _{1})}{\Lambda (1)\bigtriangledown (1)} \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}-\frac{\Lambda ({1})}{3} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \biggl\vert \frac{\bigtriangledown ({s})}{2}-\frac{\bigtriangledown ({1})}{3} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times \biggl\{ ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}-\frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl[ \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ &\qquad{} + \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}} \\ &\qquad{} + ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl[ \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ &\qquad{} + \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}} \\ &\qquad{} + ( \int _{0}^{1} \int _{0}^{1} ( \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl[ \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}} \\ & \qquad{}+ ( \int _{0}^{1} \int _{0}^{1} ( \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \\ & \qquad{}\times \biggl[ \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{} + \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2},\kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}} \biggr\} , \end{aligned}$$
where \(\Re (\kappa _{1},\kappa _{2};\kappa _{3},\kappa _{4})\) is defined as in Lemma 2.
Proof
Using the power mean inequality and coordinate-convexity of \(|\frac{\partial ^{2}\digamma }{\partial t\,\partial s} |^{q}\), we have
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2},\frac{1-s}{2} \kappa _{3}+ \frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad \leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert ^{q}\,dtds \biggr)^{\frac{1}{q}} \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl[ \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(4.10)
Similarly, we obtain
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \end{aligned}$$
(4.11)
$$\begin{aligned} & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ &\qquad{} \times ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl[ \biggl( \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \quad\qquad{}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}}, \\ & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1-s}{2} \kappa _{3}+\frac{1+s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ &\quad \leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times ( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({t})}{2}- \frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ &\qquad{} \times \biggl[ \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}}, \end{aligned}$$
(4.12)
and
$$\begin{aligned} & \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \\ & \qquad{}\times \biggl\vert \frac{\partial ^{2}}{\partial t\,\partial {s}} \digamma \biggl(\frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2}, \frac{1+s}{2} \kappa _{3}+\frac{1-s}{2}\kappa _{4} \biggr) \biggr\vert \,ds\,dt \\ & \quad\leq \biggl( \int _{0}^{1} \biggl\vert \frac{\Lambda ({1})}{3}- \frac{\Lambda ({t})}{2} \biggr\vert \,dt \biggr)^{1-\frac{1}{q}} \biggl( \int _{0}^{1} \biggl\vert \frac{\bigtriangledown ({1})}{3}- \frac{\bigtriangledown ({s})}{2} \biggr\vert \,ds \biggr)^{1-\frac{1}{q}} \\ & \qquad{}\times ( \int _{0}^{1} \int _{0}^{1} ( \biggl\vert \frac{\Lambda ({t})}{2}-\frac{\Lambda ({1})}{3} \biggr\vert \biggl\vert \frac{\bigtriangledown ({s})}{2}- \frac{\bigtriangledown ({1})}{3} \biggr\vert \\ & \qquad{}\times \biggl[ \biggl( \biggl(\frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1+t}{2} \biggr) \biggl(\frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{1},\kappa _{4}) \biggr\vert ^{q} \\ & \qquad{}+ \biggl(\frac{1-t}{2} \biggr) \biggl(\frac{1+s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{3}) \biggr\vert ^{q}+ \biggl(\frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac{\partial ^{2}\digamma }{\partial t\,\partial s}(\kappa _{2}, \kappa _{4}) \biggr\vert ^{q} \biggr]\,ds\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(4.13)
By substituting the inequalities (4.10)–(4.12) in (4.1), we obtain the desired result. □
Remark 1
By special choices of the functions ϕ and ψ in Theorem 5, one can obtain several new Simpson-type inequalities. These are left to the reader.