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Some parameterized Simpson-, midpoint- and trapezoid-type inequalities for generalized fractional integrals

Abstract

In this paper, we first obtain an identity for differentiable mappings. Then, we establish some new generalized inequalities for differentiable convex functions involving some parameters and generalized fractional integrals. We show that these results reduce to several new Simpson-, midpoint- and trapezoid-type inequalities. The results given in this study are the generalizations of results proved in several earlier papers.

Introduction

Simpson’s inequality plays an important role in many areas of mathematics. The classical Simpson’s inequality is expressed as follows for four-times continuously differentiable functions:

Theorem 1

Suppose that \(\digamma : [ \kappa _{1},\kappa _{2} ] \rightarrow \mathbb{R}\) is a four-times continuously differentiable mapping on \(( \kappa _{1},\kappa _{2} ) \), and let \(\Vert \digamma ^{ ( 4 ) } \Vert _{\infty }= \underset{x\in ( \kappa _{1},\kappa _{2} ) }{\sup } \vert \digamma ^{ ( 4 ) }(x) \vert <\infty \). Then, one has the inequality

$$ \biggl\vert \frac{1}{3} \biggl[ \frac{\digamma (\kappa _{1})+\digamma (\kappa _{2})}{2}+2 \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] -\frac{1}{\kappa _{2}-\kappa _{1}} \int _{ \kappa _{1}}^{\kappa _{2}}\digamma (x)\,dx \biggr\vert \leq \frac{1}{2880} \bigl\Vert \digamma ^{ ( 4 ) } \bigr\Vert _{\infty } ( \kappa _{2}-\kappa _{1} ) ^{4}. $$

In recent years, many writers have focused on Simpson-type inequalities in various categories of work. Specifically, some mathematicians have worked on the results of the Simpson- and Newton-type inequalities by using convex mappings, because convexity theory is an effective and powerful way to solve a large number of problems from different branches of pure and applied mathematics. For example, Dragomir et al. [11] presented new Simpson-type results and their applications to quadrature formulas in numerical integration. Also, new Newton-type inequalities for functions whose local fractional derivatives are generalized convex are given by Iftikhar et al. in [20]. For more recent developments, one can consult [15, 9, 1215, 19, 2830, 35].

The aim of this paper is to obtain several generalized inequalities for differentiable mappings by utilizing generalized fractional integrals and some nonnegative parameters. By special choice of parameters, the obtained results reduce some well-known Simpson-, midpoint- and trapezoid-type inequalities obtained by several authors in [10, 16, 17, 23, 33, 34].

Generalized fractional integral operators

In this section, we mention the generalized fractional integrals defined by Sarikaya and Ertuğral in [32].

Let \(\varphi :[0,\infty )\rightarrow {}[ 0,\infty )\) be a function satisfying the following condition:

$$ \int _{0}^{1}\frac{\varphi ( t ) }{t}\,dt< \infty . $$

The left-sided and right-sided generalized fractional integral operators are defined, respectively, as follows:

$$\begin{aligned}& {}_{\kappa _{1}+}I_{\varphi }\digamma (x)= \int _{ \kappa _{1}}^{x}\frac{\varphi ( x-t ) }{x-t} \digamma (t)\,dt,\quad x>\kappa _{1}, \end{aligned}$$
(2.1)
$$\begin{aligned}& {}_{\kappa _{2}-}I_{\varphi }\digamma (x)= \int _{x}^{\kappa _{2}}\frac{\varphi ( t-x ) }{t-x} \digamma (t)\,dt,\quad x< \kappa _{2}. \end{aligned}$$
(2.2)

These fractional operators genralize several fractional integrals such as Riemann–Liouville fractional integrals, k-Riemann–Liouville fractional integrals, Katugampola fractional integrals, conformable fractional integrals, Hadamard fractional integrals, etc. These important special cases of the integral operators (2.1) and (2.2) are mentioned below.

  1. (1)

    If we choose \(\varphi ( t ) =t\), the operators (2.1) and (2.2) reduce to the Riemann integral.

  2. (2)

    Considering \(\varphi ( t ) = \frac{t^{\alpha }}{\Gamma ( \alpha ) }\) and \(\alpha >0\), the operators (2.1) and (2.2) reduce to the Riemann–Liouville fractional integrals \(J_{{\kappa _{1}+}}^{\alpha }\digamma (x)\) and \(J_{\kappa _{2}-}^{\alpha }\digamma (x)\), respectively. Here, Γ is the Gamma function.

  3. (3)

    For \(\varphi ( t ) = \frac{1}{k\Gamma _{k} ( \alpha ) }t^{\frac{\alpha }{k}}\) and \(\alpha ,k>0\), the operators (2.1) and (2.2) reduce to the k-Riemann–Liouville fractional integrals [27] \(J_{\kappa _{1}+,k}^{\alpha }\digamma (x)\) and \(J_{\kappa _{2}-,k}^{\alpha }\digamma (x)\), respectively. Here, \(\Gamma _{k}\) is the k-Gamma function.

Sarıkaya and Ertuğral also established the following Hermite–Hadamard inequality for the generalized fractional integral operators:

Theorem 2

([32])

Let \(\digamma : [ \kappa _{1},\kappa _{2} ] \rightarrow \mathbb{R} \) be a convex function on \([ \kappa _{1},\kappa _{2} ] \) with \(\kappa _{1}<\kappa _{2}\), then the following inequalities for fractional integral operators hold:

$$ \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \leq \frac{1}{2\Lambda (1)} \bigl[ {}_{\kappa _{1}+}I_{\varphi } \digamma (\kappa _{2})+{}_{ \kappa _{2}-}I_{\varphi } \digamma (\kappa _{1}) \bigr] \leq \frac{\digamma (\kappa _{1})+\digamma (\kappa _{2})}{2} , $$
(2.3)

where the mapping \(\Lambda : [ 0,1 ] \rightarrow \mathbb{R} \) is defined by

$$ \Lambda (t)= \int _{0}^{t} \frac{\varphi ( ( \kappa _{2}-\kappa _{1} ) u ) }{u}\,du. $$

In the literature there are several papers on inequalities for generalized fractional integrals. For some of these please refer to [6, 7, 16, 18, 21, 22, 2426, 31, 36].

An identity for generalized fractional integrals

In this section, we offer a parameterized identity involving an ordinary first derivative via generalized fractional integrals.

Lemma 1

Let \(\digamma : [ \kappa _{1}, \kappa _{2} ] \rightarrow \mathbb{R} \) be a differentiable function on \(( \kappa _{1}, \kappa _{2} ) \). If \(\digamma ^{\prime }\) is continuous and integrable on \([ \kappa _{1},\kappa _{2} ] \), then for \(\rho ,\sigma \geq 0\), one has the identity

$$\begin{aligned}& ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1- \rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{\kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \int _{0}^{1} \bigl( \Delta ( t ) - \Delta ( 1 ) \rho \bigr) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2} \biggr) \,dt \\& \qquad {} + \int _{0}^{1} \bigl( \Delta ( 1 ) \sigma - \Delta ( t ) \bigr) \digamma ^{\prime } \biggl( \frac{1+t}{2} \kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \,dt \biggr] , \end{aligned}$$
(3.1)

where the mapping \(\Delta : [ 0,1 ] \rightarrow \mathbb{R} \) is defined by

$$ \Delta (t)= \int _{0}^{t} \frac{\varphi ( ( \frac{ \kappa _{2}-\kappa _{1}}{2} ) u ) }{u}\,du. $$

Proof

Applying the fundamental rules of integration, we have

$$\begin{aligned}& \int _{0}^{1} \bigl( \Delta ( t ) -\Delta ( 1 ) \rho \bigr) \digamma ^{\prime } \biggl( \frac{1-t}{2} \kappa _{1}+ \frac{1+t}{2}\kappa _{2} \biggr) \,dt \\& \quad = \frac{2}{\kappa _{2}-\kappa _{1}} \biggl[ \Delta ( 1 ) \biggl( ( 1-\rho ) \digamma ( \kappa _{2} ) +\rho \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr) -{}_{ \kappa _{2}-}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \end{aligned}$$
(3.2)

and

$$\begin{aligned}& \int _{0}^{1} \bigl( \Delta ( 1 ) \sigma -\Delta ( t ) \bigr) \digamma ^{\prime } \biggl( \frac{1+t}{2} \kappa _{1}+\frac{1-t}{2}\kappa _{2} \biggr) \,dt \\& \quad = \frac{2}{\kappa _{2}-\kappa _{1}} \biggl[ \Delta ( 1 ) \biggl( ( 1-\sigma ) \digamma ( \kappa _{1} ) +\sigma \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr) -{}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] . \end{aligned}$$
(3.3)

By adding (3.2) and (3.3), we obtain the required equality (3.1). □

Corollary 1

If we assume \(\varphi ( t ) =t\) in Lemma 1, then we obtain the following equality:

$$\begin{aligned}& \frac{1}{2} \biggl[ ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \biggr] - \frac{1}{\kappa _{2}-\kappa _{1}} \int _{ \kappa _{1}}^{\kappa _{2}}\digamma (t)\,dt \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4} \biggl[ \int _{0}^{1} ( t-\rho ) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2} \biggr) \,dt \\& \qquad {} + \int _{0}^{1} ( \sigma -t ) \digamma ^{ \prime } \biggl( \frac{1+t}{2}\kappa _{1}+\frac{1-t}{2} \kappa _{2} \biggr) \,dt \biggr] . \end{aligned}$$

Corollary 2

In Lemma 1, if we set \(\varphi ( t ) = \frac{t^{\alpha }}{\Gamma ( \alpha ) }\), then we obtain the following new identity for the Riemann–Liouville fractional integral:

$$\begin{aligned}& ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1- \rho ) \digamma ( \kappa _{2} ) \\& \qquad {} - \frac{2^{\alpha }\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2} \biggl[ \int _{0}^{1} \bigl( t^{\alpha }-\rho \bigr) \digamma ^{\prime } \biggl( \frac{1-t}{2} \kappa _{1}+\frac{1+t}{2}\kappa _{2} \biggr) \,dt \\& \qquad {}+ \int _{0}^{1} \bigl( \sigma -t^{\alpha } \bigr) \digamma ^{\prime } \biggl( \frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \,dt \biggr] . \end{aligned}$$

Corollary 3

In Lemma 1, if we take \(\varphi ( t ) = \frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we obtain the following new identity for the k-Riemann–Liouville fractional integral:

$$\begin{aligned}& ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1- \rho ) \digamma ( \kappa _{2} ) \\& \qquad {} - \frac{2^{\frac{\alpha }{k}}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}^{+},k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}^{-},k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2} \biggl[ \int _{0}^{1} \bigl( t^{\frac{\alpha }{k}}-\rho \bigr) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2} \kappa _{2} \biggr) \,dt \\& \qquad {} + \int _{0}^{1} \bigl( \sigma -t^{\frac{\alpha }{k}} \bigr) \digamma ^{\prime } \biggl( \frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \,dt \biggr] . \end{aligned}$$

Some parameterized inequalities for generalized fractional integral operators

In this section, we establish some new generalized inequalities for differentiable convex functions via generalized fractional integrals.

Theorem 3

We assume that the conditions of Lemma 1hold. If the mapping \(\vert \digamma ^{\prime } \vert \) is convex on \([ \kappa _{1},\kappa _{2} ] \), then the following inequality holds for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert \bigl( \Pi _{1}^{\varphi } ( \rho ) +\Pi _{2}^{\varphi } ( \sigma ) \bigr) + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigl( \Pi _{1}^{\varphi } ( \sigma ) +\Pi _{2}^{\varphi } ( \rho ) \bigr) \bigr] , \end{aligned}$$
(4.1)

where

$$ \Pi _{1}^{\varphi } ( \tau ) = \int _{0}^{1} ( 1-t ) \bigl\vert \Delta ( t ) -\Delta ( 1 ) \tau \bigr\vert \,dt $$

and

$$ \Pi _{2}^{\varphi } ( \tau ) = \int _{0}^{1} ( 1+t ) \bigl\vert \Delta ( t ) -\Delta ( 1 ) \tau \bigr\vert \,dt. $$

Proof

By taking the modulus in Lemma 1 and using the properties of the modulus, we obtain that

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{\varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \int _{0}^{1} \bigl\vert \Delta ( t ) - \Delta ( 1 ) \rho \bigr\vert \biggl\vert \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2} \biggr) \biggr\vert \,dt \\& \qquad {} + \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert \biggl\vert \digamma ^{ \prime } \biggl( \frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2} \kappa _{2} \biggr) \biggr\vert \,dt \biggr] . \end{aligned}$$
(4.2)

Since the mapping \(\vert \digamma ^{\prime } \vert \) is convex on \([ \kappa _{1},\kappa _{2} ] \), we have

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \biggl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert \biggl( \int _{0}^{1} ( 1-t ) \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \,dt+ \int _{0}^{1} ( 1+t ) \bigl\vert \Delta ( 1 ) \sigma -\Delta ( t ) \bigr\vert \,dt \biggr) \\& \qquad {} + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \biggl( \int _{0}^{1} ( 1+t ) \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \,dt+ \int _{0}^{1} ( 1-t ) \bigl\vert \Delta ( 1 ) \sigma -\Delta ( t ) \bigr\vert \,dt \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigl( \Pi _{1}^{\varphi } ( \sigma ) +\Pi _{2}^{\varphi } ( \rho ) \bigr) + \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert \bigl( \Pi _{1}^{\varphi } ( \rho ) +\Pi _{2}^{ \varphi } ( \sigma ) \bigr) \bigr], \end{aligned}$$

which ends the proof. □

Corollary 4

Under the assumption of Theorem 3with \(\varphi ( t ) =t\), we obtain the following inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{2} \biggl[ ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \biggr] - \frac{1}{\kappa _{2}-\kappa _{1}}\int _{\kappa _{1}}^{\kappa _{2}} \digamma (t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{8} \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert \bigl( \Pi _{1} ( \rho ) +\Pi _{2} ( \sigma ) \bigr) + \bigl\vert \digamma ^{ \prime } ( \kappa _{2} ) \bigr\vert \bigl( \Pi _{1} ( \sigma ) +\Pi _{2} ( \rho ) \bigr) \bigr] , \end{aligned}$$

where

$$ \Pi _{1} ( \tau ) =\tau ^{2}-\frac{\tau ^{3}}{3}- \frac{\tau }{2}+\frac{1}{6}, $$

and

$$ \Pi _{2} ( \tau ) =\frac{\tau ^{3}}{3}+\tau ^{2}- \frac{3\tau }{2}+\frac{5}{6}. $$

Corollary 5

Under the assumption of Theorem 3with \(\varphi ( t ) = \frac{t^{\alpha }}{\Gamma ( \kappa _{1} ) }\), we obtain the following inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} - \frac{2^{\alpha }\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{ \kappa _{1}^{+}}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}^{-}}^{ \alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert \bigl( \Pi _{1}^{\alpha } ( \rho ) + \Pi _{2}^{\alpha } ( \sigma ) \bigr) + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigl( \Pi _{1}^{\alpha } ( \sigma ) +\Pi _{2}^{ \alpha } ( \rho ) \bigr) \bigr] , \end{aligned}$$

where

$$ \Pi _{1}^{\alpha } ( \tau ) =\frac{2\alpha }{\alpha +1} \tau ^{\frac{\alpha +1}{\alpha }}-\frac{\alpha }{\alpha +2}\tau ^{ \frac{\alpha +2}{\alpha }}- \frac{\tau }{2}+ \frac{1}{ ( \alpha +2 ) ( \alpha +1 ) }, $$

and

$$ \Pi _{2}^{\alpha } ( \tau ) =\frac{\alpha }{\alpha +2} \tau ^{\frac{\alpha +2}{\alpha }}-\frac{3\tau }{2}+ \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) }+ \frac{2\alpha }{\alpha +1}\tau ^{\frac{\alpha +1}{\alpha }}. $$

Corollary 6

In Theorem 3, if we take \(\varphi ( t ) = \frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \kappa _{1} ) }\), then we obtain the following inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} - \frac{2^{\frac{\alpha }{k}}\Gamma ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}^{+},k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}^{-},k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert \bigl( \Pi _{1}^{\frac{\alpha }{k}} ( \rho ) +\Pi _{2}^{\frac{\alpha }{k}} ( \sigma ) \bigr) + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigl( \Pi _{1}^{\frac{\alpha }{k}} ( \sigma ) + \Pi _{2}^{\frac{\alpha }{k}} ( \rho ) \bigr) \bigr] , \end{aligned}$$

where

$$ \Pi _{1}^{\frac{\alpha }{k}} ( \tau ) = \frac{2\alpha }{\alpha +k}\tau ^{\frac{\alpha +k}{\alpha }}-\frac{\alpha }{\alpha +2k}\tau ^{ \frac{\alpha +2k}{\alpha }}- \frac{\tau }{2}+ \frac{k^{2}}{ ( \alpha +2k ) ( \alpha +k ) }, $$

and

$$ \Pi _{2}^{\frac{\alpha }{k}} ( \tau ) = \frac{\alpha }{\alpha +2k}\tau ^{\frac{\alpha +2k}{\alpha }}-\frac{3\tau }{2}+ \frac{2\alpha k+3k^{2}}{ ( \alpha +2k ) ( \alpha +k ) }+ \frac{2\alpha }{\alpha +k}\tau ^{\frac{\alpha +k}{\alpha }}. $$

Theorem 4

We assume that the conditions of Lemma 1hold. If the mapping \(\vert \digamma \vert ^{p_{1}}\), \(p_{1}>1\), is convex on \([ \kappa _{1},\kappa _{2} ] \), then we have the following inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \,dt \biggr) ^{1- \frac{1}{p_{1}}} \biggl( \frac{\Pi _{1}^{\varphi } ( \rho ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( \rho ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert \,dt \biggr) ^{1- \frac{1}{p_{1}}} \biggl( \frac{\Pi _{1}^{\varphi } ( \sigma ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( \sigma ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \biggr] , \end{aligned}$$

where \(\Pi _{1}^{\varphi } ( \tau )\) and \(\Pi _{2}^{\varphi } ( \tau ) \) are defined as in Theorem 3.

Proof

Reutilizing inequality (4.2) and from the power mean inequality, we have

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \,dt \biggr) ^{1- \frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \biggl\vert \digamma ^{ \prime } \biggl( \frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2} \kappa _{2} \biggr) \biggr\vert ^{p_{1}}\,dt \biggr) ^{ \frac{1}{p_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert \,dt \biggr) ^{1- \frac{1}{p_{1}}} \\& \qquad {}\times \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert \biggl\vert \digamma ^{\prime } \biggl( \frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \biggr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \biggr] . \end{aligned}$$

Using the convexity of \(\vert \digamma ^{\prime } \vert ^{p_{1}}\), we have

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \,dt \biggr) ^{1- \frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \int _{0}^{1} \biggl( \frac{1-t}{2} \biggr) \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \,dt \\& \qquad {}+ \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \int _{0}^{1} \biggl( \frac{1+t}{2} \biggr) \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \,dt \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert \,dt \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {} \times \biggl( \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \int _{0}^{1} \biggl( \frac{1+t}{2} \biggr) \bigl\vert \Delta ( 1 ) \sigma -\Delta ( t ) \bigr\vert \,dt \\& \qquad {}+ \bigl\vert \digamma ^{ \prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \int _{0}^{1} \biggl( \frac{1-t}{2} \biggr) \bigl\vert \Delta ( 1 ) \sigma -\Delta ( t ) \bigr\vert \,dt \biggr) ^{\frac{1}{p_{1}}} \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert \,dt \biggr) ^{1- \frac{1}{p_{1}}} \biggl( \frac{\Pi _{1}^{\varphi } ( \rho ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( \rho ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert \,dt \biggr) ^{1- \frac{1}{p_{1}}} \biggl( \frac{\Pi _{1}^{\varphi } ( \sigma ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( \sigma ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$

which finishes the proof. □

Corollary 7

If we assume that \(\varphi ( t ) =t\) in Theorem 4, then we obtain the following inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{2} \biggl[ ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \biggr] - \frac{1}{\kappa _{2}-\kappa _{1}}\int _{\kappa _{1}}^{\kappa _{2}} \digamma (t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{8} \bigl[ \bigl( \Pi _{1} ( \rho ) +\Pi _{2} ( \rho ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1} ( \rho ) \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}}+ \Pi _{2} ( \rho ) \bigl\vert \digamma ^{ \prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \bigl( \Pi _{1} ( \sigma ) +\Pi _{2} ( \sigma ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1} ( \sigma ) \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{2} ( \sigma ) \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \bigr] , \end{aligned}$$

where \(\Pi _{1} ( \tau ) \) and \(\Pi _{2} ( \tau ) \) are defined as in Corollary 4.

Corollary 8

If we take \(\varphi ( t ) = \frac{t^{\alpha }}{\Gamma ( \alpha ) }\) in Theorem 4, then we have the following inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} - \frac{2^{\alpha }\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{ \kappa _{1}^{+}}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \bigl[ \bigl( \Pi _{1}^{\alpha } ( \rho ) +\Pi _{2}^{\alpha } ( \rho ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1}^{ \alpha } ( \rho ) \bigl\vert \digamma ^{ \prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}}+ \Pi _{2}^{\alpha } ( \rho ) \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \bigl( \Pi _{1}^{\alpha } ( \sigma ) +\Pi _{2}^{ \alpha } ( \sigma ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1}^{\alpha } ( \sigma ) \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{\alpha } ( \sigma ) \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \bigr] , \end{aligned}$$
(4.3)

where \(\Pi _{1}^{\alpha } ( \tau ) \) and \(\Pi _{2}^{\alpha } ( \tau ) \) are defined as in Corollary 5.

Corollary 9

If we take \(\varphi ( t ) = \frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\) in Theorem 4, then we have the following inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa_{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} - \frac{2^{\frac{\alpha }{k}}\Gamma _{k} ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}^{+},k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}^{-},k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \bigl[ \bigl( \Pi _{1}^{\frac{\alpha }{k}} ( \rho ) +\Pi _{2}^{ \frac{\alpha }{k}} ( \rho ) \bigr) ^{1-\frac{1}{p_{1}}} \bigl( \Pi _{1}^{ \frac{\alpha }{k}} ( \rho ) \bigl\vert \digamma ^{ \prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}}+ \Pi _{2}^{\frac{\alpha }{k}} ( \rho ) \bigl\vert \digamma ^{ \prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \bigl( \Pi _{1}^{\frac{\alpha }{k}} ( \sigma ) +\Pi _{2}^{\frac{\alpha }{k}} ( \sigma ) \bigr) ^{1- \frac{1}{p_{1}}} \bigl( \Pi _{1}^{\frac{\alpha }{k}} ( \sigma ) \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{\frac{\alpha }{k}} ( \sigma ) \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \bigr] , \end{aligned}$$
(4.4)

where \(\Pi _{1}^{\frac{\alpha }{k}} ( \tau ) \) and \(\Pi _{2}^{\frac{\alpha }{k}} ( \tau ) \) are described in Corollary 5.

Theorem 5

We assume that the conditions of Lemma 1hold. If the mapping \(\vert \digamma ^{\prime } \vert ^{r_{1}}\), \(r_{1}>1\), is convex on \([ \kappa _{1},\kappa _{2} ] \), then we have the following inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert ^{p_{1}}\,dt \biggr) ^{ \frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \biggr] , \end{aligned}$$
(4.5)

where \(\frac{1}{p_{1}}+\frac{1}{r_{1}}=1\).

Proof

Reutilizing inequality (4.2) and from the well-known Hölder’s inequality, we have

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \biggl( \int _{0}^{1} \biggl\vert \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2} \biggr) \biggr\vert ^{r_{1}}\,dt \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert ^{p_{1}}\,dt \biggr) ^{ \frac{1}{p_{1}}} \biggl( \int _{0}^{1} \biggl\vert \digamma ^{\prime } \biggl( \frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2} \biggr) \biggr\vert ^{r_{1}}\,dt \biggr) ^{\frac{1}{r_{1}}} \biggr] . \end{aligned}$$

Using the fact that \(\vert \digamma ^{\prime } \vert ^{r_{1}}\) is convex, we have

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} -\frac{1}{\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}\times\biggl( \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{r_{1}} \int _{0}^{1} \biggl( \frac{1-t}{2} \biggr) \,dt+ \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{r_{1}} \int _{0}^{1} \biggl( \frac{1+t}{2} \biggr) \,dt \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert ^{p_{1}}\,dt \biggr) ^{ \frac{1}{p_{1}}} \\& \qquad {}\times\biggl( \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{r_{1}} \int _{0}^{1} \biggl( \frac{1+t}{2} \biggr) \,dt+ \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{r_{1}} \int _{0}^{1} \biggl( \frac{1-t}{2} \biggr) \,dt \biggr) ^{\frac{1}{r_{1}}} \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \rho \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \Delta ( 1 ) \sigma - \Delta ( t ) \bigr\vert ^{p_{1}}\,dt \biggr) ^{ \frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \biggr], \end{aligned}$$

which completes the proof. □

Corollary 10

In Theorem 5, if we set \(\varphi ( t ) =t\), then we obtain the following inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{2} \biggl[ ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \biggr] - \frac{1}{\kappa _{2}-\kappa _{1}}\int _{\kappa _{1}}^{\kappa _{2}} \digamma (t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl[ \bigl( \Pi _{3} ( \rho ) \bigr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \\& \qquad {} + \bigl( \Pi _{3} ( \sigma ) \bigr) ^{ \frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \biggr] , \end{aligned}$$

where

$$ \Pi _{3} ( \tau ) = \frac{\tau ^{p_{1}+1}+ ( 1-\tau ) ^{p_{1}+1}}{p_{1}+1}. $$

Corollary 11

In Theorem 5, if we take \(\varphi ( t ) = \frac{t^{\alpha }}{\Gamma ( \alpha ) }\), then we obtain the following inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa _{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} - \frac{2^{\alpha }\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{ \kappa _{1}+}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2} \biggl[ \biggl( \int _{0}^{1} \bigl\vert t^{\alpha }- \rho \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \sigma -t^{\alpha } \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \biggr] . \end{aligned}$$

Corollary 12

In Theorem 5, if we set \(\varphi ( t ) = \frac{t^{\frac{\alpha }{k}}}{k\Gamma _{k} ( \alpha ) }\), then we obtain the following inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert ( 1-\sigma ) \digamma ( \kappa_{1} ) + ( \sigma +\rho ) \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) + ( 1-\rho ) \digamma ( \kappa _{2} ) \\& \qquad {} - \frac{2^{\frac{\alpha }{k}}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa_{2}-,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2} \biggl[ \biggl( \int _{0}^{1} \bigl\vert t^{\frac{\alpha }{k}}- \rho \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {} + \biggl( \int _{0}^{1} \bigl\vert \sigma -t^{ \frac{\alpha }{k}} \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \biggl( \frac{3 \vert \digamma^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \biggr] . \end{aligned}$$

Special cases

In this section, we give some special cases of our main results.

Remark 1

From Lemma 1, we give the following identities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following identity:

$$\begin{aligned}& \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] -\frac{1}{2\Delta ( 1 ) } \biggl[ J_{\kappa _{1}+}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl[ \int _{0}^{1} \biggl( \frac{\Delta ( t ) }{2}- \frac{\Delta ( 1 ) }{3} \biggr) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2} \kappa _{2} \biggr) \,dt \\& \qquad {} + \int _{0}^{1} \biggl( \frac{\Delta ( 1 ) }{3}- \frac{\Delta ( t ) }{2} \biggr) \digamma ^{ \prime } \biggl( \frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \,dt \biggr], \end{aligned}$$

which is given by Ertuğral and Sarikaya in [16].

2. For \(\rho =\sigma =1\), we have the following identity:

$$\begin{aligned}& \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) - \frac{1}{2\Delta ( 1 ) } \biggl[ J_{ \kappa _{1}+}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \int _{0}^{1} \bigl[ \bigl( \Delta ( t ) - \Delta ( 1 ) \bigr) \bigr] \biggl[ \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2} \kappa _{2} \biggr) -\digamma ^{\prime } \biggl( \frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \biggr] \,dt, \end{aligned}$$

which is given by Ertuğral et al. in [17].

3. For \(\rho =\sigma =0\), we have the following identity:

$$\begin{aligned}& \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{1}{2\Delta ( 1 ) } \biggl[ J_{\kappa _{1}+}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{\alpha }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \int _{0}^{1} \biggl[ \Delta ( t ) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa_{1}+\frac{1+t}{2} \kappa_{2} \biggr) \,dt- \int _{0}^{1}\Delta ( t ) \digamma ^{\prime } \biggl( \frac{1+t}{2}\kappa _{1}+\frac{1-t}{2} \kappa_{2} \biggr) \,dt \biggr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

Remark 2

From Corollary 1, we have the following identities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following new identity:

$$\begin{aligned}& \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] -\frac{1}{\kappa _{2}-\kappa _{1}} \int _{ \kappa_{1}}^{\kappa _{2}}\digamma (t)\,dt \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2} \biggl[ \int _{0}^{1} \biggl( \frac{t}{2}- \frac{2}{3} \biggr) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2} \kappa_{2} \biggr) \,dt \\& \qquad {} + \int _{0}^{1} \biggl( \frac{2}{3}- \frac{t}{2} \biggr) \digamma^{\prime } \biggl( \frac{1+t}{2}\kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \,dt \biggr] . \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following identity:

$$\begin{aligned}& \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{1}{\kappa _{2}-\kappa _{1}} \int _{\kappa _{1}}^{ \kappa_{2}}\digamma (t)\,dt \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4} \int _{0}^{1}t \biggl[ \digamma ^{\prime } \biggl( \frac{1-t}{2} \kappa _{1}+\frac{1+t}{2}\kappa _{2} \biggr) -\digamma ^{\prime } \biggl( \frac{1+t}{2}\kappa_{1}+\frac{1-t}{2} \kappa _{2} \biggr) \biggr] \,dt. \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following identity:

$$\begin{aligned}& \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) - \frac{1}{\kappa _{2}-\kappa _{1}} \int _{\kappa _{1}}^{\kappa _{2}} \digamma (t)\,dt \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4} \int _{0}^{1} ( t-1 ) \biggl[ \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa_{1}+\frac{1+t}{2}\kappa _{2} \biggr) - \digamma ^{\prime } \biggl( \frac{1+t}{2} \kappa _{1}+\frac{1-t}{2} \kappa_{2} \biggr) \biggr] \,dt. \end{aligned}$$

Remark 3

From Corollary 2, we have the following identities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following new identity:

$$\begin{aligned}& \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {} - \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa_{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{ \kappa_{1}+}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2} \biggl[ \int _{0}^{1} \biggl( \frac{t^{\alpha }}{2}- \frac{1}{3} \biggr) \digamma^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2}\kappa _{2} \biggr) \\& \qquad {} - \int _{0}^{1} \biggl( \frac{t^{\alpha }}{2}- \frac{1}{3} \biggr) \digamma ^{\prime } \biggl( \frac{1+t}{2} \kappa _{1}+\frac{1-t}{2} \kappa _{2} \biggr) \,dt \biggr], \end{aligned}$$

which is given by Chen and Huang in [8].

2. For \(\rho =\sigma =0\), we have the following identity:

$$\begin{aligned}& \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4} \biggl[ \int _{0}^{1}t^{ \alpha }\digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2} \biggr) \,dt- \int _{0}^{1}t^{\alpha }\digamma ^{\prime } \biggl( \frac{1+t}{2}\kappa _{1}+\frac{1-t}{2}\kappa _{2} \biggr) \,dt \biggr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

3. For \(\rho =\sigma =1\), we have the following identity:

$$\begin{aligned}& \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) - \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{\alpha }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}-}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4} \biggl[ \int _{0}^{1} \bigl( t^{\alpha }-1 \bigr) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2}\kappa _{2} \biggr) \,dt \\& \qquad {} - \int _{0}^{1} \bigl( t^{\alpha }-1 \bigr) \digamma ^{\prime } \biggl( \frac{1+t}{2} \kappa _{1}+\frac{1-t}{2} \kappa_{2} \biggr) \,dt \biggr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

Remark 4

From Corollary 3, we have the following identities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following identity:

$$\begin{aligned}& \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +2\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {} - \frac{2^{\frac{\alpha }{2}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}-,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{2} \biggl[ \int _{0}^{1} \biggl( \frac{t^{\frac{\alpha }{k}}}{2}- \frac{1}{3} \biggr) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+ \frac{1+t}{2} \kappa _{2} \biggr) \,dt \\& \qquad {} - \int _{0}^{1} \biggl( \frac{t^{\frac{\alpha }{k}}}{2}- \frac{1}{3} \biggr) \digamma ^{\prime } \biggl( \frac{1+t}{2} \kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \,dt \biggr], \end{aligned}$$

which is given by Ertuğral and Sarikaya in [16].

2. For \(\rho =\sigma =0\), we have the following identity:

$$\begin{aligned}& \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{2^{\frac{\alpha }{2}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{ \kappa _{1}+,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-,k}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4} \biggl[ \int _{0}^{1} \bigl( t^{\frac{\alpha }{k}} \bigr) \digamma ^{\prime } \biggl( \frac{1-t}{2}\kappa _{1}+\frac{1+t}{2} \kappa _{2} \biggr) \,dt \\& \qquad {} - \int _{0}^{1} \bigl( t^{\frac{\alpha }{k}} \bigr) \digamma ^{\prime } \biggl( \frac{1+t}{2} \kappa _{1}+\frac{1-t}{2} \kappa_{2} \biggr) \,dt \biggr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

3. For \(\rho =\sigma =1\), we have the following identity:

$$\begin{aligned}& \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) - \frac{2^{\frac{\alpha }{2}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa_{2}-,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4} \biggl[ \int _{0}^{1} \bigl( t^{\frac{\alpha }{k}}-1 \bigr) \digamma ^{\prime } \biggl( \frac{1-t}{2} \kappa _{1}+\frac{1+t}{2} \kappa_{2} \biggr) \,dt \\& \qquad {} - \int _{0}^{1} \bigl( t^{\frac{\alpha }{k}}-1 \bigr) \digamma^{\prime } \biggl( \frac{1+t}{2} \kappa _{1}+ \frac{1-t}{2}\kappa _{2} \biggr) \,dt \biggr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

Remark 5

From Theorem 3, we have the following new inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +2\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {} -\frac{1}{2\Delta ( 1 ) } \biggl[ {}_{ \kappa _{1}+}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{8\Delta ( 1 ) } \biggl[ \Pi _{1}^{\varphi } \biggl( \frac{2}{3} \biggr) + \Pi _{2}^{\varphi } \biggl( \frac{2}{3} \biggr) \biggr] \bigl[ \bigl\vert \digamma^{\prime } ( \kappa _{2} ) \bigr\vert + \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert \bigr], \end{aligned}$$

which is given by Ertuğral and Sarikaya in [16].

2. For \(\rho =\sigma =0\), we have the following inequality:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{1}{2\Delta ( 1 ) } \biggl[ {}_{\kappa _{1}+}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) \bigr\vert \,dt \biggr) \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert + \bigl\vert \digamma^{\prime } ( \kappa _{1} ) \bigr\vert \bigr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

3. For \(\rho =\sigma =1\), we have the following inequality:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{1}{2\Delta ( 1 ) } \biggl[ {}_{\kappa _{1}+}I_{\varphi }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +{}_{ \kappa _{2}-}I_{\varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad = \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) -\Delta ( 1 ) \bigr\vert \,dt \biggr) \bigl[ \bigl\vert \digamma^{\prime } ( \kappa _{2} ) \bigr\vert + \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert \bigr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

Remark 6

From Corollary 4, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson’s inequality for Riemann integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] - \frac{1}{\kappa _{2}-\kappa _{1}}\int _{\kappa _{1}}^{\kappa _{2}} \digamma(t)\,dt \biggr\vert \\& \quad \leq \frac{5 ( \kappa _{2}-\kappa _{1} ) }{72} \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigr], \end{aligned}$$

which is given by Sarikaya et al. in [33, 34].

2. For \(\rho =\sigma =0\), we have the following trapezoid inequality for Riemann integrals:

$$ \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{1}{\kappa _{2}-\kappa _{1}} \int _{ \kappa _{1}}^{\kappa _{2}}\digamma (t)\,dt \biggr\vert \leq \frac{\kappa _{2}-\kappa _{1}}{8} \bigl[ \bigl\vert \digamma^{\prime } ( \kappa _{1} ) \bigr\vert + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigr], $$

which is proved by Dragomir and Agarwal [10].

3. For \(\rho =\sigma =1\), we have the following midpoint inequality for Riemann integrals:

$$ \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{1}{\kappa _{2}-\kappa _{1}} \int _{\kappa _{1}}^{\kappa _{2}} \digamma(t)\,dt \biggr\vert \leq \frac{\kappa _{2}-\kappa _{1}}{8} \bigl[ \bigl\vert \digamma^{\prime } ( \kappa_{1} ) \bigr\vert + \bigl\vert \digamma ^{ \prime } ( \kappa _{2} ) \bigr\vert \bigr], $$

which is given by Kirmaci in [23].

Remark 7

From Corollary 5, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson’s inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {} - \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{\alpha }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}-}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2} \biggl( \frac{\alpha }{\alpha +1} \biggl( \frac{2}{3} \biggr) ^{ \frac{\alpha +1}{\alpha }}+\frac{1}{2 ( \alpha +1 ) }- \frac{1}{3} \biggr) \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigr], \end{aligned}$$

which is given by Ertuğral and Sarikaya in [16].

2. For \(\rho =\sigma =0\), we have the following trapezoidal-type inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa_{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4 ( \alpha +1 ) } \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa_{1} ) \bigr\vert + \bigl\vert \digamma ^{ \prime } ( \kappa _{2} ) \bigr\vert \bigr] . \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{\alpha }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}-}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\alpha ( \kappa _{2}-\kappa _{1} ) }{4 ( \alpha +1 ) } \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

Remark 8

From Corollary 6, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {} - \frac{2^{\frac{\alpha }{k}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa_{2}-,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2} \biggl( \frac{\alpha }{\alpha +k} \biggl( \frac{2}{3} \biggr) ^{ \frac{\alpha +k}{\alpha }}+\frac{k}{2 ( \alpha +k ) }- \frac{1}{3} \biggr) \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigr]. \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following trapezoidal-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{2^{\frac{\alpha }{k}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{ \kappa_{1}+,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-,k}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{k ( \kappa _{2}-\kappa _{1} ) }{4 ( \alpha +k ) } \bigl[ \bigl\vert \digamma ^{ \prime } ( \kappa _{1} ) \bigr\vert + \bigl\vert \digamma^{\prime } ( \kappa _{2} ) \bigr\vert \bigr] . \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert 2\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) - \frac{2^{\frac{\alpha }{k}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\alpha ( \kappa _{2}-\kappa _{1} ) }{4 ( \alpha +k ) } \bigl[ \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert + \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert \bigr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

Remark 9

From Theorem 4, we have the following new inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] -\frac{1}{2\Delta ( 1 ) } \biggl[ {}_{ \kappa_{1}+}I_{\varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl( \int _{0}^{1} \biggl\vert \Delta ( t ) - \Delta ( 1 ) \frac{2}{3} \biggr\vert \,dt \biggr) ^{1- \frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{\Pi _{1}^{\varphi } ( \frac{2}{3} ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( \frac{2}{3} ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \frac{\Pi _{1}^{\varphi } ( \frac{2}{3} ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( \frac{2}{3} ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \biggr] . \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following trapezoidal-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{1}{2\Delta ( 1 ) } \biggl[ {}_{\kappa _{1}+}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) \bigr\vert \,dt \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{\Pi _{1}^{\varphi } ( 0 ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( 0 ) \vert \digamma^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \frac{\Pi _{1}^{\varphi } ( 0 ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( 0 ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \biggr] . \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{1}{2\Delta ( 1 ) } \biggl[ {}_{\kappa _{1}+}I_{\varphi }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +{}_{ \kappa _{2}-}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) - \Delta ( 1 ) \bigr\vert \,dt \biggr) ^{1-\frac{1}{p_{1}}} \biggl[ \biggl( \frac{\Pi _{1}^{\varphi } ( 1 ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( 1 ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \biggl( \frac{\Pi _{1}^{\varphi } ( 1 ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+\Pi _{2}^{\varphi } ( 1 ) \vert \digamma^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

Remark 10

From Corollary 7, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for Riemann integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] - \frac{1}{\kappa _{2}-\kappa _{1}}\int _{\kappa _{1}}^{\kappa _{2}} \digamma(t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{8} \biggl( \frac{5}{9} \biggr) ^{1-\frac{1}{p_{1}}} \biggl[ \biggl( \frac{29 \vert \digamma^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+61 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{162} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {}+ \biggl( \frac{29 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+61 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{324} \biggr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$

which is given by Sarikaya et al. in [33, 34]

2. For \(\rho =\sigma =0\), we have the following trapezoid-type inequality for Riemann integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{1}{\kappa _{2}-\kappa _{1}} \int _{ \kappa _{1}}^{\kappa _{2}}\digamma (t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{8} \biggl[ \biggl( \frac{ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+5 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{6} \biggr) ^{\frac{1}{p_{1}}}+ \biggl( \frac{ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+5 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{6} \biggr) ^{\frac{1}{p_{1}}} \biggr]. \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for Riemann integrals:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{1}{\kappa _{2}-\kappa _{1}} \int _{\kappa _{1}}^{\kappa _{2}} \digamma(t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{8} \biggl[ \biggl( \frac{ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+2 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{3} \biggr) ^{\frac{1}{p_{1}}}+ \biggl( \frac{ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+2 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{3} \biggr) ^{\frac{1}{p_{1}}} \biggr]. \end{aligned}$$

Remark 11

From Corollary 8, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {} - \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}^{+}}^{\alpha }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}-}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{4\alpha }{\alpha +1} \biggl( \frac{2}{3} \biggr) ^{ \frac{\alpha +1}{\alpha }}+\frac{2}{\alpha +1}- \frac{4}{3} \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \Pi _{1}^{\alpha } \biggl( \frac{2}{3} \biggr) \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{\alpha } \biggl( \frac{2}{3} \biggr) \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{ \frac{1}{p_{1}}} \\& \qquad {} + \biggl( \Pi _{1}^{\alpha } \biggl( \frac{2}{3} \biggr) \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{\alpha } \biggl( \frac{2}{3} \biggr) \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \biggr) ^{ \frac{1}{p_{1}}} \biggr]. \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following trapezoidal-type inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa_{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}^{+}}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{2\alpha }{\alpha +1} \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+ ( 2\alpha +3 ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr) ^{\frac{1}{p_{1}}}+ \biggl( \frac{ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+ ( 2\alpha +3 ) \vert \digamma ^{\prime } ( \kappa_{1} ) \vert ^{p_{1}}}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr) ^{\frac{1}{p_{1}}} \biggr]. \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}^{+}}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}-}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{2\alpha }{\alpha +1} \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{\alpha ( \alpha +3 ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+\alpha ( 3\alpha +5 ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{2 ( \alpha +1 ) ( \alpha +2 ) } \biggr) ^{ \frac{1}{p_{1}}} \\& \qquad {} + \biggl( \frac{\alpha ( \alpha +3 ) \vert \digamma^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+\alpha ( 3\alpha +5 ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{2 ( \alpha +1 ) ( \alpha +2 ) } \biggr) ^{ \frac{1}{p_{1}}} \biggr]. \end{aligned}$$

Remark 12

From Corollary 9, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {} - \frac{2^{\frac{\alpha }{k}-1}\Gamma _{k} ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa_{2}-,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{4\alpha }{\alpha +k} \biggl( \frac{2}{3} \biggr) ^{ \frac{\alpha +k}{\alpha }}+\frac{2k}{\alpha +k}- \frac{4}{3} \biggr) ^{1-\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \Pi _{1}^{\frac{\alpha }{k}} \biggl( \frac{2}{3} \biggr) \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{ \frac{\alpha }{k}} \biggl( \frac{2}{3} \biggr) \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \biggl( \Pi _{1}^{\frac{\alpha }{k}} \biggl( \frac{2}{3} \biggr) \bigl\vert \digamma ^{\prime } ( \kappa _{2} ) \bigr\vert ^{p_{1}}+\Pi _{2}^{ \frac{\alpha }{k}} \biggl( \frac{2}{3} \biggr) \bigl\vert \digamma ^{\prime } ( \kappa _{1} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \biggr] . \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following trapezoidal-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) - \frac{2^{\frac{\alpha }{k}-1}\Gamma _{k} ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{ \kappa_{1}+,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-,k}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{2\alpha }{\alpha +k} \biggr) ^{1-\frac{1}{p_{1}}} \biggl[ \biggl( \frac{k^{2} \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+k ( 2\alpha +3k ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{ ( \alpha +k ) ( \alpha +2k ) } \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \biggl( \frac{k^{2} \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+k ( 2\alpha +3k ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{ ( \alpha +k ) ( \alpha +2k ) } \biggr) ^{\frac{1}{p_{1}}} \biggr] . \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert 2\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) - \frac{2^{\frac{\alpha }{k}-1}\Gamma _{k} ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{2\alpha }{\alpha +1} \biggr) ^{1-\frac{1}{p_{1}}} \biggl[ \biggl( \frac{\alpha ( \alpha +3k ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}+\alpha ( 3\alpha +5k ) \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}}{2 ( \alpha +k ) ( \alpha +2k ) } \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} + \biggl( \frac{\alpha ( \alpha +3k ) \vert \digamma^{\prime } ( \kappa _{2} ) \vert ^{p_{1}}+\alpha ( 3\alpha +5k ) \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{p_{1}}}{2 ( \alpha +k ) ( \alpha +2k ) } \biggr) ^{ \frac{1}{p_{1}}} \biggr]. \end{aligned}$$

Remark 13

From Theorem 5, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {}-\frac{1}{2\Delta ( 1 ) } \biggl[ {}_{ \kappa_{1}+}I_{\varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{2\Delta ( 1 ) } \biggl( \int _{0}^{1} \biggl\vert \Delta ( t ) - \frac{2}{3}\Delta ( 1 ) \biggr\vert ^{p_{1}}\,dt \biggr) ^{ \frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr] . \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following trapezoidal-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{1}{2\Delta ( 1 ) } \biggl[ {}_{\kappa _{1}+}I_{ \varphi }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +{}_{\kappa _{2}-}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) \bigr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{1}{2\Delta ( 1 ) } \biggl[ {}_{\kappa _{1}+}I_{\varphi }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +{}_{ \kappa _{2}-}I_{\varphi } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4\Delta ( 1 ) } \biggl( \int _{0}^{1} \bigl\vert \Delta ( t ) - \Delta ( 1 ) \bigr\vert ^{p_{1}}\,dt \biggr) ^{ \frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr], \end{aligned}$$

which is given by Ertuğral et al. in [17].

Remark 14

From Corollary 10, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for Riemann integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] - \frac{1}{\kappa _{2}-\kappa _{1}}\int _{\kappa _{1}}^{\kappa _{2}} \digamma(t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{2^{p_{1}+1}+1}{ ( p_{1}+1 ) 3^{p_{1}+1}} \biggr) ^{ \frac{1}{p_{1}}} \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {} + \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \biggr] . \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following trapezoid-type inequality for Riemann integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{1}{\kappa _{2}-\kappa _{1}} \int _{ \kappa _{1}}^{\kappa _{2}}\digamma (t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{1}{p_{1}+1} \biggr) ^{\frac{1}{p_{1}}} \biggl[ \biggl( \frac{3 \vert \digamma^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \biggr] . \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for Riemann integrals:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{1}{\kappa _{2}-\kappa _{1}} \int _{\kappa _{1}}^{\kappa _{2}} \digamma(t)\,dt \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{1}{p_{1}+1} \biggr) ^{\frac{1}{p_{1}}} \biggl[ \biggl( \frac{3 \vert \digamma^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \\& \qquad {}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}} \biggr], \end{aligned}$$

which is proved by Kirmaci in [23].

Remark 15

From Corollary 11, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {}- \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{\alpha }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}-}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \int _{0}^{1} \biggl\vert t^{\alpha }- \frac{2}{3} \biggr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr]. \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following trapezoidal-type inequality for Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa_{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{1}{\alpha p_{1}+1} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr]. \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for generalized fractional integrals:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{2^{\alpha -1}\Gamma ( \alpha +1 ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\alpha }} \biggl[ J_{\kappa _{1}+}^{\alpha }\digamma \biggl( \frac{\kappa_{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa _{2}-}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \int _{0}^{1} \bigl( 1-t^{\alpha } \bigr) ^{p_{1}}\,dt \biggr) ^{ \frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr] . \end{aligned}$$

Remark 16

From Corollary 12, we have the following inequalities:

1. For \(\rho =\sigma =\frac{2}{3}\), we have the following Simpson-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{1}{6} \biggl[ \digamma ( \kappa _{1} ) +4\digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) +\digamma ( \kappa _{2} ) \biggr] \\& \qquad {} - \frac{2^{\frac{\alpha }{2}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{ \kappa_{2}-,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \int _{0}^{1} \biggl\vert t^{\frac{\alpha }{k}}- \frac{2}{3} \biggr\vert ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr] . \end{aligned}$$

2. For \(\rho =\sigma =0\), we have the following trapezoidal-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \frac{\digamma ( \kappa _{1} ) +\digamma ( \kappa _{2} ) }{2}- \frac{2^{\frac{\alpha }{2}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{ \kappa_{1}+,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-,k}^{ \alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \frac{k}{\alpha p_{1}+k} \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr] . \end{aligned}$$

3. For \(\rho =\sigma =1\), we have the following midpoint-type inequality for k-Riemann–Liouville fractional integrals:

$$\begin{aligned}& \biggl\vert \digamma \biggl( \frac{\kappa _{1}+\kappa_{2}}{2} \biggr) - \frac{2^{\frac{\alpha }{2}-1}\Gamma _{k} ( \alpha +k ) }{ ( \kappa _{2}-\kappa _{1} ) ^{\frac{\alpha }{k}}} \biggl[ J_{\kappa _{1}+,k}^{\alpha } \digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) +J_{\kappa _{2}-,k}^{\alpha }\digamma \biggl( \frac{\kappa _{1}+\kappa _{2}}{2} \biggr) \biggr] \biggr\vert \\& \quad \leq \frac{\kappa _{2}-\kappa _{1}}{4} \biggl( \int _{0}^{1} \bigl( 1-t^{\frac{\alpha }{k}} \bigr) ^{p_{1}}\,dt \biggr) ^{\frac{1}{p_{1}}} \\& \qquad {} \times \biggl[ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{2} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}}{4} \biggr) ^{\frac{1}{r_{1}}}+ \biggl( \frac{3 \vert \digamma ^{\prime } ( \kappa _{1} ) \vert ^{r_{1}}+ \vert \digamma ^{\prime } ( \kappa_{2} ) \vert ^{r_{1}}}{4} \biggr) ^{ \frac{1}{r_{1}}} \biggr] . \end{aligned}$$

Concluding remarks

In this study, we present some generalized inequalities for differentiable convex functions via generalized fractional integrals. It is also shown that the results proved here are the strong generalizations of some already published ones. It is an interesting and new problem that future researchers can use the techniques of this study and obtain similar inequalities for different kinds of convexity in their work.

Availability of data and materials

Not applicable.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

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HB: conceptualization, computation, writing, reviewing and editing. SKY: computation, writing original draft. MZS: computation, writing, reviewing and editing. HY: supervision and editing. All authors read and approved the final manuscript.

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Correspondence to Hüseyin Budak.

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Budak, H., Kılınç Yıldırım, S., Sarıkaya, M.Z. et al. Some parameterized Simpson-, midpoint- and trapezoid-type inequalities for generalized fractional integrals. J Inequal Appl 2022, 40 (2022). https://doi.org/10.1186/s13660-022-02773-5

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  • DOI: https://doi.org/10.1186/s13660-022-02773-5

MSC

  • 26B25
  • 26D10
  • 26D15

Keywords

  • Simpson’s \(\frac{1}{3}\) formula
  • Integral inequalities
  • Fractional calculus
  • Convex functions