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Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals

  • Humaira Kalsoom , Hüseyin Budak EMAIL logo , Hasan Kara and Muhammad Aamir Ali
Published/Copyright: November 18, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this study, we first obtain a new identity for generalized fractional integrals which contains some parameters. Then by this equality, we establish some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals. Moreover, we show that the results proved in the main section reduce to several Simpson-, trapezoid- and midpoint-type inequalities for various values of parameters.

Keywords: Simpson’s 1/3 formula; integral inequalities; fractional calculus; co-ordinated convex functions
MSC 2010: 26D07; 26D10; 26D15; 26B15; 26B25

1 Introduction

The inequalities discovered by C. Hermite and J. Hadamard for convex functions are considered significant in the literature. These inequalities state that if F : I → R is a convex function on the interval I of real numbers and κ 1 , κ 2 ∈ I with κ 1 < κ 2 , then

(1.1) F κ 1 + κ 2 2 ≤ 1 κ 2 − κ 1 ∫ κ 1 κ 2 F ( τ 1 ) d τ 1 ≤ F ( κ 1 ) + F ( κ 2 ) 2 .

Both inequalities hold in the reversed direction if F is concave.

Over the last 20 years, numerous studies have focused on obtaining trapezoid- and midpoint-type inequalities which give bounds for the right-hand side and left-hand side of the inequality (1.1), respectively. For example, Dragomir and Agarwal first obtained trapezoid inequalities for convex functions in [1], whereas Kirmacı first, established midpoint inequalities for convex functions in [2]. In [3], Sarikaya et al. generalized the inequalities (1.1) for fractional integrals and the authors also proved some corresponding trapezoid-type inequalities. Iqbal et al. presented some fractional midpoint-type inequalities for convex functions in [4]. On the other hand, Dragomir proved Hermite-Hadamard inequalities for co-ordinated convex mappings in [5]. In [6] and [7], the authors proved midpoint- and trapezoid-type inequalities for co-ordinated convex functions, respectively. Moreover, Sarikaya obtained fractional Hermite-Hadamard inequalities and fractional trapezoid for functions with two variables in [8]. Tunç et al. presented some fractional midpoint-type inequalities for co-ordinated convex functions in [9]. In [10], Sarikaya and Ertuğral first introduced new fractional integrals which are called generalized fractional integrals, and then, they proved Hermite-Hadamard inequalities and several trapezoid- and midpoint-type inequalities for generalized fractional integrals. In addition, Turkay et al. defined the generalized fractional integrals for functions with two variables and they presented Hermite-Hadamard- and trapezoid-type inequalities for this kind of fractional integrals in [11]. For the other similar inequalities, please refer to [12,13,14, 15,16,17, 18,19].

On the other hand, the following inequality is well known in the literature as Simpson’s inequality.

Theorem 1

Suppose that F : [ κ 1 , κ 2 ] → R is a four times continuously differentiable mapping on ( κ 1 , κ 2 ) , and let ∥ F ( 4 ) ∥ ∞ = sup τ 1 ∈ ( κ 1 , κ 2 ) ∣ F ( 4 ) ( τ 1 ) ∣ < ∞ . Then, one has the inequality

1 3 F ( κ 1 ) + F ( κ 2 ) 2 + 2 F κ 1 + κ 2 2 − 1 κ 2 − κ 1 ∫ κ 1 κ 2 F ( τ 1 ) d τ 1 ≤ 1 2880 ∥ F ( 4 ) ∥ ∞ ( κ 2 − κ 1 ) 4 .

In recent years, many authors have focused on Simpson-type inequalities for various classes of functions. For example, Dragomir et al. presented new Simpson-type results and their applications to quadrature formulas in numerical integration in [20]. In addition, some inequalities of Simpson-type for η -convex functions are deduced by Alomari et al. in [21]. Afterward, Sarikaya et al. observed the variants of Simpson-type inequalities based on convexity in [22]. Moreover, some papers are devoted to Simpson inequalities for co-ordinated convex functions [23,24, 25,26]. On the other hand, some authors proved several Simpson-type inequalities for fractional integrals in [27,28, 29,30]. In addition, in [31], Ertugral and Sarikaya obtained some inequalities of Simpson-type for generalized fractional integrals. For more recent developments, one can refer to [32,33, 34,35,36, 37,38].

The aim of this paper is to obtain some parameterized inequalities for co-ordinated convex functions via generalized fractional integrals. These inequalities reduce to Simpson, trapezoid and midpoint inequalities in the case of special choice of parameters. The overall structure of the study takes the form of six sections including an introduction. The remaining part of the paper proceeds as follows: In Section 2, the generalized fractional integral operators is summarized, along with some related theorems. In Section 3, an identity involving generalized fractional integrals is presented for partial differentiable functions. Then we prove several parameterized inequalities for functions whose partial derivatives in absolute value are co-ordinated convex in Section 4. Moreover, some special cases of the results in Section 4 are presented in Section 5. Finally, some conclusions and further directions of research are discussed in Section 6.

A formal definition for co-ordinated convex function may be stated as follows:

Definition 1

A function F : Δ ≔ [ κ 1 , κ 2 ] × [ κ 3 , κ 4 ] → R is called co-ordinated convex on Δ , for all ( x , u ) , ( y , v ) ∈ Δ and t , s ∈ [ 0 , 1 ] , if it satisfies the following inequality:

(1.2) F ( t x + ( 1 − t ) y , s u + ( 1 − s ) v ) ≤ t s F ( x , u ) + t ( 1 − s ) F ( x , v ) + s ( 1 − t ) F ( y , u ) + ( 1 − t ) ( 1 − s ) F ( y , v ) .

The mapping F is a co-ordinated concave on Δ if the inequality (1.2) holds in the reversed direction for all t , s ∈ [ 0 , 1 ] and ( x , u ) , ( y , v ) ∈ Δ .

2 Generalized fractional integrals

Fractional calculus and applications have application areas in many different fields such as physics, chemistry and engineering as well as mathematics. The application of arithmetic carried out in classical analysis in fractional analysis is very important in terms of obtaining more realistic results in the solution of many problems. Many real dynamical systems are better characterized by using non-integer order dynamic models based on fractional computation. While integer orders are a model that is not suitable for nature in classical analysis, fractional computation in which arbitrary orders are examined enables us to obtain more realistic approaches.

In this section, we summarize the generalized fractional integrals defined by Sarikaya and Ertuğral in [10].

Definition 2

Let F : [ κ 1 , κ 2 ] → R be an integrable function. The left-sided and right-sided generalized fractional integral operators are given by

(2.1) I φ κ 1 + F ( τ 1 ) = ∫ κ 1 τ 1 φ ( τ 1 − ξ ) τ 1 − ξ F ( ξ ) d ξ , τ 1 > κ 1 ,

and

(2.2) I φ κ 2 − F ( τ 1 ) = ∫ τ 1 κ 2 φ ( ξ − τ 1 ) ξ − τ 1 F ( ξ ) d ξ , τ 1 < κ 2 ,

respectively. Here, the function φ : [ 0 , ∞ ) → [ 0 , ∞ ) satisfies the condition

∫ 0 1 φ ( ξ ) ξ d ξ < ∞ .

The most important feature of generalized fractional integrals is that they generalize some types of fractional integrals such as Riemann-Liouville fractional integral, k -Riemann-Liouville fractional integral, Katugampola fractional integrals, conformable fractional integral, Hadamard fractional integrals, etc. These important special cases of the integral operators (2.1) and (2.2) are mentioned below.

  1. If we take φ ( ξ ) = ξ , the operators (2.1) and (2.2) reduce to the Riemann integral as follows:

    I κ 1 + F ( τ 1 ) = ∫ κ 1 τ 1 F ( ξ ) d ξ , τ 1 > κ 1 , I κ 2 − F ( τ 1 ) = ∫ τ 1 κ 2 F ( ξ ) d ξ , τ 1 < κ 2 .

  2. If we take φ ( ξ ) = ξ α Γ ( α ) , the operators (2.1) and (2.2) reduce to the Riemann-Liouville fractional integral as follows:

    J κ 1 + α F ( τ 1 ) = 1 Γ ( α ) ∫ κ 1 τ 1 ( τ 1 − ξ ) α − 1 F ( ξ ) d ξ , τ 1 > κ 1 , J κ 2 − α F ( τ 1 ) = 1 Γ ( α ) ∫ τ 1 κ 2 ( ξ − τ 1 ) α − 1 F ( ξ ) d ξ , τ 1 < κ 2 .

  3. If we take φ ( ξ ) = 1 k Γ k ( α ) ξ α k , the operators (2.1) and (2.2) reduce to the k -Riemann-Liouville fractional integral as follows:

    J κ 1 + , k α F ( τ 1 ) = 1 k Γ k ( α ) ∫ κ 1 τ 1 ( τ 1 − ξ ) α k − 1 F ( ξ ) d ξ , τ 1 > κ 1 , J κ 2 − , k α F ( τ 1 ) = 1 k Γ k ( α ) ∫ τ 1 κ 2 ( ξ − τ 1 ) α k − 1 F ( ξ ) d ξ , τ 1 < κ 2 ,

    where

    Γ k ( α ) = ∫ 0 ∞ ξ α − 1 e − ξ k k d ξ , ℛ ( α ) > 0

    and

    Γ k ( α ) = k α k − 1 Γ α k , ℛ ( α ) > 0 ; k > 0

    are given by Mubeen and Habibullah in [39].

In the literature, there are several papers on inequalities for generalized fractional integrals. Some of them please refer to [31,40,41, 42,43,44, 45,46,47].

Inspired by this definition, Turkay et al. [11] give the following definitions:

Definition 3

Let F ∈ L 1 ( [ κ 1 , κ 2 ] × [ κ 3 , κ 4 ] ) . The generalized Riemann-Liouville fractional integrals I φ , ψ κ 1 + , κ 3 + , I φ , ψ κ 1 + , κ 4 − , I φ , ψ κ 2 − , κ 3 + , I φ , ψ κ 2 − , κ 4 − are defined by

(2.3) I φ , ψ κ 1 + , κ 3 + F ( τ 1 , τ 2 ) = ∫ κ 1 τ 1 ∫ κ 3 τ 2 φ ( τ 1 − ξ ) τ 1 − ξ ψ ( τ 2 − η ) τ 2 − η F ( ξ , η ) d η d ξ , τ 1 > κ 1 , τ 2 > κ 3 ,

(2.4) I φ , ψ κ 1 + , κ 4 − F ( τ 1 , τ 2 ) = ∫ κ 1 τ 1 ∫ τ 2 κ 4 φ ( τ 1 − ξ ) τ 1 − ξ ψ ( η − τ 2 ) η − τ 2 F ( ξ , η ) d η d ξ , τ 1 > κ 1 , τ 2 < κ 4 ,

(2.5) I φ , ψ κ 2 − , κ 3 + F ( τ 1 , τ 2 ) = ∫ τ 1 κ 2 ∫ κ 3 τ 2 φ ( ξ − τ 1 ) ξ − τ 1 ψ ( τ 2 − η ) τ 2 − η F ( ξ , η ) d η d ξ , τ 1 < κ 2 , τ 2 > κ 3

and

(2.6) I φ , ψ κ 2 − , κ 4 − F ( τ 1 , τ 2 ) = ∫ τ 1 κ 2 ∫ τ 2 κ 4 φ ( ξ − τ 1 ) ξ − τ 1 ψ ( η − τ 2 ) η − τ 2 F ( ξ , η ) d η d ξ , τ 1 < κ 2 , τ 2 < κ 4 ,

where φ , ψ : [ 0 , ∞ ) → [ 0 , ∞ ) are functions which satisfy ∫ 0 1 φ ( ξ ) ξ d ξ < ∞ and ∫ 0 1 ψ ( η ) η d η < ∞ , respectively.

In this definition, known fractional integrals can be obtained by some special choices. For example,

  1. If we take φ ( ξ ) = ξ and ψ ( η ) = η , then the operators (2.3), (2.4), (2.5) and (2.6) transform into the Riemann integrals on coordinates, respectively, as follows:

    I κ 1 + , κ 3 + F ( τ 1 , τ 2 ) = ∫ κ 1 τ 1 ∫ κ 3 τ 2 F ( ξ , η ) d η d ξ , τ 1 > κ 1 , τ 2 > κ 3 , I κ 1 + , κ 4 F ( τ 1 , τ 2 ) = ∫ κ 1 τ 1 ∫ τ 2 κ 4 F ( ξ , η ) d η d ξ , τ 1 > κ 1 , τ 2 < κ 4 , I κ 2 − , κ 3 + F ( τ 1 , τ 2 ) = ∫ τ 1 κ 2 ∫ κ 3 τ 2 F ( ξ , η ) d η d ξ , τ 1 < κ 2 , τ 2 > κ 3

    and

    I κ 2 − , κ 4 − F ( τ 1 , τ 2 ) = ∫ τ 1 κ 2 ∫ τ 2 κ 4 F ( ξ , η ) d η d ξ , τ 1 < κ 2 , τ 2 < κ 4 .

  2. If we take φ ( ξ ) = ξ α Γ ( α ) , ψ ( η ) = η β Γ ( β ) , then for α , β > 0 the operators (2.3), (2.4), (2.5) and (2.6) transform into the Riemann-Liouville fractional integrals on coordinates [8], respectively, as follows:

    J κ 1 + , κ 3 + α , β F ( τ 1 , τ 2 ) = 1 Γ ( α ) Γ ( β ) ∫ κ 1 τ 1 ∫ κ 3 τ 2 ( τ 1 − ξ ) α − 1 ( τ 2 − η ) β − 1 F ( ξ , η ) d η d ξ , τ 1 > κ 1 , τ 2 > κ 3 , J κ 1 + , κ 4 α , β F ( τ 1 , τ 2 ) = 1 Γ ( α ) Γ ( β ) ∫ κ 1 τ 1 ∫ τ 2 κ 4 ( τ 1 − ξ ) α − 1 ( η − τ 2 ) β − 1 F ( ξ , η ) d η d ξ , τ 1 > κ 1 , τ 2 < κ 4 , J κ 2 − , κ 3 + α , β F ( τ 1 , τ 2 ) = 1 Γ ( α ) Γ ( β ) ∫ τ 1 κ 2 ∫ κ 3 τ 2 ( ξ − τ 1 ) α − 1 ( τ 2 − η ) β − 1 F ( ξ , η ) d η d ξ , τ 1 < κ 2 , τ 2 > κ 3

    and

    J κ 2 − , κ 4 − α , β F ( τ 1 , τ 2 ) = 1 Γ ( α ) Γ ( β ) ∫ τ 1 κ 2 ∫ τ 2 κ 4 ( ξ − τ 1 ) α − 1 ( η − τ 2 ) β − 1 F ( ξ , η ) d η d ξ , τ 1 < κ 2 , τ 2 < κ 4 ,

    where Γ is the gamma function.

  3. If we take φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = η β k k Γ k ( β ) , for α , β , k > 0 , then the operators (2.3), (2.4), (2.5) and (2.6) transform into the Riemann-Liouville k -fractional integrals on coordinates [48], respectively, as follows:

    J κ 1 + , κ 3 + α , β , k F ( τ 1 , τ 2 ) = 1 k 2 Γ k ( α ) Γ k ( β ) ∫ κ 1 τ 1 ∫ κ 3 τ 2 ( τ 1 − ξ ) α k − 1 ( τ 2 − η ) β k − 1 F ( ξ , η ) d η d ξ , τ 1 > κ 1 , τ 2 > κ 3 , J κ 1 + , κ 4 − α , β , k F ( τ 1 , τ 2 ) = 1 k 2 Γ k ( α ) Γ k ( β ) ∫ κ 1 τ 1 ∫ τ 2 κ 4 ( τ 1 − ξ ) α k − 1 ( η − τ 2 ) β k − 1 F ( ξ , η ) d η d ξ , τ 1 > κ 1 , τ 2 < κ 4 , J κ 2 − , κ 3 + α , β , k F ( τ 1 , τ 2 ) = 1 k 2 Γ k ( α ) Γ k ( β ) ∫ τ 1 κ 2 ∫ κ 3 τ 2 ( ξ − τ 1 ) α k − 1 ( τ 2 − η ) β k − 1 F ( ξ , η ) d η d ξ , τ 1 < κ 2 , τ 2 > κ 3

    and

    J κ 2 − , κ 4 − α , β , k F ( τ 1 , τ 2 ) = 1 k 2 Γ k ( α ) Γ k ( β ) ∫ τ 1 κ 2 ∫ τ 2 κ 4 ( ξ − τ 1 ) α k − 1 ( η − τ 2 ) β k − 1 F ( ξ , η ) d η d ξ , τ 1 < κ 2 , τ 2 < κ 4 ,

    where Γ k is the k -gamma function.

3 An Identity

Throughout this study, for brevity, we define

(3.1) Δ ( ξ ) = ∫ 0 ξ φ ( ( κ 2 − κ 1 ) u ) u d u , Λ ( η ) = ∫ 0 η ψ ( ( κ 4 − κ 3 ) u ) u d u .

Now we give an identity for generalized fractional integrals.

Lemma 1

Let F : Δ → R be a twice partially differentiable mapping on Δ ∘ . If ∂ 2 F ∂ ξ ∂ η ∈ L ( Δ ) , then we have the following equality:

Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) = ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 w ( ξ , η ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d η d ξ

for all λ i , μ i ≥ 0 , i = 1 , 2 , where the mapping w : [ 0 , 1 ] × [ 0 , 1 ] → R is defined by

w ( ξ , η ) = ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ( Λ ( η ) − Λ ( 1 ) λ 2 ) , 0 ≤ ξ ≤ 1 2 , 0 ≤ η ≤ 1 2 , ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ( Λ ( η ) − Λ ( 1 ) μ 2 ) , 0 ≤ ξ ≤ 1 2 , 1 2 ≤ η ≤ 1 , ( Δ ( ξ ) − Δ ( 1 ) μ 1 ) ( Λ ( η ) − Λ ( 1 ) λ 2 ) , 1 2 ≤ ξ ≤ 1 , 0 ≤ η ≤ 1 2 , ( Δ ( ξ ) − Δ ( 1 ) μ 1 ) ( Λ ( η ) − Λ ( 1 ) μ 2 ) , 1 2 ≤ ξ ≤ 1 , 1 2 ≤ η ≤ 1

and

Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) = Δ ( 1 ) ( μ 1 − λ 1 ) Λ ( 1 ) ( μ 2 − λ 2 ) F κ 1 + κ 2 2 , κ 3 + κ 4 2 + Δ ( 1 ) λ 1 Λ ( 1 ) ( μ 2 − λ 2 ) F κ 1 , κ 3 + κ 4 2 + Δ ( 1 ) ( 1 − μ 1 ) Λ ( 1 ) ( μ 2 − λ 2 ) F κ 2 , κ 3 + κ 4 2 + Δ ( 1 ) ( μ 1 − λ 1 ) Λ ( 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + Δ ( 1 ) ( μ 1 − λ 1 ) Λ ( 1 ) ( 1 − μ 2 ) F κ 1 + κ 2 2 , κ 4 + Δ ( 1 ) λ 1 Λ ( 1 ) λ 2 F ( κ 1 , κ 3 ) + Δ ( 1 ) ( 1 − μ 1 ) Λ ( 1 ) λ 2 F ( κ 2 , κ 3 ) + Δ ( 1 ) λ 1 Λ ( 1 ) ( 1 − μ 2 ) F ( κ 1 , κ 4 ) + Δ ( 1 ) ( 1 − μ 1 ) Λ ( 1 ) ( 1 − μ 2 ) F ( κ 2 , κ 4 ) − Λ ( 1 ) ( μ 2 − λ 2 ) I φ κ 1 + κ 2 2 − F κ 1 , κ 3 + κ 4 2 + I φ κ 1 + κ 2 2 + F κ 2 , κ 3 + κ 4 2 − Δ ( 1 ) ( μ 1 − λ 1 ) I ψ κ 3 + κ 4 2 − F κ 1 + κ 2 2 , κ 3 + I ψ κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 4 − Δ ( 1 ) λ 1 I ψ κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I ψ κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) − Λ ( 1 ) λ 2 I φ κ 1 + κ 2 2 − F ( κ 1 , κ 3 ) + I φ κ 1 + κ 2 2 + F ( κ 2 , κ 3 ) − Λ ( 1 ) ( 1 − μ 2 ) I φ κ 1 + κ 2 2 − F ( κ 1 , κ 4 ) + I φ κ 1 + κ 2 2 + F ( κ 2 , κ 4 ) − Δ ( 1 ) ( 1 − μ 1 ) I ψ κ 3 + κ 4 2 − F ( κ 2 , κ 3 ) + I ψ κ 3 + κ 4 2 + F ( κ 2 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 − F ( κ 2 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 + F ( κ 2 , κ 4 ) .

Proof

From the definition of the mapping w ( ξ , η ) , we have

( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 w ( ξ , η ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d η d ξ = ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∫ 0 1 2 ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ( Λ ( η ) − Λ ( 1 ) λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ∫ 0 1 2 ∫ 1 2 1 ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ( Λ ( η ) − Λ ( 1 ) μ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η

+ ∫ 1 2 1 ∫ 0 1 2 ( Δ ( ξ ) − Δ ( 1 ) μ 1 ) ( Λ ( η ) − Λ ( 1 ) λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ∫ 1 2 1 ∫ 1 2 1 ( Δ ( ξ ) − Δ ( 1 ) μ 1 ) ( Λ ( η ) − Λ ( 1 ) μ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) [ J 1 + J 2 + J 3 + J 4 ] .

Integrating by parts, one can easily obtain

J 1 = ∫ 0 1 2 ∫ 0 1 2 ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ( Λ ( η ) − Λ ( 1 ) λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = 1 ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) Δ 1 2 − Δ ( 1 ) λ 1 Λ 1 2 − Λ ( 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + κ 4 2 + Δ ( 1 ) λ 1 Λ 1 2 − Λ ( 1 ) λ 2 F κ 1 , κ 3 + κ 4 2 − Λ 1 2 − Λ ( 1 ) λ 2 × ∫ κ 1 κ 1 + κ 2 2 F τ 1 , κ 3 + κ 4 2 φ ( τ 1 − κ 1 ) τ 1 − κ 1 d τ 1 + Δ 1 2 − Δ ( 1 ) λ 1 Λ ( 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + Δ ( 1 ) λ 1 Λ ( 1 ) λ 2 F ( κ 1 , κ 3 ) − Λ ( 1 ) λ 2 ∫ 0 κ 1 + κ 2 2 φ ( τ 1 − κ 1 ) τ 1 − κ 1 F ( τ 1 , κ 3 ) d τ 1 − Δ 1 2 − Δ ( 1 ) λ 1 × ∫ κ 3 κ 3 + κ 4 2 F κ 1 + κ 2 2 , τ 2 ψ ( τ 2 − κ 3 ) τ 2 − κ 3 d τ 2 − Δ ( 1 ) λ 1 ∫ κ 3 κ 3 + κ 4 2 F ( κ 1 , τ 2 ) ψ ( τ 2 − κ 3 ) τ 2 − κ 3 d τ 2 + ∫ κ 1 κ 1 + κ 2 2 ∫ κ 3 κ 3 + κ 4 2 F ( τ 1 , τ 2 ) φ ( τ 1 − κ 1 ) τ 1 − κ 1 ψ ( τ 2 − κ 3 ) τ 2 − κ 3 d τ 2 d τ 1 .

Hence, we obtain

(3.2) J 1 = ∫ 0 1 2 ∫ 0 1 2 ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ( Λ ( η ) − Λ ( 1 ) λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = 1 ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) Δ 1 2 − Δ ( 1 ) λ 1 Λ 1 2 − Λ ( 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + κ 4 2 + Δ ( 1 ) λ 1 Λ 1 2 − Λ ( 1 ) λ 2 F κ 1 , κ 3 + κ 4 2 − Λ 1 2 − Λ ( 1 ) λ 2 I φ κ 1 + κ 2 2 − F κ 1 , κ 3 + κ 4 2 + Δ 1 2 − Δ ( 1 ) λ 1 Λ ( 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + Δ ( 1 ) λ 1 Λ ( 1 ) λ 2 F ( κ 1 , κ 3 ) − Λ ( 1 ) λ 2 κ 1 + κ 2 2 − I φ F ( κ 1 , κ 3 ) − Δ 1 2 − Δ ( 1 ) λ 1 I ψ κ 3 + κ 4 2 − F κ 1 + κ 2 2 , κ 3 − Δ ( 1 ) λ 1 κ 3 + κ 4 2 − I ψ F ( κ 1 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) .

Similarly, we get

(3.3) J 2 = ∫ 0 1 2 ∫ 1 2 1 ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ( Λ ( η ) − Λ ( 1 ) μ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = 1 ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) Δ 1 2 − Δ ( 1 ) λ 1 ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) F κ 1 + κ 2 2 , κ 4 + Δ ( 1 ) λ 1 ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) F ( κ 1 , κ 4 ) − ( Λ ( 1 ) − Λ ( 1 ) ) μ 2 I φ κ 1 + κ 2 2 − F ( κ 1 , κ 4 ) − Δ 1 2 − Δ ( 1 ) λ 1 Λ 1 2 − Λ ( 1 ) μ 2 F κ 1 + κ 2 2 , κ 3 + κ 4 2 − Δ ( 1 ) λ 1 Λ 1 2 − Λ ( 1 ) μ 2 F κ 1 , κ 3 + κ 4 2 + Λ 1 2 − Λ ( 1 ) μ 2 I φ κ 1 + κ 2 2 − F κ 1 , κ 3 + κ 4 2 − Δ 1 2 − Δ ( 1 ) λ 1 I ψ κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 4 − Δ ( 1 ) λ 1 I ψ κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) ,

(3.4) J 3 = ∫ 1 2 1 ∫ 0 1 2 ( Δ ( ξ ) − Δ ( 1 ) μ 1 ) ( Λ ( η ) − Λ ( 1 ) λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = 1 ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) Λ 1 2 − Λ ( 1 ) λ 2 F κ 2 , κ 3 + κ 4 2 − Δ 1 2 − Δ ( 1 ) μ 1 Λ 1 2 − Λ ( 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + κ 4 2 − Λ 1 2 − Λ ( 1 ) λ 2 I φ κ 1 + κ 2 2 + F κ 2 , κ 3 + κ 4 2 + ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) Λ ( 1 ) λ 2 F ( κ 2 , κ 3 ) − Δ 1 2 − Δ ( 1 ) μ 1 Λ ( 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 − Λ ( 1 ) λ 2 I φ κ 1 + κ 2 2 + F ( κ 2 , κ 3 ) − ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) I ψ κ 3 + κ 4 2 − F ( κ 2 , κ 3 ) + Δ 1 2 − Δ ( 1 ) μ 1 I ψ κ 3 + κ 4 2 − F κ 1 + κ 2 2 , κ 3 + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 − F ( κ 2 , κ 3 )

and

(3.5) J 4 = ∫ 1 2 1 ∫ 1 2 1 ( Δ ( ξ ) − Δ ( 1 ) μ 1 ) ( Λ ( η ) − Λ ( 1 ) μ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = 1 ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) [ ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) F ( κ 2 , κ 4 ) − Δ 1 2 − Δ ( 1 ) μ 1 ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) F κ 1 + κ 2 2 , κ 4 − ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) κ 1 + κ 2 2 + I φ F ( κ 2 , κ 4 ) − ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) Λ 1 2 − Λ ( 1 ) μ 2 F κ 2 , κ 3 + κ 4 2 + Δ 1 2 − Δ ( 1 ) μ 1 Λ 1 2 − Λ ( 1 ) μ 2 F κ 1 + κ 2 2 , κ 3 + κ 4 2 + Λ 1 2 − Λ ( 1 ) μ 2 I φ κ 1 + κ 2 2 + F κ 2 , κ 3 + κ 4 2 − ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) I ψ κ 3 + κ 4 2 + F ( κ 2 , κ 4 ) + Δ 1 2 − Δ ( 1 ) μ 1 I ψ κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 4 + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 + F ( κ 2 , κ 4 ) .

By the equalities (3.2)–(3.5), we have

( κ 2 − κ 1 ) ( κ 4 − κ 3 ) [ J 1 + J 2 + J 3 + J 4 ] = ( Δ ( 1 ) μ 1 − Δ ( 1 ) λ 1 ) ( Λ ( 1 ) μ 2 − Λ ( 1 ) λ 2 ) F κ 1 + κ 2 2 , κ 3 + κ 4 2 + Δ ( 1 ) λ 1 ( Λ ( 1 ) μ 2 − Λ ( 1 ) λ 2 ) F κ 1 , κ 3 + κ 4 2 + ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) ( Λ ( 1 ) μ 2 − Λ ( 1 ) λ 2 ) F κ 2 , κ 3 + κ 4 2 + ( Δ ( 1 ) μ 1 − Δ ( 1 ) λ 1 ) Λ ( 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + ( Δ ( 1 ) μ 1 − Δ ( 1 ) λ 1 ) ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) F κ 1 + κ 2 2 , κ 4 + Δ ( 1 ) λ 1 Λ ( 1 ) λ 2 F ( κ 1 , κ 3 ) + ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) Λ ( 1 ) λ 2 F ( κ 2 , κ 3 ) + Δ ( 1 ) λ 1 ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) F ( κ 1 , κ 4 ) + ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) F ( κ 2 , κ 4 ) − ( Λ ( 1 ) μ 2 − Λ ( 1 ) λ 2 ) I φ κ 1 + κ 2 2 − F κ 1 , κ 3 + κ 4 2 + I φ κ 1 + κ 2 2 + F κ 2 , κ 3 + κ 4 2 − ( Δ ( 1 ) μ 1 − Δ ( 1 ) λ 1 ) I ψ κ 3 + κ 4 2 − F κ 1 + κ 2 2 , κ 3 + I ψ κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 4 − Δ ( 1 ) λ 1 I ψ κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I ψ κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) − Λ ( 1 ) λ 2 I φ κ 1 + κ 2 2 − F ( κ 1 , κ 3 ) + I φ κ 1 + κ 2 2 + F ( κ 2 , κ 3 ) − ( Λ ( 1 ) − Λ ( 1 ) μ 2 ) I φ κ 1 + κ 2 2 − F ( κ 1 , κ 4 ) + I φ κ 1 + κ 2 2 + F ( κ 2 , κ 4 ) − ( Δ ( 1 ) − Δ ( 1 ) μ 1 ) I ψ κ 3 + κ 4 2 + F ( κ 2 , κ 3 ) + I ψ κ 3 + κ 4 2 − F ( κ 2 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 − F ( κ 2 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 + F ( κ 2 , κ 4 ) .

which completes the proof.□

Remark 1

If we assume φ ( ξ ) = ξ and ψ ( η ) = η in Lemma 1, we obtain,

Φ ( κ 1 , κ 2 ; κ 3 , κ 4 ) = ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∫ 0 1 2 ( ξ − λ 1 ) ( η − λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∫ 1 2 1 ( ξ − λ 1 ) ( η − μ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 1 2 1 ∫ 0 1 2 ( ξ − μ 1 ) ( η − λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 1 2 1 ∫ 1 2 1 ( ξ − μ 1 ) ( η − μ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ,

where

Φ ( κ 1 , κ 2 ; κ 3 , κ 4 ) = ( μ 1 − λ 1 ) ( μ 2 − λ 2 ) F κ 1 + κ 2 2 , κ 3 + κ 4 2 + λ 1 ( μ 2 − λ 2 ) F κ 1 , κ 3 + κ 4 2 + ( 1 − μ 1 ) ( μ 2 − λ 2 ) F κ 2 , κ 3 + κ 4 2 + ( μ 1 − λ 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + ( μ 1 − λ 1 ) ( 1 − μ 2 ) F κ 1 + κ 2 2 , κ 4 + λ 1 λ 2 F ( κ 1 , κ 3 ) + ( 1 − μ 1 ) λ 2 F ( κ 2 , κ 3 ) + λ 1 ( 1 − μ 2 ) F ( κ 1 , κ 4 ) + ( 1 − μ 1 ) ( 1 − μ 2 ) F ( κ 2 , κ 4 )

− λ 2 κ 2 − κ 1 ∫ κ 1 κ 2 F ( τ 1 , κ 3 ) d τ 1 − μ 2 − λ 2 κ 2 − κ 1 ∫ κ 1 κ 2 F τ 1 , κ 3 + κ 4 2 d τ 1 − 1 − μ 2 κ 2 − κ 1 ∫ κ 1 κ 2 F ( τ 1 , κ 4 ) d τ 1 − λ 1 κ 4 − κ 3 ∫ κ 3 κ 4 F ( κ 1 , τ 2 ) d τ 2 − μ 1 − λ 1 κ 4 − κ 3 ∫ κ 3 κ 4 F κ 1 + κ 2 2 , τ 2 d τ 2 − 1 − μ 1 κ 4 − κ 3 ∫ κ 3 κ 4 F ( κ 2 , τ 2 ) d τ 2 + 1 ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ κ 1 κ 2 ∫ κ 3 κ 4 F ( τ 1 , τ 2 ) d τ 2 d τ 1 ,

which is given by Budak and Ali in [49].

Corollary 1

In Lemma 1, if we set φ ( ξ ) = ξ α Γ ( α ) and ψ ( η ) = ξ β Γ ( β ) , then we obtain the following new Riemann-Liouville fractional integral identity:

Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) = ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∫ 0 1 2 ( ξ α − λ 1 ) ( η β − λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∫ 1 2 1 ( ξ α − λ 1 ) ( η β − μ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 1 2 1 ∫ 0 1 2 ( ξ α − μ 1 ) ( η β − λ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 1 2 1 ∫ 1 2 1 ( ξ α − μ 1 ) ( η β − μ 2 ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ,

where

Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) = ( μ 1 − λ 1 ) ( μ 2 − λ 2 ) F κ 1 + κ 2 2 , κ 3 + κ 4 2 + λ 1 ( μ 2 − λ 2 ) F κ 1 , κ 3 + κ 4 2 + ( 1 − μ 1 ) ( μ 2 − λ 2 ) F κ 2 , κ 3 + κ 4 2 + ( μ 1 − λ 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + ( μ 1 − λ 1 ) ( 1 − μ 2 ) F κ 1 + κ 2 2 , κ 4 + λ 1 λ 2 F ( κ 1 , κ 3 ) + ( 1 − μ 1 ) λ 2 F ( κ 2 , κ 3 ) + λ 1 ( 1 − μ 2 ) F ( κ 1 , κ 4 ) + ( 1 − μ 1 ) ( 1 − μ 2 ) F ( κ 2 , κ 4 ) − ( μ 2 − λ 2 ) Γ ( α + 1 ) ( κ 2 − κ 1 ) α J κ 1 + κ 2 2 − α F κ 1 , κ 3 + κ 4 2 + J κ 1 + κ 2 2 + α F κ 2 , κ 3 + κ 4 2 − ( μ 1 − λ 1 ) Γ ( β + 1 ) ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − β F κ 1 + κ 2 2 , κ 3 + J κ 3 + κ 4 2 + β F κ 1 + κ 2 2 , κ 4 − λ 1 Γ ( β + 1 ) ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − β F ( κ 1 , κ 3 ) + J κ 3 + κ 4 2 + β F ( κ 1 , κ 4 ) − λ 2 Γ ( α + 1 ) ( κ 2 − κ 1 ) α J κ 1 + κ 2 2 − α F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 + α F ( κ 2 , κ 3 ) − ( 1 − μ 2 ) Γ ( α + 1 ) ( κ 2 − κ 1 ) α J κ 1 + κ 2 2 − α F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + α F ( κ 2 , κ 4 ) − ( 1 − μ 1 ) Γ ( β + 1 ) ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − β F ( κ 2 , κ 3 ) + J κ 3 + κ 4 2 + β F ( κ 2 , κ 4 ) + Γ ( α + 1 ) ( κ 2 − κ 1 ) α Γ ( β + 1 ) ( κ 4 − κ 3 ) β J κ 1 + κ 2 2 − κ 3 + κ 4 2 − α , β F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 − κ 3 + κ 4 2 + α , β F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + κ 3 + κ 4 2 − α , β F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 + κ 3 + κ 4 2 + α , β F ( κ 2 , κ 4 ) ,

where Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Corollary 1.

Corollary 2

In Lemma 1, if we set φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = ξ β k k Γ k ( β ) , then we obtain the following new k -Riemann-Liouville fractional integral identity:

ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) = ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∫ 0 1 2 ξ α k − λ 1 η β k − λ 2 ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∫ 1 2 1 ξ α k − λ 1 η β k − μ 2 ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 1 2 1 ∫ 0 1 2 ξ α k − μ 1 η β k − λ 2 ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 1 2 1 ∫ 1 2 1 ξ α k − μ 1 η β k − μ 2 ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ,

where

ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) = ( μ 1 − λ 1 ) ( μ 2 − λ 2 ) F κ 1 + κ 2 2 , κ 3 + κ 4 2 + λ 1 ( μ 2 − λ 2 ) F κ 1 , κ 3 + κ 4 2 + ( 1 − μ 1 ) ( μ 2 − λ 2 ) F κ 2 , κ 3 + κ 4 2 + ( μ 1 − λ 1 ) λ 2 F κ 1 + κ 2 2 , κ 3 + ( μ 1 − λ 1 ) ( 1 − μ 2 ) F κ 1 + κ 2 2 , κ 4 + λ 1 λ 2 F ( κ 1 , κ 3 ) + ( 1 − μ 1 ) λ 2 F ( κ 2 , κ 3 ) + λ 1 ( 1 − μ 2 ) F ( κ 1 , κ 4 ) + ( 1 − μ 1 ) ( 1 − μ 2 ) F ( κ 2 , κ 4 ) − ( μ 2 − λ 2 ) Γ k ( α + 1 ) ( κ 2 − κ 1 ) α k J κ 1 + κ 2 2 − , k α F κ 1 , κ 3 + κ 4 2 + J κ 1 + κ 2 2 + , k α F κ 2 , κ 3 + κ 4 2 − ( μ 1 − λ 1 ) Γ k ( β + 1 ) ( κ 4 − κ 3 ) β k J κ 3 + κ 4 2 − , k β F κ 1 + κ 2 2 , κ 3 + J κ 3 + κ 4 2 + , k β F κ 1 + κ 2 2 , κ 4 − λ 1 Γ k ( β + 1 ) ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − , k β F ( κ 1 , κ 3 ) + J κ 3 + κ 4 2 + , k β F ( κ 1 , κ 4 ) − λ 2 Γ k ( α + 1 ) ( κ 2 − κ 1 ) α J κ 1 + κ 2 2 − , k α F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 + , k α F ( κ 2 , κ 3 ) − ( 1 − μ 2 ) Γ k ( α + 1 ) ( κ 2 − κ 1 ) α k J κ 1 + κ 2 2 − , k α F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + , k α F ( κ 2 , κ 4 ) − ( 1 − μ 1 ) Γ k ( β + 1 ) ( κ 4 − κ 3 ) β k J κ 3 + κ 4 2 − , k β F ( κ 2 , κ 3 ) + J κ 3 + κ 4 2 + , k β F ( κ 2 , κ 4 ) + Γ k ( α + 1 ) ( κ 2 − κ 1 ) α k Γ k ( β + 1 ) ( κ 4 − κ 3 ) β k J κ 1 + κ 2 2 − κ 3 + κ 4 2 − α , β , k F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 − κ 3 + κ 4 2 + α , β , k F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + κ 3 + κ 4 2 − α , β , k F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 + κ 3 + κ 4 2 + α , β , k F ( κ 2 , κ 4 ) ,

where ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Corollary 2.

4 Some new inequalities for generalized fractional integrals

In this section, we establish some new Simpson-type inequalities for differentiable co-ordinated convex functions via generalized fractional integrals.

Theorem 2

We assume that the conditions of Lemma 1 hold. If the mapping ∂ 2 F ∂ ξ ∂ η is co-ordinated convex on [ κ 1 , κ 2 ] , then the following inequality holds for generalized fractional integrals:

(4.1) ∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) × ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) ( Π 1 φ ( λ 1 ) ∇ 1 ψ ( λ 2 ) + Π 1 φ ( λ 1 ) ∇ 3 ψ ( μ 2 ) + Π 3 φ ( μ 1 ) ∇ 1 ψ ( λ 2 ) + Π 3 φ ( μ 1 ) ∇ 3 ψ ( μ 2 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) ( Π 1 φ ( λ 1 ) ∇ 2 ψ ( λ 2 ) + Π 1 φ ( λ 1 ) ∇ 4 ψ ( μ 2 ) + Π 3 φ ( μ 1 ) ∇ 2 ψ ( λ 2 ) + Π 3 φ ( μ 1 ) ∇ 4 ψ ( μ 2 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) ( Π 2 φ ( λ 1 ) ∇ 1 ψ ( λ 2 ) + Π 2 φ ( λ 1 ) ∇ 3 ψ ( μ 2 ) + Π 4 φ ( μ 1 ) ∇ 1 ψ ( λ 2 ) + Π 4 φ ( μ 1 ) ∇ 3 ψ ( μ 2 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) ( Π 2 φ ( λ 1 ) ∇ 2 ψ ( λ 2 ) + Π 2 φ ( λ 1 ) ∇ 4 ψ ( μ 2 ) + Π 4 φ ( μ 1 ) ∇ 2 ψ ( λ 2 ) + Π 4 φ ( μ 1 ) ∇ 4 ψ ( μ 2 ) ) ,

where

(4.2) Π 1 φ ( λ 1 ) = ∫ 0 1 2 ξ ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ , Π 2 φ ( λ 1 ) = ∫ 0 1 2 ( 1 − ξ ) ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ , Π 3 φ ( μ 1 ) = ∫ 1 2 1 ξ ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ , Π 4 φ ( μ 1 ) = ∫ 1 2 1 ( 1 − ξ ) ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ , ∇ 1 ψ ( λ 2 ) = ∫ 0 1 2 η ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η , ∇ 2 ψ ( λ 2 ) = ∫ 0 1 2 ( 1 − η ) ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η , ∇ 3 ψ ( μ 2 ) = ∫ 1 2 1 η ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η , ∇ 4 ψ ( μ 2 ) = ∫ 1 2 1 ( 1 − η ) ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η .

Proof

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

(4.3) ∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) × ∫ 0 1 2 ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ∫ 0 1 2 ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) ∣ d ξ d η + ∫ 1 2 1 ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η + ∫ 1 2 1 ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η .

Since the mapping ∂ 2 F ∂ ξ ∂ η is co-ordinated convex on Δ , therefore, we have

(4.4) ∫ 0 1 2 ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 0 1 2 ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ ξ η ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) + ξ ( 1 − η ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) + ( 1 − ξ ) η ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) + ( 1 − ξ ) ( 1 − η ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) d ξ d η = Π 1 φ ( λ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) + Π 1 φ ( λ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) + Π 2 φ ( λ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) + Π 2 φ ( λ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) .

Similarly, we obtain

(4.5) ∫ 0 1 2 ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = Π 1 φ ( λ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) + Π 1 φ ( λ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) + Π 2 φ ( λ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) + Π 2 φ ( λ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) ,

(4.6) ∫ 1 2 1 ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = Π 3 φ ( μ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) + Π 3 φ ( μ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) + Π 4 φ ( μ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) + Π 4 φ ( μ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) ,

(4.7) ∫ 1 2 1 ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η = Π 3 φ ( μ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) + Π 3 φ ( μ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) + Π 4 φ ( μ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) + Π 4 φ ( μ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) .

By the inequalities (4.4)–(4.7), the proof is completed.□

Remark 2

In Theorem 2, if we take φ ( ξ ) = ξ and ψ ( η ) = η , we obtain the inequality

∣ Φ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) × Ψ 1 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) + Ψ 2 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) + Ψ 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) + Ψ 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) ,

where Φ ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Remark 1 and

Ψ 1 = λ 1 3 + μ 1 3 3 − λ 1 8 − 5 μ 1 8 + 5 12 λ 2 3 + μ 2 3 3 − λ 2 8 − 5 μ 2 8 + 5 12 , Ψ 2 = λ 1 3 + μ 1 3 3 − λ 1 8 − 5 μ 1 8 + 5 12 − λ 2 3 + μ 2 3 3 + λ 2 2 + μ 2 2 − 7 μ 2 + 3 λ 2 8 + 1 3 , Ψ 3 = − λ 1 3 + μ 1 3 3 + λ 1 2 + μ 1 2 − 7 μ 1 + 3 λ 1 8 + 1 3 λ 2 3 + μ 2 3 3 − λ 2 8 − 5 μ 2 8 + 5 12

and

Ψ 4 = − λ 1 3 + μ 1 3 3 + λ 1 2 + μ 1 2 − 7 μ 1 + 3 λ 1 8 + 1 3 − λ 2 3 + μ 2 3 3 + λ 2 2 + μ 2 2 − 7 μ 2 + 3 λ 2 8 + 1 3 ,

which is given by Budak and Ali in [49].

Corollary 3

In Theorem 2, if we use φ ( ξ ) = ξ α Γ ( κ 1 ) and ψ ( η ) = ξ β Γ ( β ) , then we obtain the following parameterized Simpson-type inequality for Riemann-Liouville fractional integrals:

∣ Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) × ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) ( Π 1 α ( λ 1 ) ∇ 1 β ( λ 2 ) + Π 1 α ( λ 1 ) ∇ 3 β ( μ 2 ) + Π 3 α ( μ 1 ) ∇ 1 β ( λ 2 ) + Π 3 α ( μ 1 ) ∇ 3 β ( μ 2 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) ( Π 1 α ( λ 1 ) ∇ 2 β ( λ 2 ) + Π 1 α ( λ 1 ) ∇ 4 β ( μ 2 ) + Π 3 α ( μ 1 ) ∇ 2 β ( λ 2 ) + Π 3 α ( μ 1 ) ∇ 4 β ( μ 2 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) ( Π 2 α ( λ 1 ) ∇ 1 β ( λ 2 ) + Π 2 α ( λ 1 ) ∇ 3 β ( μ 2 ) + Π 4 α ( μ 1 ) ∇ 1 β ( λ 2 ) + Π 4 α ( μ 1 ) ∇ 3 β ( μ 2 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) ( Π 2 α ( λ 1 ) ∇ 2 β ( λ 2 ) + Π 2 α ( λ 1 ) ∇ 4 β ( μ 2 ) + Π 4 α ( μ 1 ) ∇ 2 β ( λ 2 ) + Π 4 α ( μ 1 ) ∇ 4 β ( μ 2 ) ) ,

where Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Corollary 1 and

Π 1 α ( λ 1 ) = α α + 2 λ 1 α + 2 α − λ 1 8 + 1 2 α + 2 ( α + 2 ) , Π 2 α ( λ 1 ) = 2 α α + 1 λ 1 α + 1 α − λ 1 2 + 1 2 α + 1 ( α + 1 ) − Π 1 α ( λ 1 ) , Π 3 α ( μ 1 ) = α α + 2 μ 1 α + 2 α − 5 8 μ 1 + 2 α + 2 + 1 2 α + 2 ( α + 2 ) , Π 4 α ( μ 1 ) = 2 α α + 1 μ 1 α + 1 α − 3 2 μ 1 + 2 α + 1 + 1 2 α + 1 ( α + 1 ) − Π 3 α ( μ 1 )

and

∇ 1 β ( λ 2 ) = β β + 2 λ 2 β + 2 β − λ 2 8 + 1 2 β + 2 ( β + 2 ) , ∇ 2 β ( λ 2 ) = 2 β β + 1 λ 2 β + 1 β − λ 2 2 + 1 2 β + 1 ( β + 1 ) − ∇ 1 β ( λ 2 ) , ∇ 3 β ( μ 2 ) = β β + 2 μ 2 β + 2 β − 5 8 μ 2 + 2 β + 2 + 1 2 β + 2 ( β + 2 ) , ∇ 4 β ( μ 2 ) = 2 β β + 1 μ 2 β + 1 β − 3 2 μ 2 + 2 β + 1 + 1 2 β + 1 ( β + 1 ) − ∇ 3 β ( μ 2 ) .

Corollary 4

In Theorem 2, if we use φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = η β k k Γ k ( β ) , then we obtain the following parameterized Simpson-type inequality for k -Riemann-Liouville fractional integrals:

∣ ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) × ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) Π 1 α k ( λ 1 ) ∇ 1 β k ( λ 2 ) + Π 1 α k ( λ 1 ) ∇ 3 β k ( μ 2 ) + Π 3 α k ( μ 1 ) ∇ 1 β k ( λ 2 ) + Π 3 α k ( μ 1 ) ∇ 3 β k ( μ 2 ) + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) Π 1 α k ( λ 1 ) ∇ 2 β k ( λ 2 ) + Π 1 α k ( λ 1 ) ∇ 4 β k ( μ 2 ) + Π 3 α k ( μ 1 ) ∇ 2 β k ( λ 2 ) + Π 3 α k ( μ 1 ) ∇ 4 β k ( μ 2 ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) Π 2 α k ( λ 1 ) ∇ 1 β k ( λ 2 ) + Π 2 α k ( λ 1 ) ∇ 3 β k ( μ 2 ) + Π 4 α k ( μ 1 ) ∇ 1 β k ( λ 2 ) + Π 4 α k ( μ 1 ) ∇ 3 β k ( μ 2 ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) Π 2 α k ( λ 1 ) ∇ 2 β k ( λ 2 ) + Π 2 α k ( λ 1 ) ∇ 4 β k ( μ 2 ) + Π 4 α k ( μ 1 ) ∇ 2 β k ( λ 2 ) + Π 4 α k ( μ 1 ) ∇ 4 β k ( μ 2 ) ,

where ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Corollary 2 and

Π 1 α k ( λ 1 ) = α α + 2 k λ 1 α + 2 k α − λ 1 8 + 1 2 α + 2 k k α + 2 k k , Π 2 α k ( λ 1 ) = 2 α α + k λ 1 α + k α − λ 1 2 + k 2 α + k k ( α + k ) − Π 1 α k , Π 3 α k ( μ 1 ) = α α + 2 k μ 1 α + 2 k α − 5 8 μ 1 + 2 α + 2 k k + 1 2 α + 2 k k α + 2 k k , Π 4 α k ( μ 1 ) = 2 α α + k μ 1 α + k α − 3 2 μ 1 + 2 α + k k k + k 2 α + k k ( α + k ) − Π 3 α k

and

∇ 1 β k ( λ 2 ) = β β + 2 k λ 2 β + 2 k β − λ 2 8 + 1 2 β + 2 k k β + 2 k k , ∇ 2 β k ( λ 2 ) = 2 β β + k λ 2 β + k β − λ 2 2 + k 2 β + k k ( β + k ) − ∇ 1 β k , ∇ 3 β k ( μ 2 ) = β β + 2 k μ 2 β + 2 k β − 5 8 μ 2 + 2 β + 2 k k + 1 2 β + 2 k k β + 2 k k , ∇ 4 β k ( μ 2 ) = 2 β β + k μ 2 β + k β − 3 2 μ 2 + 2 β + k k k + k 2 β + k k ( β + k ) − ∇ 3 β k .

Theorem 3

We assume that the conditions of Lemma 1 hold. If the mapping ∂ 2 F ∂ ξ ∂ η q is co-ordinated convex on Δ , q ≥ 1 , then we have the following inequality:

(4.8) ∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 − 1 q ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 − 1 q × Π 1 φ ( λ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 φ ( λ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q

(4.8) + Π 2 φ ( λ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 φ ( λ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η 1 − 1 q ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 − 1 q × Π 1 φ ( λ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 φ ( λ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 φ ( λ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 φ ( λ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ∫ 1 2 1 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 − 1 q ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ 1 − 1 q × Π 3 φ ( μ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 φ ( μ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 φ ( μ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 φ ( μ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ∫ 1 2 1 ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η 1 − 1 q ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ 1 − 1 q × Π 3 φ ( μ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 φ ( μ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 φ ( μ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 φ ( μ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q ,

where Π i φ and ∇ i ψ , i = 1 , 2 , 3 , 4 are defined as in Theorem 2.

Proof

By using power mean inequality in (4.3) and co-ordinated convexity of ∂ 2 F ∂ ξ ∂ η q , we have

(4.9) ∫ 0 1 2 ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 − 1 q ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 − 1 q × ∫ 0 1 2 ∫ 0 1 2 ∣ ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ∣ ∣ ( Λ ( η ) − Λ ( 1 ) λ 2 ) ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) q d ξ d η 1 q ≤ ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 − 1 q ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 − 1 q

(4.9) ∫ 0 1 2 ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ ξ η ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + ξ ( 1 − η ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ( 1 − ξ ) η ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ( 1 − ξ ) ( 1 − η ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q d ξ d η = ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 − 1 q ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 − 1 q Π 1 φ ( λ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 φ ( λ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 φ ( λ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 φ ( λ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q .

Similarly, we obtain

(4.10) ∫ 0 1 2 ∫ 1 2 1 ∣ ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ∣ ∣ ( Λ ( η ) − Λ ( 1 ) μ 2 ) ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η 1 − 1 q ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 − 1 q Π 1 φ ( λ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 φ ( λ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 φ ( λ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 φ ( λ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q ,

(4.11) ∫ 1 2 1 ∫ 0 1 2 ∣ ( Δ ( ξ ) − Δ ( 1 ) μ 1 ) ∣ ∣ ( Λ ( η ) − Λ ( 1 ) λ 2 ) ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 1 2 1 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 − 1 q ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ 1 − 1 q Π 3 φ ( μ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 φ ( μ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 φ ( μ 1 ) ∇ 1 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 φ ( μ 1 ) ∇ 2 ψ ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q

and

(4.12) ∫ 1 2 1 ∫ 1 2 1 ∣ ( Δ ( ξ ) − Δ ( 1 ) μ 1 ) ∣ ∣ ( Λ ( η ) − Λ ( 1 ) μ 2 ) ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 1 2 1 ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η 1 − 1 q ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ 1 − 1 q Π 3 φ ( μ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 φ ( μ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 φ ( μ 1 ) ∇ 3 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 φ ( μ 1 ) ∇ 4 ψ ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q .

By the inequalities (4.9)–(4.12), the proof is completed.□

Corollary 5

In Theorem 3, if we take φ ( ξ ) = ξ and ψ ( η ) = η , we have

∣ Φ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ( Π 1 ( λ 1 ) + Π 2 ( λ 1 ) ) 1 − 1 q ( ∇ 1 ( λ 2 ) + ∇ 2 ( λ 2 ) ) 1 − 1 q × Π 1 ( λ 1 ) ∇ 1 ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 ( λ 1 ) ∇ 2 ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 ( λ 1 ) ∇ 1 ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 ( λ 1 ) ∇ 2 ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ( ∇ 3 ( μ 2 ) + ∇ 4 ( μ 2 ) ) 1 − 1 q ( Π 1 ( λ 1 ) + Π 2 ( λ 1 ) ) 1 − 1 q × Π 1 ( λ 1 ) ∇ 3 ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 ( λ 1 ) ∇ 4 ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 ( λ 1 ) ∇ 3 ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 ( λ 1 ) ∇ 4 ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ( ∇ 1 ( λ 2 ) + ∇ 2 ( λ 2 ) ) 1 − 1 q ( Π 3 ( μ 1 ) + Π 4 ( μ 1 ) ) 1 − 1 q × Π 3 ( μ 1 ) ∇ 1 ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 ( μ 1 ) ∇ 2 ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 ( μ 1 ) ∇ 1 ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 ( μ 1 ) ∇ 2 ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ( ∇ 3 ( μ 2 ) + ∇ 4 ( μ 2 ) ) 1 − 1 q ( Π 3 ( μ 1 ) + Π 4 ( μ 1 ) ) 1 − 1 q × Π 3 ( μ 1 ) ∇ 3 ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 ( μ 1 ) ∇ 4 ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 ( μ 1 ) ∇ 3 ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 ( μ 1 ) ∇ 4 ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q ,

where Φ ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Remark 1,

Π 1 ( λ 1 ) = 1 3 λ 1 3 − λ 1 8 + 1 24 , Π 2 ( λ 1 ) = − λ 1 3 3 + λ 1 2 − 3 λ 1 8 + 1 12 , Π 3 ( μ 1 ) = 1 3 μ 1 3 − 5 8 μ 1 + 9 24 , Π 4 ( μ 1 ) = 1 3 μ 1 3 + μ 1 2 − 7 8 μ 1 + 1 4

and

∇ 1 ( λ 2 ) = 1 3 λ 2 3 − λ 2 8 + 1 24 , ∇ 2 ( λ 2 ) = − λ 2 3 3 + λ 2 2 − 3 λ 1 8 + 1 12 , ∇ 3 ( μ 2 ) = 1 3 μ 2 3 − 5 8 μ 2 + 9 24 , ∇ 4 ( μ 2 ) = 1 3 μ 2 3 + μ 2 2 − 7 8 μ 2 + 1 4 .

Corollary 6

In Theorem 3, if we use φ ( ξ ) = ξ α Γ ( α ) and ψ ( η ) = ξ β Γ ( β ) , then we obtain the following parameterized Simpson-type inequality for Riemann-Liouville fractional integrals:

∣ Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ( Π 1 α ( λ 1 ) + Π 2 α ( λ 1 ) ) 1 − 1 q ( ∇ 1 β ( λ 2 ) + ∇ 2 β ( λ 2 ) ) 1 − 1 q × Π 1 α ( λ 1 ) ∇ 1 β ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 α ( λ 1 ) ∇ 2 β ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 α ( λ 1 ) ∇ 1 β ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 α ( λ 1 ) ∇ 2 β ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ( ∇ 3 β ( μ 2 ) + ∇ 4 β ( μ 2 ) ) 1 − 1 q ( Π 1 α ( λ 1 ) + Π 2 α ( λ 1 ) ) 1 − 1 q × Π 1 α ( λ 1 ) ∇ 3 β ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 α ( λ 1 ) ∇ 4 β ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 α ( λ 1 ) ∇ 3 β ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 α ( λ 1 ) ∇ 4 β ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ( ∇ 1 β ( λ 2 ) + ∇ 2 β ( λ 2 ) ) 1 − 1 q ( Π 3 α ( μ 1 ) + Π 4 α ( μ 1 ) ) 1 − 1 q × Π 3 α ( μ 1 ) ∇ 1 β ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 α ( μ 1 ) ∇ 2 β ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 α ( μ 1 ) ∇ 1 β ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 α ( μ 1 ) ∇ 2 β ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ( ∇ 3 β ( μ 2 ) + ∇ 4 β ( μ 2 ) ) 1 − 1 q ( Π 3 α ( μ 1 ) + Π 4 α ( μ 1 ) ) 1 − 1 q × Π 3 α ( μ 1 ) ∇ 3 β ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 α ( μ 1 ) ∇ 4 β ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 α ( μ 1 ) ∇ 3 β ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 α ( μ 1 ) ∇ 4 β ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q ,

where Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Corollary 1 and Π i α , ∇ i β , i = 1 , 2 , 3 , 4 are defined in Corollary 3.

Corollary 7

In Theorem 3, if we use φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = η β k k Γ k ( β ) , then we obtain the following parameterized Simpson-type inequality for k -Riemann-Liouville fractional integrals:

∣ ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) Π 1 α k ( λ 1 ) + Π 2 α k ( λ 1 ) 1 − 1 q ∇ 1 β k ( λ 2 ) + ∇ 2 β k ( λ 2 ) 1 − 1 q × Π 1 α k ( λ 1 ) ∇ 1 β k ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 α k ( λ 1 ) ∇ 2 β k ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 α k ( λ 1 ) ∇ 1 β k ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 α k ( λ 1 ) ∇ 2 β k ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ∇ 3 β k ( μ 2 ) + ∇ 4 β k ( μ 2 ) 1 − 1 q Π 1 α k ( λ 1 ) + Π 2 α k ( λ 1 ) 1 − 1 q × Π 1 α k ( λ 1 ) ∇ 3 β k ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 1 α k ( λ 1 ) ∇ 4 β k ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 2 α k ( λ 1 ) ∇ 3 β k ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 2 α k ( λ 1 ) ∇ 4 β k ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q

+ ∇ 1 β k ( λ 2 ) + ∇ 2 β k ( λ 2 ) 1 − 1 q Π 3 α k ( μ 1 ) + Π 4 α k ( μ 1 ) 1 − 1 q × Π 3 α k ( μ 1 ) ∇ 1 β k ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 α k ( μ 1 ) ∇ 2 β k ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 α k ( μ 1 ) ∇ 1 β k ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 α k ( μ 1 ) ∇ 2 β k ( λ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q + ∇ 3 β k ( μ 2 ) + ∇ 4 β k ( μ 2 ) 1 − 1 q Π 3 α k ( μ 1 ) + Π 4 α k ( μ 1 ) 1 − 1 q × Π 3 α k ( μ 1 ) ∇ 3 β k ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + Π 3 α k ( μ 1 ) ∇ 4 β k ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + Π 4 α k ( μ 1 ) ∇ 3 β k ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + Π 4 α k ( μ 1 ) ∇ 4 β k ( μ 2 ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q ,

where ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Corollary 2 and Π i α k , ∇ i β k , i = 1 , 2 , 3 , 4 are defined in Corollary 4.

Theorem 4

We assume that the conditions of Lemma 1 hold. If the mapping ∂ 2 F ∂ ξ ∂ η q is co-ordinated convex on Δ , then we have the following inequality:

∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 p ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 p × ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 64 1 q + ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η 1 p ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 p × 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q 64 1 q + ∫ 1 2 1 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 p ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ 1 p × 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q 64 1 q + ∫ 1 2 1 ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η 1 p ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ 1 p

× 9 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 64 1 q ,

where 1 p + 1 q = 1 .

Proof

By using the Hölder inequality in (4.3) and co-ordinated convexity of ∂ 2 F ∂ ξ ∂ η q , we have

(4.13) ∫ 0 1 2 ∫ 0 1 2 ∣ ( Δ ( ξ ) − Δ ( 1 ) λ 1 ) ∣ ∣ ( Λ ( η ) − Λ ( 1 ) λ 2 ) ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ p d ξ 1 p ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ p d η 1 p × ∫ 0 1 2 ∫ 0 1 2 ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) q d ξ d η 1 q ≤ ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ p d ξ 1 p ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ p d η 1 p × ∫ 0 1 2 ∫ 0 1 2 ξ η ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + ξ ( 1 − η ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ( 1 − ξ ) η ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ( 1 − ξ ) ( 1 − η ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 1 q d ξ d η = ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ p d ξ 1 p ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ p d η 1 p × ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 64 1 q .

Similarly, we obtain

(4.14) ∫ 0 1 2 ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 0 1 2 ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η 1 p ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) λ 1 ∣ d ξ 1 p × 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q 64 1 q ,

(4.15) ∫ 1 2 1 ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 1 2 1 ∣ Λ ( η ) − Λ ( 1 ) λ 2 ∣ d η 1 p ∫ 0 1 2 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ 1 p × 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q 64 1 q

and

(4.16) ∫ 1 2 1 ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d ξ d η ≤ ∫ 1 2 1 ∣ Λ ( η ) − Λ ( 1 ) μ 2 ∣ d η 1 p ∫ 1 2 1 ∣ Δ ( ξ ) − Δ ( 1 ) μ 1 ∣ d ξ 1 p × 9 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 64 1 q .

If we substitute the inequalities (4.13)–(4.16) in (4.3), then we obtain the required result.□

Remark 3

In Theorem 4, if we take φ ( ξ ) = ξ and ψ ( η ) = η , we obtain

∣ Φ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∣ ξ − λ 1 ∣ p d ξ 1 p ∫ 0 1 2 ∣ η − λ 2 ∣ p d η 1 p × ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 64 1 q + ∫ 0 1 2 ∣ η − μ 2 ∣ p d η 1 p ∫ 1 2 1 ∣ ξ − λ 1 ∣ p d ξ 1 p × 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q 64 1 q + ∫ 1 2 1 ∣ η − λ 2 ∣ p d η 1 p ∫ 0 1 2 ∣ ξ − μ 1 ∣ P d ξ 1 p × 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q 64 1 q

+ ∫ 1 2 1 ∣ η − μ 2 ∣ P d η 1 p ∫ 1 2 1 ∣ ξ − μ 1 ∣ p d ξ 1 p × 9 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 64 1 q ,

where Φ ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Remark 1.

Corollary 8

In Theorem 4, if we use φ ( ξ ) = ξ α Γ ( κ 1 ) and ψ ( η ) = ξ β Γ ( β ) , then we obtain the following parameterized Simpson-type inequality for Riemann-Liouville fractional integrals:

∣ Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ∣ ξ α − λ 1 ∣ p d ξ 1 p ∫ 0 1 2 ∣ η β − λ 2 ∣ p d η 1 p × ∂ 2 F ∂ ξ α ∂ η β ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ α ∂ η β ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ α ∂ η β ( κ 1 , κ 4 ) q + 9 and ∂ 2 F ∂ ξ α ∂ η β ( κ 1 , κ 3 ) q 64 1 q + ∫ 0 1 2 ∣ η β − μ 2 ∣ p d η 1 p ∫ 1 2 1 ∣ ξ α − λ 1 ∣ p d ξ 1 p × 3 ∂ 2 F ∂ ξ α ∂ η β ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ α ∂ η β ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ α ∂ η β ( κ 2 , κ 3 ) q + 9 ∂ 2 F ∂ ξ α ∂ η β ( κ 1 , κ 4 ) q 64 1 q + ∫ 1 2 1 ∣ η β − λ 2 ∣ p d η 1 p ∫ 0 1 2 ∣ ξ α − μ 1 ∣ P d ξ 1 p × 3 ∂ 2 F ∂ ξ α ∂ η β ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ α ∂ η β ( κ 1 , κ 3 ) q + 9 ∂ 2 F ∂ ξ α ∂ η β ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ α ∂ η β ( κ 1 , κ 4 ) q 64 1 q + ∫ 1 2 1 ∣ η β − μ 2 ∣ P d η 1 p ∫ 1 2 1 ∣ ξ α − μ 1 ∣ p d ξ 1 p × 9 ∂ 2 F ∂ ξ α ∂ η β ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ α ∂ η β ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ α ∂ η β ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ α ∂ η β ( κ 1 , κ 3 ) q 64 1 q ,

where Ω ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Corollary 1.

Corollary 9

In Theorem 4, if we use φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = η β k k Γ k ( β ) , then we obtain the following parameterized Simpson-type inequality for k -Riemann-Liouville fractional integrals:

∣ ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 2 ξ α k − λ 1 p d ξ 1 p ∫ 0 1 2 η β k − λ 2 p d η 1 p × ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 64 1 q + ∫ 0 1 2 η β k − μ 2 p d η 1 p ∫ 1 2 1 ξ α k − λ 1 p d ξ 1 p × 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q 64 1 q + ∫ 1 2 1 η β k − λ 2 p d η 1 p ∫ 0 1 2 ξ α k − μ 1 P d ξ 1 p × 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + 9 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q 64 1 q + ∫ 1 2 1 η β k − μ 2 P d η 1 p ∫ 1 2 1 ξ α k − μ 1 p d ξ 1 p × 9 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + 3 ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q 64 1 q ,

where ℑ ( κ 1 , κ 2 ; κ 3 , κ 4 ) is defined as in Corollary 2.

Theorem 5

We assume that the conditions of Lemma 1 hold. If the mapping ∂ 2 F ∂ ξ ∂ η q is co-ordinated convex on Δ , then we have the following inequality:

∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 ∣ w ( ξ , η ) ∣ p d η d ξ 1 p × ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q 4 ,

where 1 p + 1 q = 1 .

Proof

Taking the modulus in Lemma 1 and using the Hölder inequality,

∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ = ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 w ( ξ , η ) ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d η d ξ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 ∣ w ( ξ , η ) ∣ ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) d η d ξ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 ∣ w ( ξ , η ) ∣ p d η d ξ 1 p × ∫ 0 1 ∫ 0 1 ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) q 1 q ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 ∣ w ( ξ , η ) ∣ p d η d ξ 1 p × ∫ 0 1 ∫ 0 1 ξ η ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q + ξ ( 1 − η ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ( 1 − ξ ) η ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ( 1 − ξ ) ( 1 − η ) ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q d η d ξ 1 q .

Since ∂ 2 F ∂ ξ ∂ η q , q > 1 , is a co-ordinated convex function on Δ , we obtain

(4.17) ∫ 0 1 ∫ 0 1 ∂ 2 F ∂ ξ ∂ η ( ξ κ 2 + ( 1 − ξ ) κ 1 , η κ 4 + ( 1 − η ) κ 3 ) q d η d ξ ≤ ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q 4 .

□

Remark 4

In Theorem 5, if we take φ ( ξ ) = ξ and ψ ( η ) = η , we have

∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ( A p ( λ , μ ) ) 1 p × ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q 4 1 q ,

where

A p ( λ , μ ) = 1 ( p + 1 ) 2 λ 1 p + 1 + 1 2 − λ 1 p + 1 + μ 1 − 1 2 p + 1 + ( 1 − μ 1 ) p + 1 × λ 2 p + 1 + 1 2 − λ 2 p + 1 + μ 2 − 1 2 p + 1 + ( 1 − μ 2 ) p + 1 ,

given by Budak and Ali in [49].

Corollary 10

In Theorem 5, if we use φ ( ξ ) = ξ α Γ ( α ) and ψ ( η ) = ξ β Γ ( β ) , then we obtain

∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 ∣ w 1 ( ξ , η ) ∣ p d η d ξ 1 p × ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q 4 1 q ,

where the mapping w 1 : [ 0 , 1 ] × [ 0 , 1 ] → R is defined by

w 1 ( ξ α , η β ) = ( ξ α − λ 1 ) ( η β − λ 2 ) , 0 ≤ ξ ≤ 1 2 , 0 ≤ η ≤ 1 2 , ( ξ α − λ 1 ) ( η β − μ 2 ) , 0 ≤ ξ ≤ 1 2 , 1 2 ≤ η ≤ 1 , ( ξ α − μ 1 ) ( η β − λ 2 ) , 1 2 ≤ ξ ≤ 1 , 0 ≤ η ≤ 1 2 , ( ξ α − μ 1 ) ( η β − μ 2 ) , 1 2 ≤ ξ ≤ 1 , 1 2 ≤ η ≤ 1 .

Corollary 11

In Theorem 5, if we use φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = η β k k Γ k ( β ) , then we obtain

∣ Θ ( κ 1 , κ 2 ; κ 3 , κ 4 ) ∣ ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∫ 0 1 ∫ 0 1 ∣ w 2 ( ξ , η ) ∣ p d η d ξ 1 p × ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) q + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) q 4 1 q ,

where the mapping w 2 : [ 0 , 1 ] × [ 0 , 1 ] → R is defined by

w 2 ( ξ , η ) = ξ α k − λ 1 η β k − λ 2 , 0 ≤ ξ ≤ 1 2 , 0 ≤ η ≤ 1 2 , ξ α k − λ 1 η β k − μ 2 , 0 ≤ ξ ≤ 1 2 , 1 2 ≤ η ≤ 1 , ξ α k − μ 1 η β k − λ 2 , 1 2 ≤ ξ ≤ 1 , 0 ≤ η ≤ 1 2 , ξ α k − μ 1 η β k − μ 2 , 1 2 ≤ ξ ≤ 1 , 1 2 ≤ η ≤ 1 .

5 Special cases of main results

In this section, we present some special cases of the results proved in the previous section.

Corollary 12

Under assumptions of Theorem 2 with λ 1 = λ 2 = 1 6 and μ 1 = μ 2 = 5 6 , we have the following inequality:

Δ ( 1 ) Λ ( 1 ) F κ 1 , κ 3 + κ 4 2 + F κ 2 , κ 3 + κ 4 2 + 4 F κ 1 + κ 2 2 , κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 3 + F κ 1 + κ 2 2 , κ 4 9

+ F ( κ 1 , κ 3 ) + F ( κ 2 , κ 3 ) + F ( κ 1 , κ 4 ) + F ( κ 2 , κ 4 ) 36 − 2 3 Λ ( 1 ) I φ κ 1 + κ 2 2 − F κ 1 , κ 3 + κ 4 2 + I φ κ 1 + κ 2 2 + F κ 2 , κ 3 + κ 4 2 + Δ ( 1 ) I ψ κ 3 + κ 4 2 − F κ 1 + κ 2 2 , κ 3 + I ψ κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 4 − 1 6 Δ ( 1 ) I ψ κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I ψ κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) + I ψ κ 3 + κ 4 2 + F ( κ 2 , κ 3 ) + I ψ κ 3 + κ 4 2 − F ( κ 2 , κ 4 ) − 1 6 Λ ( 1 ) I φ κ 1 + κ 2 2 − F ( κ 1 , κ 3 ) + I φ κ 1 + κ 2 2 + F ( κ 2 , κ 3 ) + I φ κ 1 + κ 2 2 − F ( κ 1 , κ 4 ) + I φ κ 1 + κ 2 2 + F ( κ 2 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 − F ( κ 2 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 + F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) Π 1 φ 1 6 + Π 3 φ 5 6 ∇ 1 ψ 1 6 + ∇ 3 ψ 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) Π 1 φ 1 6 + Π 3 φ 5 6 ∇ 2 ψ 1 6 + ∇ 4 ψ 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) Π 2 φ 1 6 + Π 4 φ 5 6 ∇ 1 ψ 1 6 + ∇ 3 ψ 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) Π 2 φ 1 6 + Π 4 φ 5 6 ∇ 2 ψ 1 6 + ∇ 4 ψ 5 6 ,

where Π i φ and ∇ i ψ , i = 1 , 2 , 3 , 4 are defined as in (4.2).

Remark 5

In Corollary 12, if we take φ ( ξ ) = ξ and ψ ( η ) = η , then Corollary 12 reduces to [24, Theorem 3].

Remark 6

In Corollary 12, if we take φ ( ξ ) = ξ α Γ ( α ) and ψ ( η ) = η β Γ ( β ) , then we obtain

( κ 2 − κ 1 ) α Γ ( α + 1 ) ( κ 4 − κ 3 ) β Γ ( β + 1 ) F κ 1 , κ 3 + κ 4 2 + F κ 2 , κ 3 + κ 4 2 + 4 F κ 1 + κ 2 2 , κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 3 + F κ 1 + κ 2 2 , κ 4 9 + F ( κ 1 , κ 3 ) + F ( κ 2 , κ 3 ) + F ( κ 1 , κ 4 ) + F ( κ 2 , κ 4 ) 36 − 2 3 Γ ( α + 1 ) ( κ 2 − κ 1 ) α J κ 1 + κ 2 2 − α F κ 1 , κ 3 + κ 4 2 + J κ 1 + κ 2 2 + α F κ 2 , κ 3 + κ 4 2 + Γ ( α + 1 ) ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − ψ F κ 1 + κ 2 2 , κ 3 + J κ 3 + κ 4 2 + ψ F κ 1 + κ 2 2 , κ 4 − 1 6 Γ ( α + 1 ) ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − β F ( κ 1 , κ 3 ) + J κ 3 + κ 4 2 + β F ( κ 1 , κ 4 ) + J κ 3 + κ 4 2 + β F ( κ 2 , κ 3 ) + J κ 3 + κ 4 2 − β F ( κ 2 , κ 4 ) − 1 6 Γ ( α + 1 ) ( κ 2 − κ 1 ) α J κ 1 + κ 2 2 − α F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 + α F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 − α F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + α F ( κ 2 , κ 4 )

+ Γ ( α + 1 ) ( κ 2 − κ 1 ) α Γ ( β + 1 ) ( κ 4 − κ 3 ) β J κ 1 + κ 2 2 − , κ 3 + κ 4 2 − α , β F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 − , κ 3 + κ 4 2 + α , β F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 − α , β F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 + α , β F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) Π 1 α 1 6 + Π 3 α 5 6 ∇ 1 β 1 6 + ∇ 3 β 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) Π 1 α 1 6 + Π 3 α 5 6 ∇ 2 β 1 6 + ∇ 4 β 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) Π 2 α 1 6 + Π 4 α 5 6 ∇ 1 β 1 6 + ∇ 3 β 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) Π 2 α 1 6 + Π 4 α 5 6 ∇ 2 β 1 6 + ∇ 4 β 5 6 .

Remark 7

In Corollary 12, if we take φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = η β k k Γ k ( β ) , then we obtain

( κ 2 − κ 1 ) α k Γ ( α + k ) ( κ 4 − κ 3 ) β k Γ ( β + k ) F κ 1 , κ 3 + κ 4 2 + F κ 2 , κ 3 + κ 4 2 + 4 F κ 1 + κ 2 2 , κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 3 + F κ 1 + κ 2 2 , κ 4 9 + F ( κ 1 , κ 3 ) + F ( κ 2 , κ 3 ) + F ( κ 1 , κ 4 ) + F ( κ 2 , κ 4 ) 36 − 2 3 Γ ( α + k ) ( κ 2 − κ 1 ) α k J κ 1 + κ 2 2 − , k α F κ 1 , κ 3 + κ 4 2 + J κ 1 + κ 2 2 + , k α F κ 2 , κ 3 + κ 4 2 + Γ ( α + k ) ( κ 4 − κ 3 ) β k J κ 3 + κ 4 2 − , k ψ F κ 1 + κ 2 2 , κ 3 + J κ 3 + κ 4 2 + , k ψ F κ 1 + κ 2 2 , κ 4 − 1 6 Γ ( α + k ) ( κ 4 − κ 3 ) β k J κ 3 + κ 4 2 − , k β F ( κ 1 , κ 3 ) + J κ 3 + κ 4 2 + , k β F ( κ 1 , κ 4 ) + J κ 3 + κ 4 2 + , k β F ( κ 2 , κ 3 ) + J κ 3 + κ 4 2 − , k β F ( κ 2 , κ 4 ) − 1 6 Γ ( α + 1 ) ( κ 2 − κ 1 ) α k J κ 1 + κ 2 2 − , k α F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 + , k α F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 − , k α F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + , k α F ( κ 2 , κ 4 ) + Γ ( α + k ) ( κ 2 − κ 1 ) α Γ ( β + k ) ( κ 4 − κ 3 ) β J κ 1 + κ 2 2 − , κ 3 + κ 4 2 − α , β , k F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 − , κ 3 + κ 4 2 + α , β , k F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 − α , β , k F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 + α , β , k F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) Π 1 α k 1 6 + Π 3 α k 5 6 ∇ 1 β k 1 6 + ∇ 3 β k 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) Π 1 α k 1 6 + Π 3 α k 5 6 ∇ 2 β k 1 6 + ∇ 4 β k 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) Π 2 α k 1 6 + Π 4 α k 5 6 ∇ 1 β k 1 6 + ∇ 3 β k 5 6 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) Π 2 α k 1 6 + Π 4 α k 5 6 ∇ 2 β k 1 6 + ∇ 4 β k 5 6 .

Corollary 13

Under assumptions of Theorem 2 with λ 1 = λ 2 = μ 1 = μ 2 = 1 2 , we have the following inequality:

( κ 2 − κ 1 ) α Γ ( α + 1 ) ( κ 4 − κ 3 ) β Γ ( β + 1 ) F ( κ 1 , κ 3 ) + F ( κ 2 , κ 3 ) + F ( κ 1 , κ 4 ) + F ( κ 2 , κ 4 ) 4 − Δ ( 1 ) 2 I ψ κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I ψ κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) + I ψ κ 3 + κ 4 2 + F ( κ 2 , κ 3 ) + I ψ κ 3 + κ 4 2 − F ( κ 2 , κ 4 ) − Λ ( 1 ) 2 I φ κ 1 + κ 2 2 − F ( κ 1 , κ 3 ) + I φ κ 1 + κ 2 2 + F ( κ 2 , κ 3 ) + κ 1 + κ 2 2 − I φ F ( κ 1 , κ 4 ) + I φ κ 1 + κ 2 2 + F ( κ 2 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 − F ( κ 2 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 + F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) Π 1 φ 1 2 + Π 3 φ 1 2 ∇ 1 ψ 1 2 + ∇ 3 ψ 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) Π 1 φ 1 2 + Π 3 φ 1 2 ∇ 2 ψ 1 2 + ∇ 4 ψ 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) Π 2 φ 1 2 + Π 4 φ 1 2 ∇ 1 ψ 1 2 + ∇ 3 ψ 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) Π 2 φ 1 2 + Π 4 φ 1 2 ∇ 2 ψ 1 2 + ∇ 4 ψ 1 2 ,

where Π i φ and ∇ i ψ , i = 1 , 2 , 3 , 4 are defined as in (4.2).

Remark 8

In Corollary 13, if we take φ ( ξ ) = ξ and ψ ( η ) = η , then Corollary 13 reduces to [7, Theorem 2].

Remark 9

In Corollary 13, if we take φ ( ξ ) = ξ α Γ ( α ) and ψ ( η ) = η β Γ ( β ) , then we obtain

F ( κ 1 , κ 3 ) + F ( κ 2 , κ 3 ) + F ( κ 1 , κ 4 ) + F ( κ 2 , κ 4 ) 4 − Γ ( α + 1 ) 2 ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − β F ( κ 1 , κ 3 ) + J κ 3 + κ 4 2 + β F ( κ 1 , κ 4 ) + J κ 3 + κ 4 2 + β F ( κ 2 , κ 3 ) + J κ 3 + κ 4 2 − β F ( κ 2 , κ 4 ) − Γ ( α + 1 ) 2 ( κ 2 − κ 1 ) α J κ 1 + κ 2 2 − α F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 + α F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 − α F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + α F ( κ 2 , κ 4 ) + Γ ( α + 1 ) ( κ 2 − κ 1 ) α Γ ( β + 1 ) ( κ 4 − κ 3 ) β J κ 1 + κ 2 2 − , κ 3 + κ 4 2 − α , β F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 − , κ 3 + κ 4 2 + α , β F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 − α , β F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 + α , β F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) Π 1 α 1 2 + Π 3 α 1 2 ∇ 1 β 1 2 + ∇ 3 β 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) Π 1 α 1 2 + Π 3 α 1 2 ∇ 2 β 1 2 + ∇ 4 β 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) Π 2 α 1 2 + Π 4 α 1 2 ∇ 1 β 1 2 + ∇ 3 β 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) Π 2 α 1 2 + Π 4 α 1 2 ∇ 2 β 1 2 + ∇ 4 β 1 2 .

Remark 10

In Corollary 13, if we take φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = η β k k Γ k ( β ) , then we obtain

F ( κ 1 , κ 3 ) + F ( κ 2 , κ 3 ) + F ( κ 1 , κ 4 ) + F ( κ 2 , κ 4 ) 4 − Γ ( α + k ) 2 ( κ 4 − κ 3 ) β k J κ 3 + κ 4 2 − β , k F ( κ 1 , κ 3 ) + J κ 3 + κ 4 2 + β , k F ( κ 1 , κ 4 ) + J κ 3 + κ 4 2 + β , k F ( κ 2 , κ 3 ) + J κ 3 + κ 4 2 − β , k F ( κ 2 , κ 4 ) − Γ ( α + k ) 2 ( κ 2 − κ 1 ) α k J κ 1 + κ 2 2 − α , k F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 + α , k F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 − α , k F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + α , k F ( κ 2 , κ 4 ) + Γ ( α + k ) ( κ 2 − κ 1 ) α k Γ ( β + k ) ( κ 4 − κ 3 ) β k J κ 1 + κ 2 2 − , κ 3 + κ 4 2 − α , β , k F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 − , κ 3 + κ 4 2 + α , β , k F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 − α , β , k F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 + α , β , k F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) Π 1 α k 1 2 + Π 3 α k 1 2 ∇ 1 β k 1 2 + ∇ 3 β k 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) Π 1 α k 1 2 + Π 3 α k 1 2 ∇ 2 β k 1 2 + ∇ 4 β k 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) Π 2 α k 1 2 + Π 4 α k 1 2 ∇ 1 β k 1 2 + ∇ 3 β k 1 2 + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) Π 2 α k 1 2 + Π 4 α k 1 2 ∇ 2 β k 1 2 + ∇ 4 β k 1 2 .

Corollary 14

Under assumptions of Theorem 2 with λ 1 = λ 2 = 0 and μ 1 = μ 2 = 1 , we have the following inequality:

Δ ( 1 ) Λ ( 1 ) F κ 1 + κ 2 2 , κ 3 + κ 4 2 − Λ ( 1 ) I φ κ 1 + κ 2 2 − F κ 1 , κ 3 + κ 4 2 + I φ κ 1 + κ 2 2 + F κ 2 , κ 3 + κ 4 2 − Δ ( 1 ) I ψ κ 3 + κ 4 2 − F κ 1 + κ 2 2 , κ 3 + I ψ κ 3 + κ 4 2 + F κ 1 + κ 2 2 , κ 4 + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 − F ( κ 1 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 − κ 3 + κ 4 2 + F ( κ 1 , κ 4 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 − F ( κ 2 , κ 3 ) + I φ , ψ κ 1 + κ 2 2 + κ 3 + κ 4 2 + F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) ( Π 1 φ ( 0 ) + Π 3 φ ( 1 ) ) ( ∇ 1 ψ ( 0 ) + ∇ 3 ψ ( 1 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) ( Π 1 φ ( 0 ) + Π 3 φ ( 1 ) ) ( ∇ 2 ψ ( 0 ) + ∇ 4 ψ ( 1 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) ( Π 2 φ ( 0 ) + Π 4 φ ( 1 ) ) ( ∇ 1 ψ ( 0 ) + ∇ 3 ψ ( 1 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) ( Π 2 φ ( 0 ) + Π 4 φ ( 1 ) ) ( ∇ 2 ψ ( 0 ) + ∇ 4 ψ ( 1 ) ) ,

where Π i φ and ∇ i ψ , i = 1 , 2 , 3 , 4 are defined as in (4.2).

Remark 11

In Corollary 14, if we take φ ( ξ ) = ξ and ψ ( η ) = η , then Corollary 13 reduces to [6, Theorem 2].

Remark 12

In Corollary 14, if we take φ ( ξ ) = ξ α Γ ( α ) and ψ ( η ) = η β Γ ( β ) , then we obtain

F κ 1 + κ 2 2 , κ 3 + κ 4 2 − Γ ( α + 1 ) ( κ 2 − κ 1 ) α J κ 1 + κ 2 2 − α F κ 1 , κ 3 + κ 4 2 + J κ 1 + κ 2 2 + α F κ 2 , κ 3 + κ 4 2 − Γ ( α + 1 ) ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − ψ F κ 1 + κ 2 2 , κ 3 + J κ 3 + κ 4 2 + ψ F κ 1 + κ 2 2 , κ 4 + Γ ( α + 1 ) ( κ 2 − κ 1 ) α Γ ( β + 1 ) ( κ 4 − κ 3 ) β J κ 1 + κ 2 2 − , κ 3 + κ 4 2 − α , β F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 − , κ 3 + κ 4 2 + α , β F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 − α , β F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 + α , β F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) ( Π 1 α ( 0 ) + Π 3 α ( 1 ) ) ( ∇ 1 β ( 0 ) + ∇ 3 β ( 1 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) ( Π 1 α ( 0 ) + Π 3 α ( 1 ) ) ( ∇ 2 β ( 0 ) + ∇ 4 β ( 1 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) ( Π 2 α ( 0 ) + Π 4 α ( 1 ) ) ( ∇ 1 β ( 0 ) + ∇ 3 β ( 1 ) ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) ( Π 2 α ( 0 ) + Π 4 α ( 1 ) ) ( ∇ 2 β ( 0 ) + ∇ 4 β ( 1 ) ) .

Remark 13

In Corollary 13, if we take φ ( ξ ) = ξ α k k Γ k ( α ) and ψ ( η ) = η β k k Γ k ( β ) , then we obtain

F κ 1 + κ 2 2 , κ 3 + κ 4 2 − Γ ( α + k ) ( κ 2 − κ 1 ) α k J κ 1 + κ 2 2 − , k α F κ 1 , κ 3 + κ 4 2 + J κ 1 + κ 2 2 + , k α F κ 2 , κ 3 + κ 4 2 − Γ ( α + k ) ( κ 4 − κ 3 ) β J κ 3 + κ 4 2 − , k ψ F κ 1 + κ 2 2 , κ 3 + J κ 3 + κ 4 2 + , k ψ F κ 1 + κ 2 2 , κ 4 + Γ ( α + k ) ( κ 2 − κ 1 ) α Γ ( β + 1 ) ( κ 4 − κ 3 ) β J κ 1 + κ 2 2 − , κ 3 + κ 4 2 − α , β , k F ( κ 1 , κ 3 ) + J κ 1 + κ 2 2 − , κ 3 + κ 4 2 + α , β , k F ( κ 1 , κ 4 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 − α , β , k F ( κ 2 , κ 3 ) + J κ 1 + κ 2 2 + , κ 3 + κ 4 2 + α , β , k F ( κ 2 , κ 4 ) ≤ ( κ 2 − κ 1 ) ( κ 4 − κ 3 ) ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 4 ) Π 1 α k ( 0 ) + Π 3 α k ( 1 ) ∇ 1 β k ( 0 ) + ∇ 3 β k ( 1 ) + ∂ 2 F ∂ ξ ∂ η ( κ 2 , κ 3 ) Π 1 α k ( 0 ) + Π 3 α k ( 1 ) ∇ 2 β k ( 0 ) + ∇ 4 β k ( 1 ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 4 ) Π 2 α k ( 0 ) + Π 4 α k ( 1 ) ∇ 1 β k ( 0 ) + ∇ 3 β k ( 1 ) + ∂ 2 F ∂ ξ ∂ η ( κ 1 , κ 3 ) Π 2 α k ( 0 ) + Π 4 α k ( 1 ) ∇ 2 β k ( 0 ) + ∇ 4 β k ( 1 ) .

Remark 14

By special choices of λ 1 , λ 2 , μ 1 and μ 2 in Theorems 3, 4 and 5, one can obtain several new Simpson-, trapezoid- and midpoint-type inequalities. Writing these situations is left to the reader as it will make the article too long.

6 Conclusion

In this paper, we present several generalized inequalities for co-ordinated convex functions via generalized fractional integrals. It is also shown that the results given here are the strong generalization of some already published ones. It is an interesting and new problem that the forthcoming researchers can use the techniques of this study and obtain similar inequalities for different kinds of co-ordinated convexity in their next works.

Acknowledgements

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

  1. Funding information: The funding of this work was supported by Zhejiang Normal University, Jinhua, China.

  2. Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

  3. Conflict of interest: Authors declare that they have no competing interests.

  4. Data availability statement: Data sharing is not applicable to this paper as no data sets were generated or analyzed during the current study.

References

[1] S. S. Dragomir and R. P. Agarwal , Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998), no. 5, 91–95. 10.1016/S0893-9659(98)00086-XSearch in Google Scholar

[2] U. S. Kirmaci , Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput. 147 (2004), no. 1, 137–146. 10.1016/S0096-3003(02)00657-4Search in Google Scholar

[3] M. Z. Sarikaya , E. Set , H. Yaldiz , and N. Basak , Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model. 57 (2013), no. 9–10, 2403–2407. 10.1016/j.mcm.2011.12.048Search in Google Scholar

[4] M. Iqbal , S. Qaisar , and M. Muddassar , A short note on integral inequality of type Hermite-Hadamard through convexity, J. Comput. Anal. Appl. 21 (2016), no. 5, 946–953. Search in Google Scholar

[5] S. S. Dragomir , On Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math. 4 (2001), 775–788. 10.11650/twjm/1500574995Search in Google Scholar

[6] M. A. Latif and S. S. Dragomir , On some new inequalities for differentiable co-ordinated convex functions, J. Inequal. Appl. 1 (2012), 1–13. 10.1186/1029-242X-2012-28Search in Google Scholar

[7] M. Z. Sarikaya , E. Set , M. E. Özdemir , and S. S. Dragomir , New some Hadamard’s type inequalities for co-ordinated convex functions, Tamsui Oxf. J. Inf. Math. Sci. 28 (2012), no. 2, 137–152. 10.1186/1029-242X-2012-28Search in Google Scholar

[8] M. Z. Sarıkaya , On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integral Transforms Spec. Funct. 25 (2014), no. 2, 134–147. 10.1080/10652469.2013.824436Search in Google Scholar

[9] T. Tunç , M. Z. Sarikaya , and H. Yaldiz , Fractional Hermite Hadamard’s type inequality for the co-ordinated convex functions, TWMS J. Pure Appl. Math. 11 (2020), 3–29. Search in Google Scholar

[10] M. Z. Sarikaya and F. Ertuğral , On the generalized Hermite-Hadamard inequalities, An. Univ. Craiova Ser. Mat. Inform. 47 (2020), no. 1, 193–213. Search in Google Scholar

[11] M. E. Turkay , M. Z. Sarikaya , H. Budak , and H. Yildirim , Some Hermite-Hadamard type inequalities for co-ordinated convex functions via generalized fractional integrals, available at:https://www.researchgate.net/publication/321803898. Search in Google Scholar

[12] H. Budak , H. Kara , and M. E. Kiriş , On Hermite-Hadamard type inequalities for co-ordinated trigonometrically ρ -convex functions, Tbilisi Math. J. 13 (2020), no. 2, 1–26, https://doi.org/10.32513/tbilisi/1593223217. Search in Google Scholar

[13] F. Chen , A note on the Hermite-Hadamard inequality for convex functions on the co-ordinates, J. Math. Inequal. 8 (2014), no. 4, 915–923. 10.7153/jmi-08-69Search in Google Scholar

[14] H. Kara , M. A. Ali , and H. Budak , Hermite-Hadamard-type inequalities for interval-valued coordinated convex functions involving generalized fractional integrals, Math. Methods Appl. Sci. 44 (2021), 104–123. 10.1002/mma.6712Search in Google Scholar

[15] S. Kermausuor , Generalized Ostrowski-type inequalities for s -convex functions on the coordinates via fractional integrals, Fract. Differ. Calc. 10 (2020), no. 2, 169–187. 10.7153/fdc-2020-10-11Search in Google Scholar

[16] M. V. Marcela , New inequalities for co-ordinated convex functions via Riemann-Liouville fractional calculus, Tamkang J. Math. 45 (2014), no. 3, 285–296. 10.5556/j.tkjm.45.2014.1365Search in Google Scholar

[17] M. E. Ozdemir , C. Yildiz , and A. O. Akdemir , On some new Hadamard-type inequalities for co-ordinated quasi-convex functions, Hacet. J. Math. Stat. 41 (2012), no. 5, 697–707. Search in Google Scholar

[18] M. Z. Sarikaya , A. Akkurt , H. Budak , M. E. Yildirim , and H. Yildirim , Hermite-Hadamard’s inequalities for conformable fractional integrals, Konuralp J. Math. 8 (2020), no. 2, 376–383. 10.11121/ijocta.01.2019.00559Search in Google Scholar

[19] T. Tunç and M. Z. Sarıkaya , On Hermite-Hadamard type inequalities via fractional integral operators, Filomat 33 (2019), no. 3, 837–854. 10.2298/FIL1903837TSearch in Google Scholar

[20] S. S. Dragomir , R. P. Agarwal , and P. Cerone , On Simpson’s inequality and applications, J. Inequal. Appl. 5 (2000), 533–579. 10.1155/S102558340000031XSearch in Google Scholar

[21] M. Alomari , M. Darus , and S. S. Dragomir , New inequalities of Simpson’s type for η -convex functions with applications, RGMIA Res. Rep. Coll. 12 (2009), no. 4, 9. Search in Google Scholar

[22] M. Z. Sarikaya , E. Set , and M. E. Özdemir , On new inequalities of Simpson-type for convex functions, RGMIA Res. Rep. Coll. 13 (2010), no. 2, 2. Search in Google Scholar

[23] H. Kalsoom , J. D. Wu , S. Hussain , and M. A. Latif , Simpson-type inequalities for co-ordinated convex functions on quantum calculus, Symmetry 11 (2019), no. 6, 768. 10.3390/sym11060768Search in Google Scholar

[24] M. E. Ozdemir , A. O. Akdemir , and H. Kavurmacı , On the Simpson’s inequality for convex functions on the coordinates, Turkish J. Anal. Number Theory 2 (2014), no. 5, 165–169. 10.12691/tjant-2-5-2Search in Google Scholar

[25] J. Park , Generalizations of the Simpson-like type inequalities for co-ordinated η -convex mappings in the second sense, Int. J. Math. Math. Sci. 2012 (2012), 715751. 10.1155/2012/715751Search in Google Scholar

[26] J. Park , Generalizations of the Simpson-like type inequalities for coordinated η -convex mappings, Far East J. Math. Sci. 54 (2011), no. 2, 225–236. Search in Google Scholar

[27] J. Chen and X. Huang , Some new inequalities of Simpson-type for η -convex functions via fractional integrals, Filomat 31 (2017), no. 15, 4989–4997. 10.2298/FIL1715989CSearch in Google Scholar

[28] S. Hussain , J. Khalid , and Y. M. Chu , Some generalized fractional integral Simpson-type inequalities with applications, AIMS Math. 5 (2020), no. 6, 5859–5883. 10.3934/math.2020375Search in Google Scholar

[29] M. Iqbal , S. Qaisar , and S. Hussain , On Simpson-type inequalities utilizing fractional integrals, J. Comput. Anal. Appl. 23 (2017), no. 6, 1137–1145. Search in Google Scholar

[30] S. Rashid , A. O. Akdemir , F. Jarad , M. A. Noor , and K. I. Noor , Simpson-type integral inequalities for κ -fractional integrals and their applications, AIMS Math. 4 (2019), no. 4, 1087–1100. 10.3934/math.2019.4.1087Search in Google Scholar

[31] F. Ertuğral and M. Z. Sarikaya , Simpson-type integral inequalities for generalized fractional integral, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), no. 4, 3115–3124. 10.1007/s13398-019-00680-xSearch in Google Scholar

[32] M. A. Ali , H. Budak , Z. Zhang , and H. Yildrim , Some new Simpson-type inequalities for co-ordinated convex functions in quantum calculus, Math. Methods Appl. Sci. 44 (2020), no. 6, 4515–4540, https://doi.org/10.1002/mma.7048. Search in Google Scholar

[33] H. Budak , S. Erden , and M. A. Ali , Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Methods Appl. Sci. 44 (2021), no. 1, 378–390. 10.1002/mma.6742Search in Google Scholar

[34] S. Erden , S. Iftikhar , R. M. Delavar , P. Kumam , P. Thounthong , and W. Kumam , On generalizations of some inequalities for convex functions via quantum integral. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2020), no. 3, 1–15, https://doi.org/10.1007/s13398-020-00841-3. Search in Google Scholar

[35] S. Erden and M. Z. Sarikaya , On the Hermite-Hadamard-type and Ostrowski type inequalities for the co-ordinated convex functions, Palest. J. Math. 6 (2017), no. 1, 257–270. Search in Google Scholar

[36] S. Iftikhar , P. Kumam , and S. Erden , Newton’s type integral inequalities via local fractional integrals, Fractals 28 (2020), no. 3, 2050037, https://doi.org/10.1142/S0218348X20500371. Search in Google Scholar

[37] J. Park , On Simpson-like type integral inequalities for differentiable preinvex functions, Appl. Math. Sci. 7 (2013), no. 121, 6009–6021. 10.12988/ams.2013.39498Search in Google Scholar

[38] M. Vivas-Cortez , M. A. Ali , A. Kashuri , I. B. Sial , and Z. Zhang , Some new Newton’s type integral inequalities for co-ordinated convex functions in quantum calculus, Symmetry 12 (2020), no. 9, 1476, https://doi.org/10.3390/sym12091476 . 10.3390/sym12091476Search in Google Scholar

[39] S. Mubeen and G. M Habibullah , k-fractional integrals and application, Int. J. Contemp. Math. Sci. 7 (2012), no. 2, 89–94. Search in Google Scholar

[40] H. Budak , H. Kara , and R. Kapucu , New midpoint-type inequalities for generalized fractional integral, Comput. Methods Differ. Equ. (2021), https://doi.org/10.22034/cmde.2020.40684.1772 . 10.18514/MMN.2019.2525Search in Google Scholar

[41] H. Budak , E. Pehlivan , and P. Kösem , On new extensions of Hermite-Hadamard inequalities for generalized fractional integrals, Sahand Commun. Math. Anal. 18 (2021), no. 1, 73–88. Search in Google Scholar

[42] J. Han , P. O. Mohammed , and H. Zeng , Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function, Open Math. 18 (2020), 794–806, https://doi.org/10.1515/math-2020-0038. Search in Google Scholar

[43] A. Kashuri , M. A. Ali , M. Abbas , and H. Budak , New inequalities for generalized m-convex functions via generalized fractional integral operators and their applications, Int. J. Nonlinear Anal. Appl. 10 (2019), no. 2, 275–299. Search in Google Scholar

[44] A. Kashuri and R. Liko , On Fejér type inequalities for convex mappings utilizing generalized fractional integrals, Appl. Appl. Math. 15 (2020), no. 1, 135–150. Search in Google Scholar

[45] P. O. Mohammed , and M. Z. Sarikaya , On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math. 372 (2020), 112740. 10.1016/j.cam.2020.112740Search in Google Scholar

[46] F. Qi , P. O. Mohammed , J.-C Yao , and Y.-H. Yao , Generalized fractional integral inequalities of Hermite-Hadamard type for ( α,m )-convex functions, J. Inequal. Appl. 2019 (2019), 135, https://doi.org/10.1186/s13660-019-2079-6. Search in Google Scholar

[47] D. Zhao , M. A. Ali , A. Kashuri , H. Budak , and M. Z. Sarikaya , Hermite-Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals, J. Inequal. Appl. 2020 (2020), 222, https://doi.org/10.1186/s13660-020-02488-5 . 10.1186/s13660-020-02488-5Search in Google Scholar

[48] G. Farid , A. Rehman , and M. Zahra , On Hadamard inequalities for k-fractional integrals, Nonlinear Funct. Anal. Appl. 21 (2016), no. 3, 463–478. Search in Google Scholar

[49] H. Budak and M. A. Ali , Some parametrized Hermite-Hadamard and Simpson-type inequalities for co-ordinated convex functions, Authorea (2021), https://doi.org/10.22541/au.161380579.94697093/v1. Search in Google Scholar
Received: 2021-04-08
Revised: 2021-06-28
Accepted: 2021-07-05
Published Online: 2021-11-18

© 2021 Humaira Kalsoom et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
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  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0072
Keywords for this article
Simpson’s 1/3 formula; integral inequalities; fractional calculus; co-ordinated convex functions
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