Body Math Simpson’s inequality is a classical result in real analysis and plays a central role in approximation theory, numerical integration, and convex analysis. It provides an explicit bound for the error of Simpson’s quadrature rule in terms of the fourth derivative of the integrand and has motivated a large body of research on quadrature-type inequalities and their refinements.

Body Math Over the last decades, many authors have obtained extensions and improvements of Simpson’s inequality under various structural assumptions on the underlying functions, including convexity, boundedness, bounded variation, s-convexity, preinvexity, and co-ordinated convexity on rectangles. For functions of two variables, Simpson-type inequalities on the co-ordinates have proved to be a powerful tool in studying integral inequalities and in analyzing numerical schemes on rectangular domains.

Body Math In parallel with these developments, fractional calculus has provided an efficient framework for generalizing classical inequalities. Generalized fractional integral operators, defined via suitable kernel functions, subsume as special cases a number of well-known fractional integrals such as the Riemann-Liouville and k-Riemann-Liouville fractional integrals, the Katugampola fractional integral, conformable fractional integrals, and the Hadamard fractional integral. This unified viewpoint allows one to derive families of integral inequalities that simultaneously cover many previously known results and yield sharper bounds in several directions.

Body Math Another important refinement in convex analysis is due to Mercer, who introduced in 2003 a sharpened version of Jensen’s inequality, now commonly referred to as the Jensen-Mercer inequality. This result provides improved estimates for convex mappings and has been successfully applied to obtain refined forms of various integral inequalities. Combining Simpson-type quadrature formulas with Mercer-type refinements naturally leads to what we call Simpson-Mercer-type inequalities, which retain the structure of Simpson’s rule while incorporating additional flexibility through Mercer-type weights.

Body Math Motivated by these ideas, and by recent contributions on Simpson-type inequalities for generalized fractional integrals, we investigate parameterized Simpson-Mercer-type inequalities for functions of two variables via generalized fractional integral operators. We first establish new integral identities for twice partially differentiable functions defined on rectangular domains, involving a pair of free parameters and double generalized fractional integrals. Under appropriate co-ordinated convexity assumptions on the mixed partial derivatives, these identities yield several families of Simpson-Mercer-type inequalities in both L p and L ∞ forms. By choosing specific kernels and parameter values, our results reduce to many known Simpson-type and fractional Simpson-type inequalities for co-ordinated convex functions and, at the same time, provide new generalizations in the planar setting.

Simpson’s inequality plays an important role in many areas of mathematics. The classical Simpson’s inequality is expressed as follows:

Theorem 1.1

Suppose that f : [ a , b ] → ℝ is a four-times continuously differentiable mapping on ( a , b ) ,and let ‖ f ( 4 ) ‖ ∞ = sup x ∈ ( a , b ) | f ( 4 ) ( x ) | < ∞ .Then one has the inequalitiy

Many researchers have studied various Simpson’inequalities, such as convex functions, bounded functions, functions of bounded variation, etc. Specifically, since convexity theory’is an effective and powerful way to solve a large number of problems from different branches of pure and applied mathematics, many papers have been dedicatesto Simpson’ inequality for convex functions.

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

Definition 1.2

A function f : [ a , b ] × [ c , d ] → ℝ is called co-ordinated convex on [ a , b ] × [ c , d ] for all ( x , u ) , ( y , v ) ∈ [ a , b ] × [ c , d ] and t , s ∈ [ 0 , 1 ] if it satisfies the following inequality:

In [1], Dragomir et al. proved the following some Simpson’s inequality for which the remainder is expressed in terms of lower derivatives than the fourth.

Theorem 1.3

Suppose f : [ a , b ] → ℝ is an absolutely continuous mapping on [ a , b ] whose derivative belongs to L p [ a , b ] .Then, the following inequality holds:

In 2003, Mercer [2] introduced a refined version of the classical Jensen inequality, now known as the Jensen-Mercer inequality, stated as follows.

For a convex mapping f : [ a , b ] → ℝ , the following inequality holds for each x j ∈ [ a , b ] :

In this section, we present some definitions and results which will be used in our main section.

Let’s define a function φ : [ 0 , ∞ ) → [ 0 , ∞ ) satisfying the follow conditions:

We define the following left-sided and right-side generalized fractional integral operators, respectively, as follows:

The most important feature of generalized fractional integrals is that they generalize some type of fractional integrals such as the Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral, Katugampola fractional integrals, conformable fractional, and Hadamard feractional integrals. These important special cases of integral operator are mentioned below.

1) If we choose φ ( x ) = x , the operators a + I φ f ( x ) and b − I φ f ( x ) are reduce to the Riemann integral.

2) Considering φ ( x ) = x α Γ ( α ) and α > 0 , the operators a + I φ f ( x ) and b − I φ f ( x ) are reduce to the Riemann-Liouville fractioal integrals I a + α f ( x ) and I b − α f ( x ) , respectively. Here, Γ is a gamma function.

3) For φ ( x ) = 1 k Γ ( α ) x α k and α , k > 0 , the operators a + I φ f ( x ) and b − I φ f ( x ) are reduce to the k-Riemann-Liouville fractional integrals I a + , k α f ( x ) and I b − , k α f ( x ) , respectively. Here, Γ k is a k-gamma function.

In [3], Sarikaya and Ertuğral established the following Hermite-Hadamard inequality for the generalized fractional integral operator.

Theorem 2.1

Let f : [ a , b ] → ℝ be a convex function on [ a , b ] with a < b .Then we have the following inequalities for fractional integral operators:

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

In [4], Ali et al. established the following the Simpson’s-type inequality for the generalized fractional integral operator.

Theorem 2.2

Assume f : [ a , b ] → ℝ be a twice-differentiable mapping in ( a , b ) such that f ″ ∈ L 1 ( [ a , b ] ) and the mapping | f ″ | is convex on [ a , b ] .Then we have the following inequalities for fractional integral operators:

where

and

In [5], Budak et al. established the following the parametered Simpson’s-type inequality for the generalized fractional integral operator.

Theorem 2.3

Assume f : [ a , b ] → ℝ be a differentiable function on ( a , b ) .If f ′ is continuous and | f ′ | is convex on [ a , b ] ,then for ρ , σ ≥ 0 ,we have the following inequalities for fractional integral operators:

where

In [6], Khan and Budak established the following the parametered Simpson’s-type inequality for the generalized fractional integral operator.

Theorem 2.4

Assume f : Δ : [ a , b ] × [ c , d ] → ℝ be a twice partially differentiable function on Δ o ( Δ o is the interior of Δ ).If ∂ 2 ∂ t ∂ a f ∈ L ( Δ ) and | ∂ 2 f ∂ t ∂ s | is convex on Δ ,then we have the following inequalities for fractional integral operators:

where

We hope to incorporate parameter settings, in line with the spirit of Theorme 2.4, so that the Simpson-Mercer inequality can yield results different from the traditional Simpson inequality with the addition of different parameters.

Throughout this study, for briefly, we define

The aim of this paper is to obtain some Simpson type inequalities for coordinates involving generalized fractional integrals, we define the double generalized fractional intergals as follows:

and

where φ , ψ : [ 0 , ∞ ) → [ 0 , ∞ ) satisfying ∫ 0 1 φ ( t ) t d t < ∞ and ∫ 0 1 ψ ( t ) t d t < ∞ , respectively.

Lemma 3.1

Let f : [ a , b ] × [ c , d ] → ℝ be a absolutely continuous function such that the partial derivative of order ∂ 2 ∂ t ∂ s f ( t , s ) is continuous and integrable on [ a , b ] × [ c , d ] and a ≤ m < n ≤ b , c ≤ k < l ≤ d .Then for σ , ρ ≥ 0 ,the following equality holds:

where

and

proof:

By using integration by parts twice and changing variables by x = a + b − [ t m + ( 1 − t ) n ] and y = c + d − [ s k + ( 1 − s ) l ] , we have

By the same way, we get