Convexity has long been recognized as a fundamental concept in analysis and optimization, with classical applications to inequalities, approximation theory, and mathematical physics. Among its most celebrated consequences, the Hermite-Hadamard and Fejér inequalities provide sharp bounds for integrals of convex functions, and have motivated numerous refinements and generalizations. Many authors developed multidimensional analogues, emphasizing their importance in both numerical integration and the study of special functions therein [1]-[11].

To illustrate the motivation, we provide an example of a ( η 1 , η 2 ) -convex function that fails to be preinvex, thereby demonstrating the advantage of this generalization. This example highlights the necessity of developing integral inequalities in the ( η 1 , η 2 ) -setting, which leads naturally to stronger results with potential applications.

The aim of this paper is to establish new Hermite-Hadamard-Fejér type inequalities for functions that are ( η 1 , η 2 ) -convex on the coordinates. Our contributions include:

1) Theoretical development: Several new inequalities based on weighted symmetric functions and generalized invexity.

2) Comparative refinements: Explicit demonstration that our bounds are sharper than those derived under convexity or preinvexity.

3) Applications: Implications for numerical integration error analysis and bounding special functions such as Beta and hypergeometric functions.

This work thereby advances the literature on generalized convexity by bridging classical inequalities with modern generalizations.

The Fejér integral inequality for convex functions has been proved in [12].

Theorem 1.1 [12]

Let f : [ a , b ] → ℝ be a convex function. Then

where g : [ a , b ] → [ 0 , ∞ ) is integrable and symmetric to x = a + b 2 (that is g ( x ) = g ( a + b − x ) , ∀ x ∈ [ a , b ] ).

Furthermore, Dragomir [13] gave the Hermite-Hadamard-Fejér inequality on a retangle in plane.

Theorem 1.2 [13]

Let f : [ a , b ] × [ c , d ] → ℝ be a co-ordinated convex function. Then the double inequality holds

where g : [ a , b ] → [ 0 , ∞ ) is integrable and symmetric to x = a + b 2 , y = c + d 2 . (that is g ( x , y ) = g ( a + b − x , c + d − y ) , ∀ x ∈ [ a , b ] , y ∈ [ c , d ] ).

Theorem 1.3 [13]

It is well-known that the convexity theory has wide applications in special functions, differential equations and bivariate mean. Recently, extensions, generalizations, refinements, and variants of convexity have attracted attention.

Motivated by these works we introduce the notion of η -convex functions as generalization of convex functions.

Definition 1.4

A set I ⊆ ℝ is said to be invexwith toa real bifunction η : I × I → ℝ , if

Recently, the concept of ( η 1 , η 2 ) -convex has been introduced in [14] as a generalization of preinvex function and η -convex function [15]. In the following we can find the definition of ( η 1 , η 2 ) -convex function with some basic results.

Let I ⊆ ℝ be an invex set with respect to η 1 : I × I → ℝ . Consider f : I → ℝ and η 2 : f ( I ) × f ( I ) → ℝ . The function f is said to be ( η 1 , η 2 ) -convex if

for all x , y ∈ I and λ ∈ [ 0 , 1 ] .

Remark 1.5

An ( η 1 , η 2 ) -convex function reduces to

(i) an η -convex function if we consider η 1 ( x , y ) = x − y for all x , y ∈ I .

(ii) a preinvex function if we consider η 2 ( x , y ) = x − y for all x , y ∈ f ( I ) .

(iii) a convex function if it satisfies (i) and (ii).

Consider the function f : ℝ + → ℝ + by

Define two bifunction η 1 : ℝ + × ℝ + → ℝ and η 2 : ℝ + × ℝ + → ℝ + by

and

Then f is a ( η 1 , η 2 ) -convex function. But f is not preinvex with respect to η 1 and ai is not convex (consider x = 0 , y = 2 and λ > 0 ).

Definition 1.6

Let I 1 , I 2 ⊆ ℝ be two intervals, and f : I 1 × I 2 → ℝ and η : ℝ × ℝ → ℝ be two real-valued functions. Then f is said to be η -convex on the coordinates if the inequality

holds for all ( x , y ) , ( z , w ) ∈ I 1 × I 2 and λ , μ ∈ [ 0 , 1 ] .

Definition 1.7 [16]

Let I 1 , I 2 ⊆ ℝ be two intervals, and f : I 1 × I 2 → ℝ and η 1 , η 2 : ℝ × ℝ → ℝ be three real-valued functions, and the partial mappings f y : I 1 → ℝ and f x : I 2 → ℝ be defined by

Then f is said to be coordinate ( η 1 , η 2 ) -convex on I 1 × I 2 if f y is η 1 -convex on I 1 and f x is η 2 -convex on I 2. In particular, if η 1 = η 2 = η , then Fis said to be η -convex.

Definition 1.8

Let I 1 , I 2 ⊆ ℝ be two intervals, and f : I 1 × I 2 → ℝ and η : ℝ × ℝ → ℝ be two real-valued functions. Then f is said to be ( η 1 , η 2 ) -convex on the coordinates if the inequality

holds for all ( x , y ) , ( z , w ) ∈ I 1 × I 2 and λ ∈ [ 0 , 1 ] .

Remark 1.9 (Comparison of Coordinate Convexities).

Definitions 1.6 - 1.8 present different but related notions. Definition 1.6 introduces η -convexity on the coordinates, Definition 1.7 specifies coordinate ( η 1 , η 2 ) -convexity via separate mappings, and Definition 1.8 generalizes to invex structures with bifunctions η 1 , η 2 . In particular, Definition 1.7 can be viewed as a special case of Definition 1.8 when η 1 ( x , y ) = x − y and η 2 ( u , v ) = u − v . Throughout the main theorems, we adopt Definition 1.8, since it provides the most general framework.

Over the last decades, efforts have been made to extend convexity to more flexible frameworks. Concepts such as invexity, preinvexity, and η -convexity have been introduced to capture broader families of functions. The notion of ( η 1 , η 2 ) -convexity provides a unifying perspective that includes both preinvex and η -convex functions as special cases. Importantly, as shown in Remark 1.1, this class is strictly larger than preinvex functions, thereby allowing sharper integral inequalities that cannot be obtained under the narrower classical frameworks.

Throughout, ( η 1 , η 2 ) -convex on the coordinates means Definition 1.8 is in force, so lines parallel to each axis obey an invex interpolation via η 1 on the domain and η 2 on the range, strictly generalizing Definition 1.7 and the usual convex or preinvex cases. Readers may simply read “ ( η 1 , η 2 ) -convex” as “coordinate-wise convex under a generalized (invex) interpolation rule” unless a specific structure of ( η 1 , η 2 ) is needed later.

“Weighted symmetric” always refers to symmetry with respect to x = a + b 2 and y = c + d 2 for g , as in the classical Fejér’s setting.

Each theorem below is immediately followed by a short plain-language explanation highlighting: what quantity is bounded, what assumptions buy the estimate, and how it collapses to the classical convex case (by taking η 1 ( x , y ) = x − y and η 2 ( u , v ) = u − v . We also point out how the constants and kernels should be interpreted for numerical integration or for bounding special functions.

Theorem 2.1

Let 0 ≤ a < b , 0 ≤ c < d and f : [ a , b ] × [ c , d ] → ℝ be a η -convex function and g : [ a , b ] × [ c , d ] → ℝ be an integrable, positive and weighted symmetric function with respect to x = a + b 2 and y = c + d 2 , then the following inequalities are valid:

Proof:

By the η -convexity of f on [ a , b ] × [ c , d ] , it gives

for all x , z ∈ [ a , b ] and y , w ∈ [ c , d ] .

Setting x = λ a + ( 1 − λ ) b , z = ( 1 − λ ) a + λ b , y = μ c + ( 1 − μ ) d and w = ( 1 − μ ) c + μ d , we have

By integrating over [ 0 , 1 ] × [ 0 , 1 ] , we get

By evaluating the weighted fractional operators, we have

Since g ( t , s ) = g ( a + b − t , c + d − s ) , then we get

Now, we will prove the right side of inequality by using convexity.

By integrating over [ 0 , 1 ] × [ 0 , 1 ] , we get

By evaluating the weighted fractional operators, we have

We bound the g -weighted average of f over [ a , b ] × [ c , d ] by simple “surrogate values” built from the midpoint ( a + b 2 , c + d 2 ) , the opposite point ( a + b − t , c + d − s ) , and the corners.

The novelty is that the surrogates are allowed to interact through the bracket η ( ⋅ , ⋅ ) , which encodes our generalized (invex) interpolation.

When η 1 ( x , y ) = x − y and η 2 ( u , v ) = u − v , the inequality collapses to the classical Fejér-type double integral bound.

If g = 1 , the result compares the rectangle average of f directly with its midpoint and corner data, providing a practical control for quadrature error on meshes symmetric about the rectangle center.

Lemma 2.2

Suppose that I 1 , I 2 ⊆ ℝ is an invexset with respect to η 1 and f : I 1 × I 2 → ℝ is a twice differentiable mapping on I 1 × I 2 . For any a , b ∈ I 1 , c , d ∈ I 2 with η 1 ( b , a ) > 0 , η 1 ( d , c ) > 0 , if g : [ a , a + η 1 ( b , a ) ] × [ c , c + η 1 ( d , c ) ] → [ 0 , ∞ ) is differentiable mapping on I 1 × I 2 and

where

Proof:

Using the integration by part, we obtain

Using the change of variables x = a + t η 1 ( b , a ) and y = c + s η 1 ( d , c ) for t , s ∈ [ 0 , 1 ] . Thus we complete the proof of lemma 2.2.

This identity rewrites the deviation between the g -weighted surface integral of f and a balanced combination of its edge and corner traces as the integral of the mixed curvature f ( t , s ) against a nonnegative kernel p ( t , s ) determined by g .

It is the engine that converts smoothness or convexity information on f ( t , s ) into quantitative error bounds.

Theorem 2.3

Suppose that I 1 , I 2 ⊆ ℝ is an invexset with respect to η 1 and f : I 1 × I 2 → ℝ is a twice differentiable mapping on I 1 × I 2 . For any a , b ∈ I 1 , c , d ∈ I 2 with η 1 ( b , a ) > 0 , η 1 ( d , c ) > 0 , if g : [ a , a + η 1 ( b , a ) ] × [ c , c + η 1 ( d , c ) ] → [ 0 , ∞ ) is in-tegrable and symmetric with respect to x = a + 1 2 η 1 ( b , a ) and y = c + 1 2 η 1 ( d , c ) mapping on I 1 × I 2 and ∂ 2 ∂ t ∂ s f ( t , s ) ∈ L 2 [ a , a + η 1 ( b , a ) ] × [ c , c + η 1 ( d , c ) ] . If | ∂ 2 ∂ t ∂ s f ( t , s ) | is a ( η 1 , η 2 ) -convex mapping on [ a , a + η 1 ( b , a ) ] × [ c , c + η 1 ( d , c ) ] , then

Proof:

From the lemma 2.2 and the ( η 1 , η 2 ) -convexity of | ∂ 2 ∂ t ∂ s f ( t , s ) | , we have