New version of midpoint-type inequalities for co-ordinated convex functions via generalized conformable integrals

In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some ﬁndings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals. Mathematics Subject Classiﬁcation: 26B25; 26D07; 26D10; 26D15


Introduction
Convex functions are a fundamental and widely-used mathematical concept in various fields of analysis and optimization.A function is considered convex if the line segment connecting any two points on its graph lies either below or on the graph itself, indicating a curve that is upward-curving.Convex functions have notable properties, including the fact that the slope between any two points is either increasing or constant, making them valuable in optimization problems to find minimum or maximum values.The definition known for convex functions is as follows: Definition 1 [12] Let I be convex set on R. The function χ : I → R is said to be convex on I if it satisfies the following inequality: for all λ 1 , λ 2 ∈ I and t ∈ [0, 1].The mapping χ is a concave on I if the inequality (1.1) holds in reversed direction for all t ∈ [0, 1] and λ 1 , λ 2 ∈ I.
It is clear that all convex functions are convex on co-ordinates.However, not every function that is a convex function in coordinates has to be convex (see, [11]).
In the realm of inequalities, one prominent result is the Hermite-Hadamard inequality, which holds for convex functions.This inequality gives upper and lower bounds for the average value of a convex function over an interval.It serves as a powerful tool in various mathematical analyzes and has applications in diverse scientific fields (see, e.g., [12], [25, p.137]).Hermite-Hadamard inequality is stated that if χ : I → R is a convex function on the interval I of real numbers and λ 1 , λ 2 ∈ I with λ 1 < λ 2 , then If χ is concave, the inequality that is stated above is provided reversely.The references may be seen for the examples of Hermite-Hadamard's inequality for some convex function on the co-ordinates in mathematics literature [3-5, 7, 8, 10, 22, 23, 28, 31].Recently, this inequality has been expanded by many researchers.The left side of the Hermite-Hadamard inequality, namely the midpoint type inequality, has been the focus of many studies.Midpoint type inequalities for convex functions were first derived by Kırmacıin [21].In [32], Sarikaya et al. generalized the inequalities (1.2) for fractional integrals.Iqbal et al. proved corresponding midpoint type inequalities for Riemann-Liouville fractional integrals in [15].
In [11], Dragomir proved the Hermite-Hadamard inequality, which formed the basis of this article and is valid for co-ordinated convex functions on the rectangle from the plane R 2 .
Theorem 1 Suppose that χ : → R is co-ordinated convex, then we have the following inequalities: The inequalities in (1.3) hold in reverse direction if the mapping χ is a co-ordinated concave mapping.
The fractional calculus [16,19,24,26,27] is defined as any random real number or derivative and integral calculus in complex order.As a result of having various uses in other branches besides mathematics it is an updated study area.These definitions are the most notable definitions of Caputo, Riemann-Liouville, Grünwald-Letnikov play an important role in many fields such as physics, biology, and engineering.However, it is known that these definitions have some difficulties despite their availability.For instance, unless derivative of order in Riemann-Liouville fractional derivative definition is a natural number, derivative of fixed function is not 0. Likewise, the function f must be differentiable in Caputo fractional derivatives.Moreover, many definitions of fractional derivatives do not provide the quotient formula, the product of two functions, and the chain rule.In order to overcome these and similar difficulties, conformable fractional derivative was defined by Khalil et al. in [17].Khalil et al. described the higher order (α > 1) fractional derivative and the fractional integral of order (0 < α ≤ 1).They also proved important theorems such as the product rule, the fractional mean value theorem.They solved conformable fractional differential equations for fractional exponential functions (see, [2,13,17,33]).Thus, conformable fractional integrals became an important field of study for many researchers.For some papers on conformable fractional integrals, please see [1,6,14,17,18,29,30].
The definitions and mathematical underpinnings of conformable fractional calculus principles that are used later in this study are provided below: Definition 3 [19] For ξ ∈ L 1 [η 1 , η 2 ], the Riemann-Liouville integrals of order α > 0 are given by and respectively.Here is the Gamma function.The Riemann-Liouville integrals will be equal to the classical Riemann integrals for the condition α = 1.

Midpoint type inequalities for co-ordinated convex functions
), then the following identity holds: Proof By integration by parts, we get ∂ξ ∂s In (2.3), using the change of the variables, we can write (2.4) Thus, similarly, by integration by parts it follows that ) and (2.7) By the equalities (2.4)-(2.7),we obtain This completes the proof.
Next, we start to state the first theorem containing the midpoint type inequality for generalized conformable fractional integrals.
Theorem 3 Assume that the assumptions of Lemma 1 hold.If | ∂ 2 ξ (t,s) ∂t∂s | is a co-ordinated convex function on , then the following inequality holds.
where A is defined by (2.2) and B(•, •) refers to the Beta function.
Proof From Lemma 1, we acquire (2.9) ∂t∂s | is co-ordinated convex function on , then one has: which finishes the proof.
Remark 3 In Theorem 3, if we choose γ 1 = 1 and γ 2 = 1, then the following inequality for Riemann-Liouville fractional integrals is achieved where The inequality (2.10) is the same of [10, Remark 5].Theorem 4 Assume that the assumptions of Lemma 1 hold.If | ∂ 2 ξ ∂t∂s | q , q > 1, is a coordinated convex function on , then the following inequality holds.
Proof By using the well-known Hölder's inequality for double integrals, since | ∂ 2 ξ ∂t∂s | q is convex functions on the co-ordinates on , we get Here, we take advantage of the fact that (σ ) j ≤ jσ j , for any > σ ≥ 0 and j ≥ 1.
Remark 5 If we take γ 1 = 1 and γ 2 = 1 in Theorem 4, then the following inequality for Riemann-Liouville fractional integrals is achieved Theorem 5 Assume that the assumptions of Lemma 1 hold.If | ∂ 2 ξ ∂t∂s | q , q ≥ 1, is a coordinated convex function on , then we have the following inequality: Here, A is defined as in (2.2).
Proof By using power-mean inequality, we get .
Similarly, we have

Conclusion
In this research, we acquired some inequality of midpoint type for co-ordinated convex functions by means of conformable fractional integrals.In the future studies, researchers can obtain some new inequalities with the aid of the different kinds of co-ordinated convex mappings or other types of fractional integral operators.