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New Hadamard-type inequality for new class of geodesic convex functions

This paper aims to introduce the concept of ( E , µ , κ )-convex function by using special inequality. Hadamard integral inequality for this new class of geodesic convex function in the case of Lebesgue and Sugeno integrals is given.


Introduction
It is commonly known that convexity is used in modern analysis, either directly or indirectly [18]. The idea of convexity has been developed and generalized in numerous directions due to its uses and significance, see [7,8,15,16]. E-convexity of sets and functions, which is a broader function than invexity, was introduced in 1999 [28]. However, Young [27] claims that some of the results in [28] are inaccurate. In [4], the E-convexity was expanded to a semi-E-convexity. See [5,6,23] for further information on the E-convex or semi-E-convex functions. Furthermore, Youness and Emam in [29] discuss a novel class of functions called as strongly E-convex functions. In particular, semi-strong E-convexity as well as quasi and pseudo semi-strong E-convexity was added to this class of functions [30].
A manifold is not a linear space, and extensions of concepts and techniques from linear spaces to Riemannian manifolds are natural. Many authors, including Udrist [24] and Rapcsak [20], have studied generalized convex functions in Riemannian manifolds Geodesic E-convex sets and geodesic E-convex functions on Riemannian manifolds are investigated in 2012 [10]. Moreover, geodesic semi E-convex functions are introduced in [9]. Recently, geodesic strongly E-convex functions have been introduced, and some of their properties [11].
Based on these ideas, a new class of functions, which are called geodesic semi strongly E-convex functions, are defined and some of their properties are presented in [12]. A class of functions on Riemannian manifolds, which are called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions, are given in [13]. In [21], geodesic E-b-vex sets and geodesic E-b-vex functions on a Riemannian manifold are extended to geodesic strongly E-b-vex sets and geodesic strongly E-b-vex functions.
Sugeno integrals are a type of nonlinear integral invented by Sugeno [22] to capture and integrate interactions between criteria of various phenomena. The most well-known integral inequalities for Sugeno integral have been proven, see [1,25].

Preliminaries
In this section, we present some definitions and properties that can be found in many books on differential geometry, such as [24].
Suppose that ℵ is a C ∞ n-dimensional Riemannian manifold, and T t ℵ is the tangent space to ℵ at t. Also, assume that µ t (y 1 , y 2 ) is a positive inner product on the tangent space T t ℵ ( y 1 , y 2 ∈ T t ℵ), which is given for each point of ℵ. Then, a C ∞ map µ : t −→ µ t , which assigns a positive inner product µ t to T t ℵ, ∀t ∈ ℵ is called a Riemannian metric.
A set A in a Riemannian manifold ℵ is called t-convex if A contains every geodesic η y 1 ,y 2 of N whose endpoints y 1 and y 2 belong to A.
Note that the whole of the manifold ℵ is t-convex, and conventionally, so is the empty set. The minimal circle in a hyperboloid is t-convex, but a single point is not. Also, any proper subset of a sphere is not necessarily t-convex.
The following theorem was proved in [24].

Theorem 2.1 ([24]
). The intersection of any number of t-convex sets is t-convex.
Remark 2.2. In general, the union of a t-convex set is not necessarily t-convex.

Definition 2.3 ([24]). A function
Now let M be a non-empty set and ξ be a σalgebra of subsets of M.
Assume that (M, ξ, N), which is said to be a sugeno measure space, is a fuzzy measure space. By Definition 2.5 ( [17,22]). Assuming (M, ξ, N) is a fuzzy measure space, h ∈ χ ξ (M) and X ∈ ξ, then the Sugeno integral of h on A w.r.t. the N is defined by Some properties of the Sugeno integral can be found in [17,26] such as following.

The main results
In this part of the paper, let us take (M, ξ) be a fuzzy measure space for a given h ∈ H N (M) and In the next definition, the concept of (E, µ, κ)-convexity is given.
Next, some generalizations of Hadamard in inequality for different geodesic convex functions are given.
then the next inqualities hold.
Proof. The first inequality is the well-known Hadamard's inequality for E-convex functions. If we use the inequality (3.1), then we have the following inequalities.
If we take in Theorem 2.6, then the Corollary 3.6 can be proved.

If we take
in Theorem 2.6, then the Corollary 3.7 can be proved.

Conclusion
This paper introduces a new class of geodesic convex functions, namely (E, µ, κ)-convex function, and considers and generalizes the Hadamard inequality for (E, µ, κ)-convex function.