HERMITE–HADAMARD TYPE INEQUALITY FOR ( 𝐸, 𝐹 )-CONVEX FUNCTIONS AND GEODESIC ( 𝐸, 𝐹 )-CONVEX FUNCTIONS

. The main aim of the present paper is to introduce geodesic ( 𝐸, 𝐹 )-convex sets and geodesic ( 𝐸, 𝐹 )-functions on a Riemannian manifold. Furthermore, some basic properties of these mappings are investigated. Moreover, the Hadamard-type inequalities for ( 𝐸, 𝐹 )-convex functions are proven.


Introduction
Convex optimization has an increasing impact on many areas of mathematics, practical applications, and applied sciences.The idea of convexity has been developed and generalized in numerous directions due to its uses and significance, see [1,10,19,20].-convexity of sets and functions was introduced in 1999 [22].

Notations and preliminaries
In this section, some definitions and known results of convex, -convex and (,  )-functions in real numbers sets are presented.Also, geodesic convex, geodesic -convex functions and some results about Riemannian manifolds, which will be used throughout the paper, are given.Definition 2.1.Let  ⊆ R be an interval, then  : (2.1) for more results on this kind of function, see [14,22].
If we replace the space R  by a Riemannian manifold  .Assume that (,  ) is a complete -dimensional Riemannian manifold with Riemannian connection ▽.Given a piecewise For any two points  1 ,  2 ∈  , we define Then  is a metric which induces the original topology on  .Every Riemannian manifold there is a unique determined Riemannian connection, called a Levi-Civita connection, denoted by ▽ 1  2 , for any vector fields  1 ,  2 ∈  .Also, a smooth path  is a geodesic if and only if its tangent vector is a parallel vector field along the path , i.e.,  satisfies the equation ▽  ′  ′ = 0. Any path  joining  1 and  2 in  such that () = ( 1 ,  2 ) is a geodesic and is called a minimal geodesic.Finally, let  as a  ∞ complete -dimensional Riemannian manifold with metric  and Levi-Civita connection ▽.Moreover, considering that the points  1 ,  2 ∈  and  : [0, 1] −→  is a geodesic joining  1 ,  2 , i.e.,  1,2 (0) =  2 and  1,2 (1) =  1 .

Definition 2.5 ([21]
).A set  is totally convex if  contains every geodesic  1,2 of  whose end points  1 and  2 are in  .

Definition 2.6 ([21]
).A subset  ⊆  is called totally convex if and only if  contains every geodesic  1,2 of  whose endpoints  1 and  2 are in  .

Definition 2.9 ([4]). A function 𝑓
The next section is devoted to the study of some properties of (,  )-convex functions like Hermite-Hadamard-type inequalities.In Section 4, the concepts of geodesic (,  )-convex set and geodesic (,  )-convex function on  are introduced.Also, some properties of the geodesic (,  )-convex function are given.
The following definition is generalized from the definition of (,  )-convex function which is called a geodesic (,  )-convex function on a geodesic (,  )-convex sumset of a Riemannian manifold.Definition 4.10.Let  ⊆  be a geodesic (,  )-convex set.A real-valued function If the inequality above is strict The following remark shows that some special cases of the geodesic (,  )-convex function.
Hence,  is a geodesic (,  )-convex function.The proof of this corollary is directly from Proposition 4.2 and Theorem 4.16.