A Note on Generalized Strongly p-Convex Functions of Higher Order

Generalized strongly p-convex functions of higher order is a new concept of convex functions which introduced by Saleem et al. in 2020. The Schur type inequality for generalized strongly pconvex functions of higher order also studied by them. This paper aims to revise Schur type inequality for generalized strongly p-convex functions of higher order in their paper. In order to revise it, we show that the contradiction was true. This paper showed that Schur type inequality for generalized strongly p-convex functions of higher order previously is not valid and we give the correct Schur type inequality for generalized strongly p-convex functions of higher order.


INTRODUCTION
Convexity is a basic notion in many branches of applied mathematics. Convexity is an important thing on Functional analysis, Geometry, Mathematical programming, Probability, and Statistics. On Functional analysis, convexity has intended to ensure existence and uniqueness of solutions of problems of Calculus of variations and optimal control. On Mathematical programming, convexity has intended to ensure convergence of optimization algorithms [1].
Convexity appears in ancient Greek Geometry. Archimedes (ca. 250 BC) used convexity on study of the area and arch length. Archimedes has been the first person who gave a definition of convexity, similar to the geometric definition which used till today, a set is said to be convex if it contains all line segments between each of its points [1].
Some geometric properties of convex sets and functions have studied before 1960 by great mathematicians Hermann Minkowski and Werner Fenchel. In 1891, Minkowski proved that, in Euclidean space ℝ , every compact convex set with center at the origin and volume greater than 2 contains at least one point with integer coordinates different from the origin [2]. Afterwards, in 1951, Werner Fenchel's monograph stimulated the development of convexity theory.
In [3] discussed definition and properties of generalized strongly -convex functions of higher order also some of type inequalities which are Hermite-Hadamard, Fejér, and Schur. In Schur type inequality for generalized strongly -convex funtions of higher order in [3] indirectly mentioned that So, this paper revises the correction of Schur type inequality in [3].

METHODS
In this section, we discussed about some definitions which are implemented with generalized strongly -convex functions of higher order. We also give examples of strongly -convex funtions of higher order and its generalized.

Example:
Function in example of strongly -convex functions of higher order with respect to ( , ) = − is generalized strongly -convex functions of higher order with rescpect to .

RESULTS AND DISCUSSION
In this section, we discussed about revised Schur type inequality for generalized strongly -convex functions of higher order. We showed that
It's clear that , so from (8) can get From (9), we can write After that, because is generalized strongly -convex functions of higher order, then from (10) and = 3 − 2 3 − 1 we can get If all segments on (11) times by 3 − 1 , then we have After that, we have to take = 3 − 1 = 1 2 and = 3 − 2 = 1 4 , then its clear that 3 − 2 =

CONCLUSION
Schur type inequality for generalized strongly -convex functions of higher order on [3] has a correction.