Generalized Convex Functions on Fractal Sets and Two Related Inequalities

Abstract

In the paper, we introduce the generalized convex function on fractal sets (0 1)real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen's inequality and generalized Hermite-Hadamard's inequality. Furthermore, some applications are given.

On some inequalities for generalized s-convex functions and applications on fractal sets

“…In the same paper, [12], Mo and Sui proved that all functions which are generalized s-convex in the second sense, for s ∈ (0, 1), are non-negative.…”

“…In [12], the following example was given: let 0 < s < 1 and a α 1 , a α 2 , a α 3 ∈ R α . Define for x ∈ R + ,…”

“…holds, then f is called a generalized convex function on U [14]. In α = 1, we have convex function, which means that if P 1 , P 2 , and P 3 are three distinct points on the graph of f with P 2 between P 1 and P 3 , then P 2 is on or below the chord P 1 P 3 [7].…”

“…is known as generalized Hermite-Hadmard's inequality [14]. Many authers paid attention to the study of generalized Hermite-Hadmard's inequality and generalized convex function, see [9,13].…”

“…In [12], Mo and Sui introduced the definitions of two kinds of generalized s-convex functions on fractal sets such as follows.…”