Studies On Properties For Three Types Of H-Convex Functions

Convex functions are very important in mathematics, they have many good properties. This class of functions not only plays an important role in pure mathematics, but also is widely used in many mathematical branches. Since the theory of convex functions was established at the beginning of20century, its study has been being very active research subject. Many applied inequalities are derived from the convex function. In recent years, many mathematicians at home and abroad research all kinds of convex function, and research the inner relations between convex function and inequality, then abtain a lot of important inequalities. At the same time, they prove some complex inequalities using certain properties of convex function. Obviously, convex function plays an irreplaceable role in the inequality research.Convex function concept is put forward by Jensen ([1. p.59]).Let f:I∈Râ†'R, for any x1,x2∈I, we have Then f is said to be J convex function on I(Also called the midpoint convex function).The general form of the concept of convex function (see [1, p.60;2, p.1;3, p.413]).Let f:I∈Râ†'R, for any x,y∈I,λ∈[0,1], we have f (λX+(1-λ)y)≤λf (x)+(1-λ)f (y),Then f is said to be convex function on I.Since the definition of convex function was proposed by Jensen, the mathematicians studyed on convex function actively in more than one hundred years. They promote several concepts of convex function, such as geometric convex function, logarithmic convex function, harmonic convex function, GA-convex function, r-convex function[4-5], m-convex function[6-7],(a,m)-convex function[8], P-convex function[9], h-convex function[10]generalized convex function etc.This paper defines two kinds of new convex function, and discusses their basic properties and integral inequality. It also studies some properties of (h,m)-convex function and integral inequality.The thesis is divided into four chapters:In Chapter1, we briefly describe the development situation, the main researching contents and the present researching situation.In Chapter2, We introduce the concept of h-geometric convex function, and also discusse its fundamental properties and integral inequality.In Chapter3, We introduce the concept of h-logarithmic convex function, and discuss its fundamental properties and integral inequality.In Chapter4, We discuss some properties of (h,m)-convex function and integral inequality.