Applications Of Lipschitz Condition In Convex Function

The mathematicians in the word have researched the convex functions extensively and widely since the middle stage of the 20th century. The convex functions have a lot of superior properties. The definition and the convexity of it are established on the base of inequality. The fact makes convex functions be important tools to prove inequality. So we can establish and improve lots of inequalities. The inequalities on convex functions take an important part in the application and the basic theory of mathematics. The Hadamard inequality and the Jensen inequality are the important and classic contents of convex functions. Lipschitz condition is given by German mathematician Rudolf Lipschitz. It is a widely used. It has wide application in the uniform of functions of one variable and multivariate function and in the fixed point theory of functional analysis.In 2000, S. S.Dragomir found that convex functions of one variable under certain conditions satisfied Lipschitz condition. But the functions satisfied Lipschitz conditions are not all the convex functions. S.S.Dragomir and professor Wang Liangcheng applied Lipschitz condition to Hadamard inequality of convex functions successively and established a group of Hadamard inequality satisfied Lipschitz condition.In the second part of this paper, the jobs of S.S.Dragomir and professor Wang Liangcheng are continued. We convert the convex function of Hadamard inequality into the function satisfied Lipschitz condition and establish a new kind of Hadamard inequality with Lipschitz condition.In the third part of this paper, we use the methods of thought in the second part and apply Lipschitz condition to Jensen inequality, and then we get a kind of Jensen inequality with Lipschitz condition.In the fourth part of this paper, we revert the functions satisfied Lipschitz condition in the second and third part into the convex functions and create some new kind inequality based on Hadamard inequality and Jensen inequality of convex functions.The theory about Hadamard inequality and Jensen inequality of convex functions is greatly enriched in this paper.