Several Kinds Of Generalized Convex Functions And Their Integral Inequalities

Convex function is a kind of important function and has wide applications in pure and applied mathematics. Since convex analysis appeared in 1960s, there has been tremendous interest in generalizing convex function. In recent years, the generalized convex function and its application have been hot issues. In this paper, we study s-logarithmically preinvex functions and several kinds of generalized operator convex functions in Hilbert space.Firstly, in Chapter 3, we define s-logarithmically preinvex function which promote prein-vex function and establish an equality of n-times differentiable function. Utilizing this equality and the properties of s-logarithmically preinvex function, we obtain some integral inequalities and study their error estimation.Secondly, in Chapter 4, we define the operator s-preinvex functions in Hilbert space and give a example and a necessary and sufficient condition of which the function is opera-tor s-preinvex convex function. Then we establish Hermite-Hadamard type integral inequality and give the estimation on both sides of this inequality. Finally, we establish Hermite-Hadamard type inequalities for the product of two operator s-preinvex convex functions.In Chapter 5, we define operator m-convex function and operator (?, m)-convex function in Hilbert space which generalize m-convex function and (?, m)-convex function, give some ex-amples, and prove their some properties. Meanwhile, we establish some new Hermite-Hadamard type inequalities.Finally, in Chapter 6, we generalize the convex function on the co-ordinates in rectangular domains of the plane to Hilbert space and define operator convex function on the co-ordinates. It is pointed out that every operator convex function is operator convex function on the co-ordinates, but the converse is not generally true, and a counterexample is given. In particular, we give a necessary and sufficient condition of which the function is operator convex function on the co-ordinates in Hilbert space. Finally, we establish some Hermite-Hadamard type integral inequalities for operator convex function on the co-ordinates.