General Inequalities for Generalized Convex Functions

Abstract

It is a fact that, the theory of inequalities, priding on a history of more than two cen- turies, plays a significant role in almost all fields of mathematics and in major areas of science. In the present dissertation, we will study the general inequalities, namely integral inequalities and discrete inequalities for generalized convex functions. There- fore, we will introduce some generalized convex functions which include functions −convex functions, and n−convex func- with nondecreasing increments, ∆− and tions of higher orders. By using these functions, we will provide a generalization of the Brunk’s theorem, of the Levinson-type inequalities, of the Burkill-Mirsky-Peˇari ́’s re- c c sult and of the result related to arithmetic integral mean. We will also discuss the Popoviciu-type characterization of positivity of sums and integrals for higher order convex functions of n variables and we will give some related results. Our disserta- tion also provides generalizations of some of the celebrated and fundamental identities ˇ and inequalities including Montgomery’s identities, Ostrowski-, Gr ̈ss-, Cebyˇev- and u s Fan-type inequalities. Moreover, we will also apply an elegant method of producing n−exponentially and logarithmically convex functions for positive linear function- als constructed with the help of majorization-type results, Favard-, Berwald- and Jensen-type inequalities. The generalization and the following refinements of Jensen- Mercer’s inequalities are also provided with some applications. The Lagrange- and Cauchy-type mean value theorems are also proved and shown to be useful in studying Stolarsky-type means defined for the positive linear functionals.