ON GENERALIZATION OF INEQUALITIES FOR MONOTONE FUNCTIONS

Abstract

The thesis comprises of generalized inequalities for monotone functions from which we deduce important inequalities such as reversed Hardy type inequalities, general- ized Hermite-Hadamard’s inequalities etc by putting suitable functions. The present thesis is divided into three chapters. The first chapter includes generalized inequalities given for C-monotone functions and multidimensional monotone functions. As a result of these inequalities, we de- duce reversed Hardy inequalities for C-monotone functions and multidimensional re- versed Hardy type inequalities with the optimal constant. Furthermore, we construct functionals from the differences of above inequalities and gives their n-exponential convexity and exponential convexity. By using log-convexity of these functionals we give refinements of these inequalities. Also we give mean-value theorems for these functionals and deduce Cauchy means for them. The second chapter consists of inequalities valid for monotone functions of the form f /h and f /h. These are also very interesting as by putting suitable functions we get one side of Hermite-Hadamard’s inequality and generalized Hermite-Hadamard’s inequality. Similarly as in the first chapter, we make functionals of these inequalities and gives results regarding n-exponential convexity and exponential convexity. Also we give mean value theorems of Lagrange and Cauchy type as well as we obtain non- symmetric Stolarsky means with and without parameter. In the third and the last chapter we consider Petrovi ́ type functionals obtained from c Petrovi ́ type inequalities and investigate their properties like superadditivity, sub- c additivity, monotonicity and n-exponential convexity. Also at the end of each chapter we discuss examples in which we construct further exponential convex functions and their relative properties.