INEQUALITIES VIA MONTGOMERY IDENTITY AND RELATED RESULTS

Abstract

In this thesis, we will study the generalizations of some of the most famous and fundamental inequalities that is Jensen’s inequality, Jensen-Steffensen’s inequality, Sherman’s inequality and majorization inequality. The generalizations of these inequalities are given by using Montgomery identity and Green functions. The first chapter contains some basic definitions, concepts and several elementary results from the theory of mathematical inequalities. These includes the notion of convex functions, n−convex functions, Jensen’s inequality, Sherman’s inequality, majorization inequality and exponential convexity. In the second chapter, we present the generalizations of Jensen’s inequality and Jensen-Steffensen’s inequality by using Montgomery identity. The generalizations of the converse of Jensen’s inequality are given and we construct bounds for the remainders associated with the generalized inequalities. The linear functionals related to the generalized inequalities are defined and mean value theorems for the functionals are proved. Furthermore, the results related with the exponential convexity and n−exponential convexity for the functionals are established. At the end of the chapter, some applications of our obtained results are also given. The third chapter contains the generalizations of the Jensen, Jensen-Steffensen’s, and the converse of Jensen’s inequalities by using Green function and Montgomery identity. We also present some bounds for the generalized inequalities, construct linear functionals from the generalized differences and give mean value theorems for the functionals. The exponential convexity and log-convexity are also given. At the 4 5 end of the chapter, the applications of our obtained results are given. In the fourth chapter, we give general identity for the difference of Sherman’s inequality viaMontgomery identity. From this identity, the generalizations of Sherman’s theorem are obtained. The generalized Sherman’s inequality for n-convex functions are established. The bounds for the remainders associated with the generalized Sherman’s inequalities are also given. We prove mean value theorems and n-exponential convexity which leads to exponential convexity and give applications of our obtained results. In the last chapter, we give the identities for twice differentiable functions and Green functions. By applying these identities, we obtain the equivalent conditions for Sherman’s inequality. We give the generalizations of Sherman and majorization inequalities by using Green functions and Montgomery identity and present bounds for the remainders related with the generalized inequalities. The linear functionals associated with the generalized inequalities are also constructed and give mean value theorems and related results for the functionals. We also define a family of functions which support our results for exponentially convex functions and construct a class of means.