EXPONENTIAL CONVEXITY AND CAUCHY MEANS INTRODUCED BY INEQUALITIES FOR MONOTONE AND RELATED FUNCTIONS

Abstract

There is a lot of literature available on convexity of functions. In contrast, the litera- ture on the exponential convexity is hardly available as there is no operative criteria to recognize exponential convexity. It is not easy to find and construct exponen- tially convex functions even-though it is very important sub-class of convex functions in many ways. For example, Laplace transform of a non-negative finite measure is an exponentially convex function. Moreover, one can derive results about positive definite functions from the properties of exponentially convex functions. We consider the differences of Petrovi ́c and related inequalities, Giaccardi and related inequalities, Chebyshev’s inequality, inequality introduced by Lupa ̧s and in- equality introduced by Levin-Steckin to construct positive semi-definite matrices. We derive the classes of exponentially convex functions for the differences and discuss their properties. We introduce Cauchy means and prove the monotonicity of these means by using the important property of exponentially convex functions. As an application, we establish the mean value theorem of Cauchy type. In the first chapter, we organize some basic notions and results. In the second chapter, we use the Jensen-Petrovi ́c’s inequality for star-shaped functions, generalized Petrovi ́c inequality and inequality introduced by Vasi ́c and Peˇcari ́c for increasing functions to give results related to power sums. We consider the difference of these inequalities to construct positive semi-definite matrices for certain classes of functions to derive families of exponential and logarithmic convex functions. We introduce new means of Cauchy type related to power sums and estab- lish comparison between them. We, also illustrate integral analogs for some results viiviii and prove related mean value theorems of Cauchy type. In the third chapter, we prove the Giaccardi’s type inequality for star-shaped type functions and the Giaccardi’s inequality for convex-concave antisymmetric functions. We assume the differences of Giaccardi’s type inequality, Giaccardi’s inequality for special case and inequality introduced by Vasi ́c and Stankovi ́c. By using different classes of functions, we formulate families of exponentially convex functions related to these differences. We introduce new means of Cauchy type and prove monotonicity of these means. We, also exhibit related mean value theorems of Cauchy type. In the fourth chapter, we consider the non-negative difference of Chebyshev’s inequality as Chebyshev functional. We construct symmetric matrices generated by Chebyshev functional for a class of increasing functions and prove positive semi- definiteness of matrices which implies the exponential and logarithmic convexity of the Chebyshev functional. Moreover, we demonstrate mean value theorems of Cauchy type for the Chebyshev functional and its generalized form. In the last chapter, we start by considering an inequality related to the Cheby- shev’s inequality given by A. Lupa ̧s in 1972 but instead of monotone functions there are convex functions. In addition to that we consider the reverse of Chebyshev’s in- equality without weights introduced by Levin-Steckin; here one function is symmetric increasing and other is continuous convex. By taking the non-negative differences of each inequality, we construct families of exponentially convex functions. We introduce related Cauchy means and prove related mean value theorems of Cauchy type.