Refinements of Jensen-Steffensen and Related Inequalities

Abstract

As the title suggests, the present dissertation deals with the refinements of Jensen- Steffensen and related inequalities. We extend some classical results and give some new refinements of several well known inequalities including Jensen’s inequality, Jensen- Steffensen inequality, Hermite-Hadamard inequality, majorization-type inequality, generalized weighted Favard and Berwald inequalities. We also provide the refine- ments of some companion inequalities to the Jensen’s inequality, namely Slater’s inequality and the inequalities obtained by M. Mati ́ and J. Peˇari ́ in [44]. We c c c sharpen the lower bounds of the Jensen’s functional. Some inequalities in terms of Gˆteaux derivatives for convex functions are also provided. Finally, we present not aonly the generalizations of Hardy-Littlewood-P ́lya inequality [26, Theorem 134] but o also generalizations of some results given in [8] and [36].