Khuram Ali Khan

Differences of Weighted Mixed Symmetric Means and Related Results

Some improvements of classical Jensens inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. Related Cauchy means are also defined, and their monotonicity is established as an application.

Cyclic refinements of the discrete and integral form of Jensen’s inequality with applications

Abstract In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using this result, we obtain new refinements of the classical Jensen’s inequality. We investigate m-exponential convexity of some functionals coming from the new refinements. To apply our results we define some new mixed symmetric means, generalized means, and Cauchy means, and study their properties.

Cyclic Refinements of the Different Versions of Operator Jensen's Inequality

Refinements of the operator Jensen inequality for convex and operator convex func- tions are given by using cyclic refinements of the discrete Jensen inequality. Similar refinements are fairly rare in the literature. Some applications of the results to norm inequalities, to the Holder- McCarthy inequality and to generalized weighted power means for operators are presented.