Around Jensen's inequality for strongly convex functions

TL;DR

This paper develops new inequalities for strongly convex functions, including Jensen's type and Jensen-Mercer type, and applies them to operator inequalities, improving classical results like H"older-McCarthy inequality.

Contribution

It introduces novel inequalities for strongly convex functions and extends Jensen's inequality to operators, with applications to mean inequalities and operator bounds.

Findings

Derived new Jensen-type inequalities for strongly convex functions.

Extended Jensen's inequality to the operator setting for positive operators.

Improved H"older-McCarthy inequality under certain spectral conditions.

Abstract

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen's type and Jensen-Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen's operator inequality for strongly convex functions. As a corollary, we improve H\"older-McCarthy inequality under suitable conditions. More precisely we show that if $Sp\left( A \right)\subset I\subseteq \left( 1,\infty \right)$, then \[{{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right),\quad r\ge 2\] and if $Sp\left( A \right)\subset I\subseteq \left( 0,1 \right)$, then \[\left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right),\quad 0<r<1\] for each positive operator $A$ and $x\in \mathcal{H}$ with $\left\| x \right\|=1$.