Around Jensen's inequality for strongly convex functions

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 Hölder-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 $\Vert x\Vert =1$.

arXiv admin note: text overlap with arXiv:1611.01084, to appear in Aequationes Math