arXiv subjects

M. E. Omidvar

Publications and source records attributed to M. E. Omidvar.

5 recordsLinked to original sources

arXiv2018

New inequalities for operator concave functions involving positive linear maps

The purpose of this paper is to present some general inequalities for operator concave functions which include some known inequalities as a particular case. Among other things, we prove that if $A\in \mathcal{B}\left( \mathcal{H} \right)$ is a positive operator such that $mI\le A\le MI$ for some scalars $0<m<M$ and $Φ$ is a normalized positive linear map on $\mathcal{B}\left( \mathcal{H} \right)$, then \[\begin{aligned} {{\left( \frac{M+m}{2\sqrt{Mm}} \right)}^{r}}&\ge {{\left( \frac{\frac{1}{\sqrt{Mm}}Φ\left( A \right)+\sqrt{Mm}Φ\left( {{A}^{-1}} \right)}{2} \right)}^{r}} & \ge \frac{\frac{1}{{{\left( Mm \right)}^{\frac{r}{2}}}}Φ{{\left( A \right)}^{r}}+{{\left( Mm \right)}^{\frac{r}{2}}}Φ{{\left( {{A}^{-1}} \right)}^{r}}}{2} & \ge Φ{{\left( A \right)}^{r}}\sharpΦ{{\left( {{A}^{-1}} \right)}^{r}}, \end{aligned}\] where $0\le r\le 1$, which nicely extend the operator Kantorovich inequality.

math.FA↗

arXiv2018

Sharpening Some Classical Numerical Radius Inequalities

New upper and lower bounds for the numerical radii of Hilbert space operators are given. Among our results, we prove that if $A\in \mathcal{B} \left( \mathcal{H}\right) $ is a hyponormal operator, then for all non-negative non-decreasing operator convex $f$ on $ [0,\infty ),$ we have \[f\left( ω\left( A \right) \right)\le \frac{1}{2}\left\| f\left( \frac{1}{1+\frac{ξ_{\left| A \right|}^{2}}{8}}\left| A \right| \right)+f\left( \frac{1}{1+\frac{ξ_{\left| A \right|}^{2}}{8}}\left| {{A}^{*}} \right| \right) \right\|,\] where ${{ξ}_{\left| A\right| }}=\underset{\left| x\right| =1}{\mathop{\inf }}\,\left\{ \frac{\left\langle \left( \left| A\right| -\left| {{A}^{\ast }}\right| \right) x,x\right\rangle }{ \left\langle \left( \left| A\right| +\left| {A^{\ast }} \right| \right) x,x\right\rangle }\right\} $. Our results refine and generalize earlier inequalities for hyponormal operator.

math.FA↗

arXiv2017

A note on some inequalities for positive linear maps

We improve and generalize some operator inequalities for positive linear maps. It is shown, among other inequalities, that if $0<m\le B\le m'<M'\le A\le M$ or $0<m\le A\le m'<M'\le B\le M$, then for each $2\le p<\infty $ and $ν\in \left[ 0,1 \right]$, \begin{equation*} {{Φ}^{p}}\left( A{{\nabla }_{ν}}B \right)\le {{\left( \frac{K\left( h \right)}{{{4}^{\frac{2}{p}-1}}{{K}^{r}}\left( h' \right)} \right)}^{p}}{{Φ}^{p}}\left( A{{\#}_{ν}}B \right), \end{equation*} and \begin{equation*} {{Φ}^{p}}\left( A{{\nabla }_{ν}}B \right)\le {{\left( \frac{K\left( h \right)}{{{4}^{\frac{2}{p}-1}}{{K}^{r}}\left( h' \right)} \right)}^{p}}{{\left( Φ\left( A \right){{\#}_{ν}}Φ\left( B \right) \right)}^{p}}, \end{equation*} where $r=\min \left\{ ν,1-ν\right\}$, $h=\frac{M}{m}$ and $h'=\frac{M'}{m'}$. We also obtain an improvement of operator Pólya-Szegö inequality.

math.FA↗

arXiv2017

Around Jensen's inequality for strongly convex functions

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 $\left\| x \right\|=1$.

math.FA↗

arXiv2017

Complementary Inequalities to Improved AM-GM Inequality

Following an idea of Lin, we prove that if $A$ and $B$ be two positive operators such that $0<mI\le A\le m'I\le M'I\le B\le MI$, then \begin{equation*} {{Φ}^{2}}\left( \frac{A+B}{2} \right)\le \frac{{{K}^{2}}\left( h \right)}{{{\left( 1+\frac{{{\left( \log \frac{M'}{m'} \right)}^{2}}}{8} \right)}^{2}}}{{Φ}^{2}}\left( A\#B \right), \end{equation*} and \begin{equation*} {{Φ}^{2}}\left( \frac{A+B}{2} \right)\le \frac{{{K}^{2}}\left( h \right)}{{{\left( 1+\frac{{{\left( \log \frac{M'}{m'} \right)}^{2}}}{8} \right)}^{2}}}{{\left( Φ\left( A \right)\#Φ\left( B \right) \right)}^{2}}, \end{equation*} where $K\left( h \right)=\frac{{{\left( h+1 \right)}^{2}}}{4h}$ and $h=\frac{M}{m}$ and $Φ$ is a positive unital linear map.

math.FA↗