arXiv subjects
Publications and source records attributed to H. R. Moradi.
Matrix inequalities that extend certain scalar ones have been at the center of numerous researchers' attention. In this article, we explore the celebrated subadditive inequality for matrices via concave functions and present a reversed version of this result. Our approach will tackle concave function properties and some delicate manipulations of matrices and inner products.
In this article, we explore the celebrated Grüss inequality, where we present a new approach using the Grüss inequality to obtain new refinements of operator means inequalities. We also present several operator Grüss-type inequalities with applications to the numerical radius and entropies.
We show the following result: Let $A,B\in \mathbb{B}\left( \mathcal{H} \right)$ be two strictly positive operators such that $A\le B$ and $m{\mathbf{1}_{\mathcal{H}}}\le B\le M{\mathbf{1}_{\mathcal{H}}}$ for some scalars $0<m<M$. Then \[{{B}^{p}}\le \exp \left( \frac{M{\mathbf{1}_{\mathcal{H}}}-B}{M-m}\ln {{m}^{p}}+\frac{B-m{\mathbf{1}_{\mathcal{H}}}}{M-m}\ln {{M}^{p}} \right)\le K\left( m,M,p,q \right){{A}^{q}}\quad\text{ for }p\le 0,-1\le q\le 0\] where $K\left( m,M,p,q \right)$ is the generalized Kantorovich constant with two parameters. In addition, we obtain Kantorovich type inequalities for the chaotic order.
Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called $g-$convexity as a generalization of $\log-$convexity. Then we prove that $g-$convex functions have better estimates in certain known inequalities like the Hermite-Hadard inequality, super additivity of convex functions, the Majorization inequality and some means inequalities. Strongly related to this, we define the index of convexity as a measure of ``how much the function is convex". Applications including Hilbert space operators, matrices and entropies will be presented in the end.
The original Ando-Hiai and Golden-Thompson inequalities present comparisons for the operator geometric mean $\sharp_v$ when $0\leq v\leq 1.$ Our main target in this article is to study these celebrated inequalities for means other than the geometric mean and for the geometric mean when $v\not\in [0,1].$
Mercer inequality for convex functions is a variant of Jensen's inequality, with an operator version that is still valid without operator convexity. This paper is two folded. First, we present a Mercer-type inequality for operators without assuming convexity nor operator convexity. Yet, this form refines the known inequalities in the literature. Second, we present a log-convex version for operators. We then use these results to refine some inequalities related to quasi-arithmetic means of Mercer's type for operators.
In this article, we present some new inequalities for numerical radius of Hilbert space operators via convex functions. Our results generalize and improve earlier results by El-Haddad and Kittaneh. Among several results, we show that if $A\in \mathbb{B}\left( \mathcal{H} \right)$ and $r\ge 2$, then \[{{w}^{r}}\left( A \right)\le {{\left\| A \right\|}^{r}}-\underset{\left\| x \right\|=1}{\mathop{\inf }}\,{{\left\| {{\left| \left| A \right|-w\left( A \right) \right|}^{\frac{r}{2}}}x \right\|}^{2}}\] where $w\left( \cdot \right)$ and $\left\| \cdot \right\|$ denote the numerical radius and usual operator norm, respectively.
This paper is mainly devoted to studying operator Jensen inequality. More precisely, a new generalization of Jensen inequality and its reverse version for convex (not necessary operator convex) functions have been proved. Several special cases are discussed as well.
We give an alternative lower bound for the numerical radii of Hilbert space operators. As a by-product, we find conditions such that \begin{equation*} ω\left(\left[\begin{array}{cc} 0 & R \\ S & 0 \end{array}\right]\right)=\frac{\Vert R \Vert +\Vert S\Vert }{2} \end{equation*} where $R, S \in \mathbb{B}(\mathcal{H})$.
We establish a reverse inequality for Tsallis relative operator entropy involving a positive linear map. In addition, we present converse of Ando's inequality, for each parameter. We give examples to compare our results with the known results by Furuta and Seo. In particular, we establish an extension and a reverse of the Löwner-Heinz inequality under certain condition. Some interesting consequences of inner product spaces and norm inequalities are also presented.
In this article, we present exponential-type inequalities for positive linear mappings and Hilbert space operators, by means of convexity and the Mond-Pe\v carić method. The obtained results refine and generalize some known results. As an application, we present extensions for operator-like geometric and harmonic means.
We show the following result: Let $A$ be a positive operator satisfying $0<m{\mathbf{1}_{\mathcal{H}}}\le A\le M{\mathbf{1}_{\mathcal{H}}}$ for some scalars $m,M$ with $m<M$ and $Φ$ be a normalized positive linear map, then \[Φ\left( {{A}^{-1}} \right)\le Φ\left( {{m}^{\frac{A-M{\mathbf{1}_{\mathcal{H}}}}{M-m}}}{{M}^{\frac{m{\mathbf{1}_{\mathcal{H}}}-A}{M-m}}} \right)\le \frac{{{\left( M+m \right)}^{2}}}{4Mm}Φ{{\left( A \right)}^{-1}}.\]
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.
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.
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.
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 $\left\| x \right\|=1$.
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.
We give a Jensen operator inequality for strongly convex functions. As a corollary, we improve operator Holder-McCarthy inequality under suitable conditions.