arXiv subjects

Mohammad Sababheh

Publications and source records attributed to Mohammad Sababheh.

47 records · Page 3Linked to original sources

arXiv2019

More accurate numerical radius inequalities (II)

In a recent work of the authors, we showed some general inequalities governing numerical radius inequalities using convex functions. In this article, we present results that complement the aforementioned inequalities. In particular, the new versions can be looked at as refined and generalized forms of some well known numerical radius inequalities. Among many other results, we show that \[\left\| f\left( \frac{{{A}^{*}}A+A{{A}^{*}}}{4} \right) \right\|\le \left\| \int_{0}^{1}{f\left( \left( 1-t \right){{B}^{2}}+t{{C}^{2}} \right)dt} \right\|\le f\left( {{w}^{2}}\left( A \right) \right),\] when $A$ is a bounded linear operator on a Hilbert space having the Cartesian decomposition $A=B+iC.$ This result, for example, extends and refines a celebrated result by kittaneh.

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arXiv2019

More accurate numerical radius inequalities

In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine some known results in this field. Among other inequalities, it is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then \[{{w}^{2}}\left( A \right)\le \left\| \int_{0}^{1}{{{\left( t\left| A \right|+\left( 1-t \right)\left| {{A}^{*}} \right| \right)}^{2}}dt} \right\|\le \frac{1}{2}\left\| \;{{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right\|\] where $w\left( A \right)$ and $\left\| A \right\|$ are the numerical radius and the usual operator norm of $A$, respectively.

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arXiv2019

Subadditive inequalities for operators

In this article, we present a new subadditivity behavior of convex and concave functions, when applied to Hilbert space operators. For example, under suitable assumptions on the spectrum of the positive operators $A$ and $B$, we prove that \[{{2}^{1-r}}{{\left( A+B \right)}^{r}}\le {{A}^{r}}+{{B}^{r}}\quad\text{ for }r>1\text{ and }r<0,\] and \[{{A}^{r}}+{{B}^{r}}\le {{2}^{1-r}}{{\left( A+B \right)}^{r}}\quad\text{ for }r\in \left[ 0,1 \right].\] These results provide considerable generalization of earlier results by Aujla and Silva. Further, we present several extensions of the subadditivity idea initiated by Ando and Zhan, then extended by Bourin and Uchiyama.

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arXiv2018

Some inequalities for interpolational operator means

Using the properties of geometric mean, we shall show for any $0\le α,β\le 1$, \[f\left( A{{\nabla }_{α}}B \right)\le f\left( \left( A{{\nabla }_{α}}B \right){{\nabla }_{β}}A \right){{\sharp}_{α}}f\left( \left( A{{\nabla }_{α}}B \right){{\nabla }_{β}}B \right)\le f\left( A \right){{\sharp}_{α}}f\left( B \right)\] whenever $f$ is a non-negative operator log-convex function, $A,B\in \mathcal{B}\left( \mathcal{H} \right)$ are positive operators, and $0\le α,β\le 1$. As an application of this operator mean inequality, we present several refinements of the Aujla subadditive inequality for operator monotone decreasing functions. Also, in a similar way, we consider some inequalities of Ando's type. Among other things, it is shown that if $Φ$ is a positive linear map, then \[Φ\left( A{{\sharp}_{α}}B \right)\le Φ\left( \left( A{{\sharp}_{α}}B \right){{\sharp}_{β}}A \right){{\sharp}_{α}}Φ\left( \left( A{{\sharp}_{α}}B \right){{\sharp}_{β}}B \right)\le Φ\left( A \right){{\sharp}_{α}}Φ\left( B \right).\]

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arXiv2018

On the matrix harmonic mean

The main goal of this article is to present new types of inequalities refining and reversing inequalities of the harmonic mean of scalars and matrices. Furthermore, implementing the spectral decomposition of positive matrices, we present a new type of inequalities treating certain harmonic matrix perturbation.

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arXiv2018

New sharp inequalities for operator means

New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.

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arXiv2017

Quadratic interpolation of the Heinz means

The main goal of this article is to present several quadratic refinements and reverses of the well known Heinz inequality, for numbers and matrices, where the refining term is a quadratic function in the mean parameters. The proposed idea introduces a new approach to these inequalities, where polynomial interpolation of the Heinz function plays a major role. As a consequence, we obtain a new proof of the celebrated Heron-Heinz inequality proved by Bhatia, then we study an optimization problem to find the best possible refinement. As applications, we present matrix versions including unitarily invariant norms, trace and determinant versions.

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arXiv2016

Unitarily invariant norm inequalities for elementary operators involving $G_{1}$ operators

In this paper, motivated by perturbation theory of operators, we present some upper bounds for $|||f(A)Xg(B)+ X|||$ in terms of $|||\,|AXB|+|X|\,|||$ and $|||f(A)Xg(B)- X|||$ in terms of $|||\,|AX|+|XB|\,|||$, where $A, B$ are $G_{1}$ operators, $|||\cdot|||$ is a unitarily invariant norm and $f, g$ are certain analytic functions. Further, we find some new upper bounds for the the Schatten $2$-norm of $f(A)X\pm Xg(B)$. Several special cases are discussed as well.

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arXiv2016

Advanced refinements of Young and Heinz inequalities

In this article, we prove several multi-term refinements of Young type inequalities for both real numbers and operators improving several known results. Among other results, we prove \begin{eqnarray*} A\#_νB&+&\sum_{j=1}^{N}s_{j}(ν)\left(A\#_{α_j(ν)}B+A\#_{2^{1-j}+α_j(ν)}B-2A\#_{2^{-j}+α_j(ν)}B\right)\leq A\nabla_νB, \end{eqnarray*} for the positive operators $A$ and $B$, where $0\leq ν\leq 1, N\in\mathbb{N}$ and $α_j(ν)$ is a certain function. Moreover, some new Heinz type inequalities involving the Hilbert-Schmidt norm are established.

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arXiv2016

Convex functions and means of matrices

In this article, we prove that convex functions and log-convex functions obey certain general refinements that lead to several refinements and reverses of well known inequalities for matrices, including Young's inequality, Heinz inequality, the arithmetic-harmonic and the geometric-harmonic mean inequalities.

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arXiv2016

Means refinements via convexity

The main goal of this article is to find the exact difference between a convex function and its secant, as a limit of positive quantities. This idea will be expressed as a convex inequality that leads to refinements and reversals of well established inequalities treating different means. The significance of these inequalities is to write one inequality that brings together and refine almost all known inequalities treating the arithmetic, geometric, harmonic and Heinz means, for numbers and operators.

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