We prove several numerical radius inequalities for linear operators in Hilbert spaces. It is shown, among other inequalities, that if $A$ is a bounded linear operator on a complex Hilbert space, then \[ω\left( A \right)\le \frac{1}{2}\sqrt{\left\| {{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right\|+\left\| \left| A \right|\left| {{A}^{*}} \right|+\left| {{A}^{*}} \right|\left| A \right| \right\|},\] where $ω\left( A \right)$, $\left\| A \right\|$, and $\left| A \right|$ are the numerical radius, the usual operator norm, and the absolute value of $A$, respectively. This inequality provides a refinement of an earlier numerical radius inequality due to Kittaneh, namely, \[ω\left( A \right)\le \frac{1}{2}\left( \left\| A \right\|+{{\left\| {{A}^{2}} \right\|}^{\frac{1}{2}}} \right).\] Some related inequalities are also discussed.
In this work, some generalizations and refinements inequalities for numerical radius of the product of Hilbert space operators are proved. New inequalities for numerical radius of block matrices of Hilbert space operators are also established.
In this paper, we give a new inequality for convex functions of real variables, and we apply this inequality to obtain considerable generalizations, refinements, and reverses of the Young and Heinz inequalities for positive scalars. Applications to unitarily invariant norm inequalities involving positive semidefinite matrices are also given.
We give some new refinements and reverses Young inequalities in both additive-type and multiplicative-type for two positive numbers/operators. We show our advantages by comparing with known results. A few applications are also given. Some results relevant to the Heron mean are also considered.
In this paper, sharp results on operator Young's inequality are obtained. We first obtain sharp multiplicative refinements and reverses for the operator Young's inequality. Secondly, we give an additive result, which improves a well-known inequality due to Tominaga. We also provide some estimates for $A{{\sharp}_{v}}B-A{{\nabla }_{v}}B$ in which $v\notin \left[ 0,1 \right]$.
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.
We present sharp lower bounds for the A-numerical radius of semi-Hilbertian space operators. We also present an upper bound. Further we compute new upper bounds for the $B$-numerical radius of $2 \times 2$ operator matrices where $B = \textit{diag}(A,A)$, $A$ being a positive operator. As an application of the A-numerical radius inequalities, we obtain a bound for the zeros of a polynomial which is quite a bit improvement of some famous existing bounds for the zeros of polynomials.
In this work, a refinement of the Cauchy--Schwarz inequality in inner product space is proved. A more general refinement of the Kato's inequality or the so called mixed Schwarz inequality is established. Refinements of some famous numerical radius inequalities are also pointed out. As shown in this work, these refinements generalize and refine some recent and old results obtained in literature. Among others, it is proved that if $T\in\mathscr{B}\left(\mathscr{H}\right)$, then \begin{align*} ω^{2}\left(T\right) &\le \frac{1}{12} \left\| \left| T \right|+\left| {T^* } \right|\right\|^2 + \frac{1…
In this paper, several refinements of the Berezin number inequalities are obtained. We generalize inequalities involving powers of the Berezin number for product of two operators acting on a reproducing kernel Hilbert space $\mathcal H=\mathcal H(Ω)$ and also improve them. Among other inequalities, it is shown that if $A,B\in {\mathcal B}(\mathcal H)$ such that $|A|B=B^{*}|A|$, $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ satisfying $f(t)g(t)=t\,(t\geq 0)$, then \begin{align*} &\textbf{ber}^{p}(AB)\leq r^{p}(B)\times\\&\left(\textbf{ber} \big(\frac{1}αf^{αp}(|A|)+\frac{1}βg…
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.
In this paper, we give some refinements for the second inequality in $\frac{1}{2}\|A\| \leq w(A) \leq \|A\|$, where $A\in B(H)$. In particular, if $A$ is hyponormal by refining the Young inequality with the Kantorovich constant $K(\cdot, \cdot)$, we show that $w(A)\leq \dfrac{1}{\displaystyle {2\inf_{\| x \|=1}}ζ(x)}\| |A|+|A^{*}|\|\leq \dfrac{1}{2}\| |A|+|A^*|\|$, where $ζ(x)=K(\frac{\langle |A|x,x \rangle}{\langle |A^{*}|x,x \rangle},2)^{r},~~~r=\min\{λ,1-λ\}$ and $0\leq λ\leq 1$ . We also give a reverse for the classical numerical radius power inequality $w(A^{n})\leq w^{n}(A)$ for any oper…
We establish in this paper some inequalities for analytic and convex functions on an open interval and positive normalized functionals defined on a Hermitian unital Banach *-algebra. Reverses and refinements of Jensen's and Slater's type inequalities are provided. Some examples for particular convex functions of interest are given as well.
In this article we study the Heinz and Hermite-Hadamard inequalities. We derive the whole series of refinements of these inequalities involving unitarily invariant norms, which improve some recent results, known from the literature. We also prove that if $A , B, X\in M_n(\mathbb{C})$ such that $A$ and $B$ are positive definite and $f$ is an operator monotone function on $(0,\infty)$. Then \begin{equation*} |||f(A)X-Xf(B)|||\leq \max\{||f'(A)||, ||f'(B)||\} |||AX-XB|||. \end{equation*} Finally we obtain a series of refinements of the Heinz operator inequalities, which were proved by Kittaneh an…
In this work, an improvement of Hölder-McCarty inequality is established. Based on that, several refinements of the generalized mixed Schwarz inequality are obtained. Consequently, some new numerical radius inequalities are proved. New inequalities for numerical radius of $n\times n$ matrix of Hilbert space operators are proved as well. Some refinements of some earlier results were proved in literature are also given. Some of the presented results are refined and it shown to be better than earlier results were proved in literature.
In this paper, we show some refinements of generalized numerical radius inequalities involving the Young and Heinz inequalities. In particular, we present \begin{align*} w_{p}^{p}(A_{1}^{*}T_{1}B_{1},...,A_{n}^{*}T_{n}B_{n})\leq\frac{n^{1-\frac{1}{r}}}{2^{\frac{1}{r}}}\Big\|\sum_{i=1}^{n}[B_{i}^{*} f^{2}(|T_{i}|)B_{i}]^{rp}+[A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i}]^{rp}\Big\|^{\frac{1}{r}} -\inf_{\|x\|=1}η(x), \end{align*} where $T_{i}, A_{i}, B_{i} \in {\mathbb B}({\mathscr H})\,\,(1\leq i\leq n)$, $f$ and $g$ are nonnegative continuous functions on $[0, \infty)$ satisfying $f(t)g(t)=t$ for all $t\i…
In this article, we study the further refinements and reverses of the Young and Heinz inequalities with the Kantorovich constant. These modified inequalities are used to establish corresponding operator inequalities on Hilbert space and Hilbert-Schmidt norm inequalities.
We give new inequalities for $A$-operator seminorm and $A$-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space $\mathcal{H}$ and a non-zero positive bounded linear operator $A$ on $\mathcal{H},$ we show with among other seminorm inequalities, if $S,T,X\in \mathcal{B}_A(\mathcal{H})$, i.e., if $A$-adjoint of $S,T,X$ exist then $$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, we prove that if $T\in \mathcal{B}_A(\mathcal{H})$ then \begin{eqnarr…
This article improves the triangle inequality for complex numbers, using the Hermite-Hadamard inequality for convex functions. Then, applications of the obtained refinement are presented to include some operator inequalities. The operator applications include numerical radius inequalities and operator mean inequalities.
In this paper, we present various inequalities for generalized numerical radius of $2 \times 2$ block matrices for Schatten p-norm. Moreover, we give a refinement of the triangle inequality for the Schatten p-generalized numerical radius.
Refinements of some recent reverse inequalities for the celebrated Cauchy-Bunyakovsky-Schwarz inequality in 2-inner product spaces are given. Using this framework, applications for determinantal integral inequalities are also provided.
Motivated by the refinements and reverses of arithmetic-geometric mean and arithmetic-harmonic mean inequalities for scalars and matrices, in this article, we generalize the scalar and matrix inequalities for the difference between arithmetic mean and harmonic mean. In addition, relevant inequalities for the Hilbert-Schmidt norm and determinant are established.
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.
In this paper, we introduce the concept of operator geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results.
The Cauchy-Schwarz, Buzano and Kre\uın inequalities are three inequalities about inner product. The main goal of this article is to present refinements of Buzano and Cauchy-Schwarz inequalities, and to present a new proof of a refined version of a Kre\uın-type inequality. Applications that include Buzano-type inequalities for certain operators, operator norm and numerical radius inequalities of Hilbert space operators will be presented.
In this article we present some new comparisons between the Heinz and Heron operator means, which improve some recent results known from the literature. We derive some refinements of these inequalities for unitarily invariant norms with the help of the contractive maps.
In this paper, we introduce the concept of operator arithmetic-geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results. Moreover, some unitarily invariant norm inequalities are established.
In this paper we first introduce the Heron and Heinz means of two convex functionals. Afterwards, some inequalities involving these functional means are investigated. The operator versions of our theoretical functional results are immediately deduced. We also obtain new refinements of some known operator inequalities via our functional approach in a fast and nice way.
In this paper, we provide some inequalities for $P$-class functions and self-adjoint operators on a Hilbert space including an operator version of the Jensen's inequality and the Hermite-Hadamard's type inequality. We improve the Hölder-MacCarthy inequality by providing an upper bound. Some refinements of the Jensen type inequality for $P$-class functions will be of interest.
In this paper, we study refinements of some inequalities related to Young inequality for scalar and for operator. As our main results, we show refined Young inequalities for two positive operators. This results refine the ordering relations among the arithmetic mean, the geometric mean and the harmonic mean. Finally, we give supplements for refined Young inequalities for two positive real numbers. And then we also give operator inequalities based on the supplemental inequalities.
We give some new refinements of Heinz inequality and an improvement of the reverse Young's inequality for scalars and we use them to establish new inequalities for operators and the Hilbert-Schmidt norm of matrices. We give a uniformly and abbreviated form of the inequalities presented by Kittaneh and Mansarah, and the inequalities presented by Kai and we obtain some of their operator and matrix versions.
Motivated by some recently established operator Jensen-type inequalities related to a usual convexity, in the present paper we derive several more accurate operator Jensen-type inequalities for certain subclasses of convex functions. More precisely, we obtain interpolating series of Jensen-type inequalities for log-convex and non-negative superquadratic functions. In particular, we obtain the corresponding refinements of the Jensen-Mercer operator inequality for such classes of functions.
In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and refinements of some well known inequalities, including multiplicative behavior of the numerical radius and norm bounds. Among many other applications, it is shown that if $T$ is accretive-dissipative, then \[\frac{1}{\sqrt{2}}\left\| T \right\|\le ω\left( T \right),\] where $ω\left( \cdot \right)$ and $\left\| \cdot \right\|$ denote the numerical radius and the usual operator norm, respectively. This inequality…
In this note, some refinements of Young inequality and its reverse for positive numbers are proved and using these inequalities some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.
In this article, we obtain several new weighted bounds for the numerical radius of a Hilbert space operator. The significance of the obtained results is the way they generalize many existing results in the literature; where certain values of the weights imply some known results, or refinements of these results. In the end, we present some numerical examples that show how our results refine the well known results in the literature, related to this topic.
We prove some refinements of an inequality due to X. Zhan in an arbitrary complex Hilbert space by using some results on the Heinz inequality. We present several related inequalities as well as new variants of the Corach--Porta--Recht inequality. We also characterize the class of operators satisfying $\left\Vert SXS^{-1}+S^{-1}XS+kX\right\Vert \geq (k+2)\left\Vert X\right\Vert$ under certain conditions.
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 study refinements between spectral resolutions in an arbitrary II$_1$ factor $\M$ and obtain diffuse (maximal) refinements of spectral resolutions. We construct models of operators with respect to diffuse spectral resolutions. As an application we obtain new characterizations of sub-majorization and spectral preorder between positive operators in $\M$ and new versions of some known inequalities involving these preorders.
Let ($\mathcal{H}, \langle . , .\rangle )$ be a complex Hilbert space and $A$ be a positive bounded linear operator on it. Let $w_A(T)$ be the $A$-numerical radius and $\|T\|_A$ be the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $(\mathcal{H}, \langle .,.\rangle_A),$ where $\langle x, y\rangle_A:=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. In this article, we establish several upper and lower bounds for $B$-numerical radius of $2\times 2$ operator matrices, where $B=\begin{bmatrix} A & 0 0 & A \end{bmatrix}$. Further, we prove some refinements of earlie…
A considerable amount of literature in the theory of inequality is devoted to the study of Jensen's and Young's inequality. This article presents a number of new inequalities involving the log-convex functions and the geometrically convex functions. As their consequences, we derive the refinements for Young's inequality and Jensen's inequality. In addition, the operator Jensen's type inequality is also developed for conditioned two functions. Utilizing these new inequalities, we investigate the operator inequalities related to the relative operator entropy.
We propose a refinement of the Robertson-Schrodinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product …
Several new improvements of the $A$-numerical radius inequalities for operators acting on a semi-Hilbert space, i.e., a space generated by a positive operator $A$, are proved. In particular, among other inequalities, we show that \begin{align*} \frac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A \leq\frac{1}{4}\Big(2ω_A^2(T)+γ(T)\Big) \leq ω_A^2(T), \end{align*} where $$γ(T)=\sqrt{\left(\|\Re_A(T)\|_A^2-\|\Im_A(T)\|_A^2\right)^2+4\|\Re_A(T)\Im_A(T)\|_A^2}.$$ Here $ω_A(X)$ and $\|X\|_A$ denote respectively the $A$-numerical radius and the $A$-seminorm of an operator $X$. Also, $\Re_A(T):=\frac{T+T^{\…
In this article, we present several inequalities treating operator means and the Cauchy-Schwarz inequality. In particular, we present some new comparisons between operator Heron and Heinz means, several generalizations of the difference version of the Heinz means and further refinements of the Cauchy-Schwarz inequality. The techniques used to accomplish these results include convexity and Löwner matrices.
Let $Ψ$ and $Σ$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove several refinements of the known inequalities $$ρ\left(Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} \right) \le ρ(ΨΣ) ^{\frac{1}{2}} \;\; \mathrm{and}\;\; \hatρ \left(Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} \right) \le \hatρ (ΨΣ) ^{\frac{1}{2}} $$ for the generalized spectral radius $ρ$ and the joint spectral radius $\hatρ$, where $Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} $ denotes the Hadamard (Schur) geometric mean…
We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These constants are shown to be optimal and are computed explicitly in most practical cases.
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 …
In this paper we prove the Hardy inequalities for the quadratic form of the Laplacian with the Landau Hamiltonian magnetic field. Moreover, we obtain Poincaré type inequality and inequalities with more general families of weights, all with estimates for the remainder terms of these inequalities. Furthermore, we establish weighted Hardy inequalities for the quadratic form of the magnetic Baouendi-Grushin operator for the magnetic field of Aharonov-Bohm type. For these, we show refinements of the known Hardy inequalities for the Baouendi-Grushin operator involving radial derivatives in some of t…
The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in the literature. In this article, we present many new bounds for the numerical radius of accretive matrices. The importance of this study is the presence of a new approach that treats a specific class of matrices, namely the accretive ones. The new bounds provide a new set of inequalities, some of which can be considered as refinements of other existing ones, w…
