Heinz Inequalities Research Articles

Some results related to the Heinz inequality in C*-algebra

Some results related to the Heinz inequality in C*-algebra

ON STABILITY OF SOLITONS FOR MAXWELL–LORENTZ SYSTEM WITH SPINNING PARTICLE

Abstract. We consider stability of solitons of the Maxwell–Lorentz system with extended charged spinning particle. The solitons are solutions which correspond to a particle moving with a constant velocity v with |v| < 1 and rotating with a constant angular velocity ω, [1]–[4]. Our main results are the orbital stability of moving solitons with ω = 0 and a linear orbital stability of rotating solitons with v = 0. The Hamilton–Poisson structure of the Maxwell–Lorentz system is degenerate and admits the Casimir invariants, [1]. We construct the Lyapunov function as a linear combination of the Hamiltonian with a suitable Casimir invariant. The key point is a lower bound for this function. The proof of the bound in the case ω ≠ 0 relies on angular momentum conservation and suitable spectral arguments including the Heinz inequality and closed graph theorem

Convexity and inequalities of some generalized numerical radius functions

In this paper, we prove that each of the following functions is convex on R: f(t) = wN(AtXA1?t ? A1?tXAt), g(t) = wN(AtXA1?t), and h(t) = wN(AtXAt) where A > 0, X ? Mn and N(.) is a unitarily invariant norm onMn. Consequently, we answer positively the question concerning the convexity of the function t ? w(AtXAt) proposed by in (2018). We provide some generalizations and extensions of wN(.) by using Kwong functions. More precisely, we prove the following wN(f(A)X1(A) + g(A)Xf (A)) ? wN(AX+XA) ? 2wN(X)N(A), which is a kind of generalization of Heinz inequality for the generalized numerical radius norm. Finally, some inequalities for the Schatten p-generalized numerical radius for partitioned 2 ? 2 block matrices are established, which generalize the Hilbert-Schmidt numerical radius inequalities given by Aldalabih and Kittaneh in (2019).

Some Extensions of the Young and Heinz Inequalities for Matrices

In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert–Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two positive semidefinite matrices A and B we show that $$\begin{aligned}&\Big \Vert A^{\nu }XB^{1-\nu }+A^{1-\nu }XB^{\nu }\Big \Vert _{2}^{2}\le \Big \Vert AX+XB\Big \Vert _{2}^{2}- 2r\Big \Vert AX-XB\Big \Vert _{2}^{2}\\&\quad -r_{0}\left( \Big \Vert A^{\frac{1}{2}}XB^{\frac{1}{2}}-AX\Big \Vert _{2}^{2}+ \Big \Vert A^{\frac{1}{2}}XB^{\frac{1}{2}}-XB\Big \Vert _{2}^{2}\right) , \end{aligned}$$ where X is an arbitrary $$n\times n$$ matrix, $$0<\nu \le \frac{1}{2}$$ , $$r=\min \{\nu , 1-\nu \}$$ and $$r_{0}=\min \{2r, 1-2r\}$$ .

Refining Hermite-Hadamard integral inequality by two parameters

A new family of refinements of Hermite-Hadamard integral inequality is obtained by using two parameters, which can be used to refine the Heinz inequalities of matrices.

Heinz-type numerical radii inequalities

ABSTRACTThe main aim of this article is to present inequalities for the numerical radius of commutators of positive matrices. Some of these inequalities are analogues of known inequalities for unitarily invariant norms. In particular, variants, but weaker forms, of the well-known Heinz inequality and its generalizations are extended to the context of numerical radius.

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.

Two variables functionals and inequalities related to measurable operators

In this paper, we introduce two variables norm functionals of τ-measurable operators and establish their joint log-convexity. Applications of this log-convexity will include interpolated Young, Heinz and Trace inequalities related to τ-measurable operators. Additionally, interpolated versions and their monotonicity will be presented as well.

Numerous refinements of Pólya and Heinz operator inequalities

ABSTRACTIn this article, we present an infinite number of refinements of the Heinz inequality for real numbers and operators. Making use of them, infinitely many refinements of the classical Pólya inequality and their operator versions are deduced.

Generalized Reverse Young and Heinz Inequalities

In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of $v$, namely $v\in \mathbb{R}$. These modified inequalities are used to establish corresponding operator inequalities on a Hilbert space.

The converse of the Loewner–Heinz inequality via perspective

The Loewner–Heinz inequality is not only the most essential one in operator theory, but also a fundamental tool for treating operator inequalities. The aim of this paper is to investigate the converse of the Loewner–Heinz inequality in the view point of perspective and generalized perspective of operator monotone and multiplicative functions. Indeed, we give perspective inequalities equivalent to the Loewner–Heinz inequality.

Some means inequalities for positive operators in Hilbert spaces

In this paper, we obtain two refinements of the ordering relations among Heinz means with different parameters via the Taylor series of some hyperbolic functions and by the way, we derive new generalizations of Heinz operator inequalities. Moreover, we establish a matrix version of Heinz inequality for the Hilbert-Schmidt norm. Finally, we introduce a weighted multivariate geometric mean and show that the weighted multivariate operator geometric mean possess several attractive properties and means inequalities.

On Young and Heinz inequalities for faithful tracial states

Young and Heinz inequalities in two variables are formulated so as to apply to positive and invertible operators. We present the Young and Heinz-type operator inequalities in the context of von Neumman algebras, and apply to singular values and traces of positive operators. As an application, a Heinz inequality relative to tracial states on unital C-algebras is obtained. Moreover, if the tracial state is faithful, the case of equality are completely determined.

On the reverse Young and Heinz inequalities

On the reverse Young and Heinz inequalities

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.

Norm inequalities for operators related to the Cauchy-Schwarz and Heinz inequalities

We present some refinements of the Cauchy-Schwarz and Heinz inequalities for operators by utilizing a refinement of the Hermite-Hadamard inequality.

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 that for all 0≤ν≤1 and each N∈N,A#νB+∑j=1Nsj(ν)(A#αj(ν)B+A#21−j+αj(ν)B−2A#2−j+αj(ν)B)≤A∇νB, for the positive operators A and B, where sj(ν) and αj(ν) are certain functions. Moreover, some new Heinz type inequalities involving the Hilbert–Schmidt norm are established.

Further Generalizations, Refinements, and Reverses of the Young and Heinz Inequalities

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.

Estimates of operator convex and operator monotone functions on bounded intervals

Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,\infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the L\"owner--Heinz inequality.

A certain generalization of the Heinz inequality for positive operators

Norm inequalities of the form [Formula: see text] with [Formula: see text] and [Formula: see text] are studied. Here, [Formula: see text] are operators with [Formula: see text] and [Formula: see text] is an arbitrary unitarily invariant norm. We show that the inequality holds true if and only if [Formula: see text].