Related papers: Some extensions of the operator entropy type inequ…

In this paper, we investigate the relative operator entropies in the more general settings of C*-algebras, real C*-algebras and JC-algebras. We show that all the operator inequalities on relative operator entropies still hold in these…

Let $f \in M_+(\R_+)$, the class of nonnegative, Lebesgure-measurable functions on $\R_+=(0, \infty)$. We deal with integral operators of the form \[ (T_Kf)(x)=\int_{\R_+}K(x,y)f(y)\, dy, \quad x \in \R_+, \] with $K \in M_+(\R_+^2)$. We…

We consider a randomly forced Ginzburg-Landau equation on an unbounded domain. The forcing is smooth and homogeneous in space and white noise in time. We prove existence and smoothness of solutions, existence of an invariant measure for the…

An equivalent formulation of the von Neumann inequality states that the backward shift $S^*$ on $\ell_{2}$ is extremal, in the sense that if $T$ is a Hilbert space contraction, then $\|p(T)\| \leq \|p(S^*)\|$ for each polynomial $p$. We…

Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)ds \bigg)^q u(t) dt \bigg)^{\frac{r}{q}}…

The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent…

We establish a discrete analog of the R\'enyi entropy comparison due to Bobkov and Madiman. For log-concave variables on the integers, the min entropy is within log e of the usual Shannon entropy. Additionally we investigate the entropic…

Following a growing number of studies that, over the past 15 years, have established entropy inequalities via ideas and tools from additive combinatorics, in this work we obtain a number of new bounds for the differential entropy of sums,…

Recently, Haddad, Jim\'enez, and Montenegro introduced the affine $p$-Laplace operator, $p>1$, and studied associated affine versions of the isoperimetric inequalities for the first eigenvalue of the affine $p$-Laplace operator, including…

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…

We initiate the study of relative operator entropies and Tsallis relative operator entropies in the setting of JB-algebras. We establish their basic properties and extend the operator inequalities on relative operator entropies and Tsallis…

In this paper, we give some reverse-types of Ando's and H\"older-McCarthy's inequalities for positive linear maps, and positive invertible operators. For our purpose, we use a recently improved Young inequality and its reverse.

We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form \[ \|P - \Pi_{\mathbb{B}} \Pi_{\ker\mathscr{A}} P\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} \le c \,…

We propose an inertial variant of the strongly convergent inexact proximal-point (PP) method of Solodov and Svaiter (2000) for monotone inclusions. We prove strong convergence of our main algorithm under less restrictive assumptions on the…

We establish a general class of entropy inequalities that take the concise form of Gaussian comparisons. The main result unifies many classical and recent results, including the Shannon-Stam inequality, the Brunn-Minkowski inequality, the…

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or Onofri inequality for brevity. In dimension two this inequality plays a role similar to the Sobolev inequality in higher dimensions. After justifying this…

Let $S(\rho)$ be the von Neumann entropy of a density matrix $\rho$. Weak monotonicity asserts that $S(\rho_{AB}) - S(\rho_A) + S(\rho_{BC}) - S(\rho_C)\geq 0$ for any tripartite density matrix $\rho_{ABC}$, a fact that is equivalent to the…

Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequality. A general argument is given using matrix perspectives of operator convex functions. A matrix…

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and…

We investigate weighted inequalities for fractional maximal operators and fractional integral operators. We work within the innovative framework of "entropy bounds" introduced by Treil--Volberg. Using techniques developed by Lacey and the…