Related papers: Concavity of certain matrix trace and norm functio…

The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems…

We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent…

We provide comparison principles for convex functions through its proximal mappings. Consequently, we prove that the norm of the proximal operator determines a convex the function up to a constant. A new characterization of Lipschitzianity…

Given a closed orientable surface (\Sigma) of genus at least two, we establish an affine isomorphism between the convex compact set of isotopy-invariant topological measures on (\Sigma) and the convex compact set of additive functions on…

Let f be a non-negative concave function on the positive half-line. Let A and B be two positive matrices. Then, for all symmetric norms, || f(A+B) || is less than || f(A)+f(B) ||. When f is operator concave, this was proved by Ando and…

An operator convex function on (0,\infty) which satisfies the symmetry condition k(1/x) = x k(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of…

We study monotonicity and convexity properties of functions arising in the theory of elliptic integrals, and in particular in the case of a Schwarz-Christoffel conformal mapping from a half-plane to a trapezoid. We obtain sharp monotonicity…

The radii of convexity of some Lommel and Struve functions of the first kind are determined. For both of Lommel and Struve functions three different normalizations are applied in such a way that the resulting functions are analytic in the…

To prove that a measure, linearly representable by means of a finite set of nonnegative matrices $\mathcal M$, has the weak-Gibbs property, one check the uniform convergence (on $\mathcal M^\mathbb N$) of the sequence of vectors…

Consider the Mills ratio $f(x)=\big(1-\Phi(x)\big)/\phi(x), \, x\ge 0$, where $\phi$ is the density function of the standard Gaussian law and $\Phi$ its cumulative distribution.We introduce a general procedure to approximate $f$ on the…

Several matrix/operator inequalies are given. Most of them are unexpected extensions of the Araki Log-majorization theorem, obtained thanks to a new log-majorization for positive linear maps and normal operators (Theorem 2.9). The main idea…

We investigate a rearrangement inequality for pairs of n-square matrices: Let |A\|_p denote the C^p trace norm of an n-square matrix A. Consider the quantity |A+B|_p^p + |A-B|_p^p. Under certain positivity conditions, we show that this is…

Let $\ast$ and $\widetilde {\ast}$ denote the convolution of two analytic maps and that of an analytic map and a harmonic map respectively. Pokhrel [1] proved that if $f = h+\overline{g}$ is a harmonic map convex in the direction of…

In the paper, we analyze the Lebesgue exponents $p_\Phi$ and $q_\Phi$, and show that for any $p_\Phi< p < \infty$ and $1< q<q_\Phi$, there exists an equivalent Young function $\Psi$ with $p < p_\Psi < \infty$ and $1<q_\Psi < q$. This type…

Caffarelli's contraction theorem bounds the derivative of the optimal transport map between a log-convex measure and a strongly log-concave measure. We show that an analogous phenomenon holds on the level of the trace: The trace of the…

Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $\rho_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the…

We consider the problems of \emph{learning} and \emph{testing} real-valued convex functions over Gaussian space. Despite the extensive study of function convexity across mathematics, statistics, and computer science, its learnability and…

We show that many important convex matrix functions can be represented as the partial infimal projection of the generalized matrix fractional (GMF) and a relatively simple convex function. This representation provides conditions under which…

We obtain operator concavity (convexity) of some functions of two or three variables by using perspectives of regular operator mappings of one or several variables. As an application, we obtain, for $ 0<p < 1,$ concavity, respectively…

This paper investigates the traces of functions belonging to the inhomogeneous Besov spaces B $\xi$ p,q , where $\xi$ is a product of capacities defined as powers of Gibbs measures. We first establish that the traces of functions in B $\xi$…