Related papers: The function $(\cosh\sqrt{at^2+b})$ is exponential…

The BMV conjecture states that for \(n\times n\) Hermitian matrices \(A\) and \(B\) the function \(f_{A,B}(t)=\tr e^{tA+B}\) is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is…

We prove that the inequality $$\cosh \left( \mathrm{arcosh}(2 \cosh u) \cdot \tanh u \right) < \exp \left( u \cdot \tanh u \right)$$ holds for all $u > 0$. We check with the computation program Mathematica that the ratio between the…

We show that a concavity property of the exponential function is a direct consequence of the convexity of the continued Erlang loss function.

Let $\nu_0(t),\nu_1(t),\,\ldots\,,\nu_n(t)$ be the roots of the equation $R(z)=t$, where $R(z)$ is a rational function of the form \[R(z)=z+\sum\limits_{k=1}^n\frac{\alpha_k}{z-\mu_k},\] $\mu_k$ are pairwise different real numbers,…

In this note we show that $k$-convex functions on $\Bbb R^n$ are twice differentiable almost everywhere for every positive integer $k>n/2$. This generalizes the classical Alexsandrov's theorem for convex functions.

Take a random variable X with some finite exponential moments. Define an exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for admissible values of the parameter t. Denote the weighted expectation of X itself by r(t) =…

We investigate the convexity property on $(0,1)$ of the function $$f_a(x)=\frac{{\cal K}{(\sqrt x)}}{a-(1/2)\log(1-x)}.$$ We show that $f_a$ is strictly convex on $(0,1)$ if and only if $a\geq a_c$ and $1/f_a$ is strictly convex on $(0,1)$…

We prove that strictly convex surfaces moving by $K^{\alpha/2}$ become spherical as they contract to points, provided $\alpha$ lies in the range $[1,2]$. In the process we provide a natural candidate for a curvature pinching quantity for…

We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map),…

Let $X$ be an arbitrary continuous random variable and $Z$ be an independent Gaussian random variable with zero mean and unit variance. For $t~>~0$, Costa proved that $e^{2h(X+\sqrt{t}Z)}$ is concave in $t$, where the proof hinged on the…

We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by \[S_\nu(a)=\sum_{n\geq 1} (\frac{1}{2} an^2)^{-\nu} K_\nu(an^2)\qquad (|\arg\,a|<\pi/2)\] as the parameter $|a|\to 0$. It…

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…

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…

We prove a sharp square function estimate for the cone in $\mathbb{R}^3$ and consequently the local smoothing conjecture for the wave equation in $2+1$ dimensions.

In this paper, authors study the convexity and concavity properties of real-valued function with respect to the classical means, and prove a conjecture posed by Bruce Ebanks in \cite{e}.

Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$ f((x+y)/2) \le (f(x)+f(y))/2 + 1. $$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper…

We derive two types of representation results for increasing convex functionals in terms of countably additive measures. The first is a max-representation of functionals defined on spaces of real-valued continuous functions and the second a…

In this brief note, it is shown that the function p^TW log(p) is convex in p if W is a diagonally dominant positive definite M-matrix. The techniques used to prove convexity are well-known in linear algebra and essentially involves…

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive…