Related papers: Monotonicity of a Class of Integral Functionals

Sufficient boundary asymptotic conditions are established for a generalized Schur function $f$ to be identically equal to a given rational function $g$ unimodular on the unit circle. Similar rigidity statements are presented for generalized…

Let $f$ and $g$ be both continuous functions on $\left( 0,\infty \right) $ with $g\left( t\right) >0$ for $t\in \left( 0,\infty \right) $ and let $ F\left( x\right) =\mathcal{L}\left( f\right) $, $G\left( x\right) =\mathcal{L }\left(…

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(T^*(f(x),f(y)))$ where $T^*:[0,1]^2\rightarrow[0,1]$ is an associative function with neutral element in $[0,1]$, $f: [0,1]\rightarrow [0,1]$ is…

Let $f$ be a measurable, real function defined in a neighbourhood of infinity. The function $f$ is said to be of generalised regular variation if there exist functions $h \not\equiv 0$ and $g > 0$ such that $f(xt) - f(t) = h(x) g(t) +…

By applying a high-dimensional parabolic-to-elliptic transformation, we establish a monotonicity formula for the extension problem of the fractional parabolic semilinear equation $(\partial_t -\Delta)^s u = |u|^{p-1}u$, where $0<s<1$. This…

We prove that a continuous function $f:(0,\infty) \to (0,\infty)$ is operator monotone increasing if and only if $f(A \: !_t \: B) \leqs f(A) \: !_t \: f(B)$ for any positive operators $A,B$ and scalar $t \in [0,1]$. Here, $!_t$ denotes the…

The so-called generalized associativity functional equation G(J(x,y),z) = H(x,K(y,z)) has been investigated under various assumptions, for instance when the unknown functions G, H, J, and K are real, continuous, and strictly monotonic in…

Let $\mathcal{L}(x,a)$ be defined on $\left( -1,\infty \right) \times \left( 4/15,\infty \right) $ or $\left( 0,\infty \right) \times \left( 1/15,\infty \right) $ by the formula% \begin{equation*} \mathcal{L}(x,a)=\tfrac{1}{90a^{2}+2}\ln…

We first introduce the generic versions of the fraction rules for monotonicity, i.e. the one that involves integrals known as the Gromov theorem and the other that involves derivatives known as L'H\^opital rule for monotonicity, which we…

In the paper, we establish an inequality involving the gamma and digamma functions and use it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.

This note contains some observations on primary matrix functions and different notions of monotonicity with relevance towards constitutive relations in nonlinear elasticity. Focussing on primary matrix functions on the set of symmetric…

Let $-\infty \leq a<b\leq \infty $. Let $f$ and $g$ be differentiable functions on $(a,b)$ and let $g^{\prime }\neq 0$ on $(a,b)$. By introducing an auxiliary function $H_{f,g}:=\left( f^{\prime }/g^{\prime }\right) g-f$, we easily prove…

In this paper we consider several families of potential non-isochronous systems and study their associated period functions. Firstly, we prove some properties of these functions, like their local behavior near the critical point or…

In this paper we introduce the new class of generalized Volterra functions. We prove some integral representations for them via Fox-Wright H-functions and Meijer G-functions. From positivity conditions on the weight in these…

We offer an alternative and shorter proof to a result by Jan J.Ub{\o}e about monotonicity properties of a one-dimensional function that appeared in the Mathematical Intelligencer in 2015. Our proof is based on reducing the problem to…

We present some completely monotonic functions involving the $q$-gamma function that are inspired by their analogues involving the gamma function.

An inequality, which combines the concept of completely monotone functions with the theory of divided differences, is proposed. It is a straightforward generalization of a result, recently introduced by two of the present authors.

We study stochastic monotonicity and propagation of order for Markov processes with respect to stochastic integral orders characterized by cones of functions satisfying $\Phi f \geq 0$ for some linear operator $\Phi$. We introduce a new…

In this note we study the monotonicity of the function $x\mapsto \psi(1 +bx)^a/\psi(1 + ax)^b$. We also give the several inequalities involving the psi function, whic is the logarithmic derivative of the gamma function.

An important tool to quantify the likeness of two probability measures are f-divergences, which have seen widespread application in statistics and information theory. An example is the total variation, which plays an exceptional role among…