Related papers: On multiply monotone functions

We obtain necessary and sufficient conditions on a function in order that it be the Laplace transform of an absolutely monotonic function. Several closely related results are also given.

We describe the idempotent Fourier multipliers that act contractively on $H^p$ spaces of the $d$-dimensional torus $\mathbb{T}^d$ for $d\geq 1$ and $1\leq p \leq \infty$. When $p$ is not an even integer, such multipliers are just…

Necessary and sufficient conditions for positive numbers $M_{k_1}, M_{k_2}, M_{k_3}, M_{k_4}$, $0 = k_1 < k_2<k_2\leq r-2$, $k_4=r$, to guarantee the existence of an $r-1$-monotone function defined on the negative half-line and such that…

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal…

Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the "moments" F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a…

According to a celebrated result by L\"owner, a real-valued function $f$ is operator monotone if and only if its L\"owner matrix, which is the matrix of divided differences $L_f=(\frac{f(x_i)-f(x_j)}{x_i-x_j})_{i,j=1}^N$, is positive…

The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a H\"older continuous function. The Fourier transform is written as the difference of $\int_{-1}^1…

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds. In this paper, we…

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form $\Phi(x)=\phi(|x_1|, \dots, |x_m|)$, $x_i\in \mathbf R^{n_i}$, in terms of the Fourier transform of the function $\phi$ on $\mathbf…

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…

We introduce and study weighted spaces of functions with mixed norm on the upper half-plane, defined in terms of Fourier transform. We give a characterization of analytic functions within these spaces, and in particular, we provide an…

The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$.…

A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It…

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(…

The problem is considered as to whether a monotone function defined on a subset P of a Euclidean space can be strictly monotonically extended to the whole space. It is proved that this is the case if and only if the function is {\em…

Motivated by existing results, we present some completely monotonic functions involving the polygamma functions.

We characterize real functions $f$ on an interval $(-\alpha,\alpha)$ for which the entrywise matrix function $[a_{ij}] \mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional…

In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)=\psi^{(k)}(x+a) - \psi^{(k)}(x) - \frac{ak!}{x^{k+1}}$, where $a\in(0,1)$ and $k\in \mathbb{N}_0$. Specifically, we consider the cases for $k\in \{ 2n:…

We mainly establish a monotonicity property between some special Riemann sums of a convex function $f$ on $[a,b]$, which in particular yields that $\frac{b-a}{n+1}\sum_{i=0}^n f\left(a+i\frac{b-a}{n}\right)$ is decreasing while…

In this paper we give necessary and sufficient conditions for the system of positive numbers $ M_{k_1}, M_{k_2},..., M_{k_{d}},$ $0\leq k_1<...<k_{d} {\leq} r$, to guarantee the existence of an $r$-monotone function defined on the negative…