Related papers: Complete Monotonicity of Special Functions

Let $\mathcal{F}_{n}^*$ be the set of Boolean functions depending on all $n$ variables. We prove that for any $f\in \mathcal{F}_{n}^*$, $f|_{x_i=0}$ or $f|_{x_i=1}$ depends on the remaining $n-1$ variables, for some variable $x_i$. This…

Let $D$ be a proper domain in the extended complex plane ${\mathbb C}_{\infty}:={\mathbb C}\cup \{\infty\}$, $M=M_+-M_-\not\equiv \pm \infty$ be a difference of non-trivial subharmonic functions $M_{\pm}\not\equiv \mp \infty$ on $D$,…

Let a function $F: [0,1]^2\rightarrow [0,1]$ be given by $F(x,y)= f^{(-1)}(T(f(x), f(y)))$ where $f :[0,1]\rightarrow [0,1]$ is a monotone function, $f^{(-1)}$ is the pseudo-inverse of $f$ and $T$ is a triangular norm. This article…

We prove that if $f(x) = \sum_{k=0}^\infty a_k x^k,$ $a_k >0, $ is an entire function such that the sequence $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$ is non-decreasing and $\frac{a_1^2}{a_{0}a_{2}} \geq 2\sqrt[3]{2},$…

It is shown that if $f$ or $1/f$ is a real entire function of infinite order of growth, with only real zeros, then $f''+\omega f$ has infinitely many non-real zeros for any $\omega > 0$.

Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For…

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j…

We consider two operations on the Mittag-Leffler function which cancel the exponential term in the expansion at infinity, and generate a completely monotonic function. The first one is the action of a certain differential-difference…

We study the entire functions $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ with non-monotonic second quotients of Taylor coefficients, namely, such that $\frac{a_{2m-1}^2}{a_{2m-2}a_{2m}} = a>1$ and $\frac{a_{2m}^2}{a_{2m-1}a_{2m+1}} = b>1$…

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose…

A function $\rho:[0,\infty)\to(0,1]$ is a completely monotonic function if and only if $\rho(\Vert\mathbf{x}\Vert^2)$ is positive definite on $\mathbb{R}^d$ for all $d$ and thus it represents the correlation function of a weakly stationary…

We study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are…

Let $\phi(x)=\sum \alpha_n x^n$ be a formal power series with real coefficients, and let $D$ denote differentiation. It is shown that "for every real polynomial $f$ there is a positive integer $m_0$ such that $\phi(D)^mf$ has only real…

We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal…

This paper systematically investigates the absolute monotonicity of two function families associated with the Gaussian hypergeometric function $F(a, b; c; x)$ (where $a,b,c\in\mathbb{R}_+$): $\mathcal{F}_p(x)=(1-x)^pF(a,b;c;x)$ and…

We present a new procedure to count the number of real zeros of a class of univariate Pfaffian functions of order $1$. The procedure is based on the construction of Sturm sequences for these functions and relies on an oracle for sign…

If $f(x,y)$ is a real function satisfying $y>0$ and $\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$ is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz…

In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-\delta}$, for some $\delta>0$, and if the Dirichlet series…

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 first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$…