Related papers: A generalization of Bohr-Mollerup's theorem for hi…

In this paper, we generalize the Bohr-Rogosinski sum for the Ma-Minda classes of starlike and convex functions. Also the phenomenon is studied for the classes of starlike functions with respect to symmetric points and conjugate points along…

The normalized incomplete beta function can be defined either as cumulative distribution function of beta density or as the Gauss hypergeometric function with one of the upper parameters equal to unity. Logarithmic concavity/convexity of…

Monotonicity properties of the ratio $$ \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, $$ where $f$ is an entire function are investigated. Earlier results for Euler's gamma function and other entire functions of genus 1 are…

Let $ \mathcal{H}(\Omega) $ be the class of complex-valued functions harmonic in $ \Omega\subset\mathbb{C} $ and each $f=h+\overline{g}\in \mathcal{H}(\Omega)$, where $ h $ and $ g $ are analytic. In the study of Bohr phenomenon for certain…

Let $ \mathcal{B}:=\{f(z)=\sum_{n=0}^{\infty}a_nz^n\; \mbox{with}\; |f(z)|<1\;\mbox{for all}\; z\in\mathbb{D}\} $. The improved version of the classical Bohr's inequality \cite{Bohr-1914} states that if $ f\in\mathcal{B} $, then the…

The logarithmic coefficients $\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…

The following extension of Bohr's theorem is established: If a somewhere convergent Dirichlet series $f$ has an analytic continuation to the half-plane $\mathbb{C}_\theta = \{s = \sigma+it\,:\, \sigma>\theta\}$ that maps $\mathbb{C}_\theta$…

Given a real function $f$, the rate function for the large deviations of the diffusion process of drift $\nabla f$ given by the Freidlin-Wentzell theorem coincides with the time integral of the energy dissipation for the gradient flow…

We prove the existence of holomorphic functions $f$ defined on any open convex subset ${\rm \Omega}\subset {{\mathbb C}}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact…

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms $a\mapsto\sum{f_k}(a)_kx^k$, $a\mapsto\sum{f_k}\Gamma(a+k)x^k$ and $a\mapsto\sum{f_k}x^k/(a)_k$. The most useful examples of such functions are…

First we recall the notion of conxity and log-convexity for real-valued. Then we generalize the trick used by Artin in his famous paper on the Gamma function to find log-convex solutions to the functional equations f(x+1)=g(x)f(x). This…

Suppose $F: \mathbb{R}^{N} \rightarrow [0, +\infty)$ be a convex function of class $C^{2}(\mathbb{R}^{N} \backslash \{0\})$ which is even and positively homogeneous of degree 1. We denote $\gamma_1=\inf\limits_{u\in W^{1,…

In this article, we first establish a generalized Bohr inequality and examine its sharpness for a class of analytic functions $f$ in a simply connected domain $\Omega_\gamma,$ where $0\leq \gamma<1$ with a sequence $\{\varphi_n(r)…

The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_\Omega F(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(\Omega , \mathbb R^N )\cap K$, where $K$ is a closed convex…

The aim of this paper is to develop analytic techniques to deal with certain monotonicity of combinatorial sequences. (1) A criterion for the monotonicity of the function $\sqrt[x]{f(x)}$ is given, which is a continuous analog for one…

In \cite{d4}, we gave a method to construct a continued fraction of the function $F(x):=e^{x}E_{1}(x)$. More precisely we define $F_{1}(x)$ as the reciprocal of $F(x)$ and we inductively define $F_{m}(x)$ as the reciprocal of ``$F_{m-1}(x)$…

Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of a particular form, then $F(s)=L_f(s)$ for some…

Let $\Gamma$ be a convex cocompact thin subgroup of an arithmetic lattice in $\operatorname{SO}(n, 1)$. We generalize Selberg's $\frac{3}{16}$ theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a…

Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual…

After H\"older proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic…