Related papers: On the Logarithmic Coefficients of Close to Convex…

Let $T$ be the map defined on $\N=\{1,2,3, ...\}$ by $T(n) = \frac{n}{2} $ if $n$ is even and by $T(n) = \frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system $(\N, 2^{\N}, T,\mu)$ where $\mu$ is the counting measure. This dynamical…

\noindent In the present investigation, we find the sharp bound of fifth coefficient of analytic normalized function $f$ satisfying $z f'(z)/f(z) \prec \varphi(z)$ when coefficients of $\varphi$ satisfy certain conditions. For an…

In this paper we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent…

The finite n-th polylogarithm li_n(z) in Z/p[z] is defined as the sum on k from 1 to p-1 of z^k/k^n. We state and prove the following theorem. Let Li_k:C_p to C_p be the p-adic polylogarithms defined by Coleman. Then a certain linear…

This expository article proves some results of Ferguson, on the approximation of continuous functions on a compact subset of R by polynomials with integral coefficients.

We use a method, first developed for the Riemann zeta-function by Masser in ["Rational values of the Riemann zeta function", Journ. Num. Th. 131 (2011), 2037-2046], to prove a new zero estimate for polynomials in z and 1/Gamma(z). This…

The classes of analytic univalent functions on the unit disk defined by $$ \mathcal{S}^*(\varphi)= \bigg\{ f \in \mathcal{A}: \frac{z f'(z)}{f(z)} \prec \varphi(z)\bigg\}$$ and $$ \mathcal{C}(\varphi)=\bigg\{ f \in \mathcal{A}: 1 + \frac{z…

Let $\mathscr{C}_\mathbb{Z}([0,1])$ be the metric space of real-valued continuous functions on $[0,1]$ with integer values at $0$ and $1$, equipped with the uniform (supremum) metric $d_\infty$. It is a classical theorem in approximation…

Let $\{P_{\theta}:\theta \in {\mathbb R}^d\}$ be a log-concave location family with $P_{\theta}(dx)=e^{-V(x-\theta)}dx,$ where $V:{\mathbb R}^d\mapsto {\mathbb R}$ is a known convex function and let $X_1,\dots, X_n$ be i.i.d. r.v. sampled…

We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.

Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of normalized analytic functions $f$ defined in the domain $|z|<1$ satisfying \begin{align*} {\rm Re\,}…

Let $f: {\Bbb R}^n \longrightarrow {\Bbb R}$ be a positive definite quadratic form and let $y \in {\Bbb R}^n$ be a point. We present a fully polynomial randomized approximation scheme (FPRAS) for computing $\sum_{x \in {\Bbb Z}^n}…

A motivation comes from {\em M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77--84} to study a generalization of close-to-convex functions by means of a $q$-analog of a difference…

In this note, we prove that for all $x \in (0 , 1)$, we have: $$ \log\Gamma(x) = \frac{1}{2} \log\pi + \pi \boldsymbol{\eta} \left(\frac{1}{2} - x\right) - \frac{1}{2} \log\sin(\pi x) + \frac{1}{\pi} \sum_{n = 1}^{\infty} \frac{\log n}{n}…

We study the convolution of functions of the form \[ f_\alpha (z) := \dfrac{\left( \frac{1 + z}{1 - z} \right)^\alpha - 1}{2 \alpha}, \] which map the open unit disk of the complex plane onto polygons of 2 edges when $\alpha\in(0,1)$. We…

Let $\ID$ denote the open unit disk and $f:\,\ID\TO\BAR\IC$ be meromorphic and univalent in $\ID$ with the simple pole at $p\in (0,1)$ and satisfying the standard normalization $f(0)=f'(0)-1=0$. Also, let $f$ have the expansion…

In this work we consider a family of function classes constructed by means of the Gauss hypergeometric function $_2F_1(1,1;2;z) =-\frac{\log(1-z)}{z}$. We demonstrate that this family, in fact, constitutes classes of analytic functions…

We obtain matching direct and inverse theorems for the degree of weighted $L_p$-approximation by polynomials with the Jacobi weights $(1-x)^\alpha (1+x)^\beta$. Combined, the estimates yield a constructive characterization of various…

In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient…

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(\alpha;z)=\sum_{n\geq 2}(\log n)^{\alpha}(\eta_n+{\rm i} \theta_n)/n^z$, properly scaled and normalized, where…