Related papers: Bernstein-Remez inequality for Nash functions: A c…

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\lambda_{0},...,\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex…

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)…

In this paper, we extend the Brown-Halmos theorems to the Fock space and investigate the range of the Berezin transform. We observe that there are non-pluriharmonic functions $u$ that can be written as a finite sum…

Let $X$ be a separable real Hilbert space. We show that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and for every $\epsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that…

Let $T$ be a power-bounded operator on a Banach space $X$, $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the unit disc $\mathbb{D}$, and assume that there is a bounded functional calculus for the operator $T$, so…

Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For…

For an analytic function $f$ defined on the unit disk $|z|<1$, let $\Delta(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<r\le 1$. In 1990, Yamashita conjectured that $\Delta(r,z/f)\le \pi r^2$ for convex…

We obtain the following dimension independent Bernstein-Markov inequality in Gauss space: for each $1\leq p<\infty$ there exists a constant $C_p>0$ such that for any $k\geq 1$ and all polynomials $P$ on $\mathbb{R}^{k}$ we have $$ \| \nabla…

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

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we…

Rational functions of total degree $l$ in n variables have a representation in the Bernstein form defined over $n$ dimensional simplex. The range of a rational function is bounded by the smallest and the largest rational Bernstein…

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}…

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if $f\in A_t^p(\calU)$ and $\bfL^{\alpha} f\equiv 0$ for some nonnegative multi-index $\alpha$, then $f\equiv 0$,…

We study locally univalent functions $f$ analytic in the unit disc $\mathbb{D}$ of the complex plane such that $|{f"(z)/f'(z)}|(1-|z|^2)\leq 1+C(1-|z|)$ holds for all $z\in\mathbb{D}$, for some $0<C<\infty$. If $C\leq 1$, then $f$ is…

The purpose of this paper is to prove a new general result about rings of complex analytic functions. Let $\Omega$ be an arbitrary nonempty open subset of the complex plane $\mathbb C$, $\mathcal{A}(\Omega)$ be the set of holomorphic…

We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold K in C^n is of polynomial growth (e.g, in dimension one, K is a finite union of transverse Jordan…

We prove a single-value version of Reshetnyak's theorem. Namely, if a non-constant map $f \in W^{1,n}_{\text{loc}}(\Omega, \mathbb{R}^n)$ from a domain $\Omega \subset \mathbb{R}^n$ satisfies the estimate $\lvert Df(x) \rvert^n \leq K…

We introduce the $f$-crucial function $\operatorname{Crucial}_f$ associated to a rational function $f\in K(z)$ of degree $>1$ over an algebraically closed field $K$ of possibly positive characteristic that is complete with respect to a…

In 1994, M. M. Popov [On integrability in F-spaces, Studia Math. no 3, 205-220] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the lp-spaces for 0<p<1,…