Related papers: Uniqueness theorems for Korenblum type spaces

We study radial behavior of analytic and harmonic functions, which admit a certain majorant in the unit disk. We prove that extremal growth or decay may occur only along small sets of radii and give precise estimates of these exceptional…

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. In 2016, we obtained a result on the existence of an entire function $f\neq 0$ satisfying the estimate $\log|f|\leq {\sf B}_u$ on $\mathbb C$, where…

In this very short paper, we provide a strong motivation for the study of the following problem: given a real normed space $E$, a closed, convex, unbounded set $X\subseteq E$ and a function $f:X\to X$, find suitable conditions under which,…

We investigate certain classes of shift invariant subspaces in growth spaces on the unit disc of the complex plane determined by a majorant $w$, which include the classical Korenblum growth spaces. Our main result provides a complete…

We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^\alpha$, where $-1 < \alpha < 1$. We also give periodic versions of these results in which the majorants…

We present an approximation theorem for continuous non-decreasing functions on compact preordered spaces, leading to an algebraic characterization of their corresponding function spaces. As an application, we prove that the family of…

Let $X$ be a locally compact topological space, $(Y,d)$ be a boundedly compact metric space and $LB(X,Y)$ be the space of all locally bounded functions from $X$ to $Y$. We characterize compact sets in $LB(X,Y)$ equipped with the topology of…

Let $E$ be an arbitrary subset of the unit circle $T$ and let $f$ be a function defined on $E$. When there exist polynomials $P_n$ which are uniformly bounded by a number $M > 0$ on $T$ and converge (pointwise) to $f$ at each point of $E$?…

We study the vanishing cycle complex $\varphi_fA_X$ for a holomorphic function $f$ on a reduced complex analytic space $X$ with $A$ a Dedekind domain (for instance, a localization of the ring of integers of a cyclotomic field, where the…

Let $\mathcal{S}$ denote the family of all functions that are analytic and univalent in the unit disk $\mathbb{D}:=\{z: |z|<1\}$ and satisfy $f(0)=f^{\prime}(0)-1=0$. In the present paper, we consider certain subclasses of univalent…

In this paper, we introduce a class of function spaces called K\"othe-Herz spaces $E(\mathcal{X})$. These spaces are similar to amalgam spaces and are characterized by a local component given by a countable family $\mathcal{X}=\left(…

In this short note we prove an optimal version of a classical result. Given a majorant determining a growth restriction on functions in the unit disk $\mathbb{D}$, we say that a set $E$ on the unit circle $\mathbb{T}$ is a uniqueness set,…

It is known that, if $S(z)$ is a non-constant, singular inner function defined on the unit disk, then $\min_{|z|\le r}|S(z)|\to0$ as $r\to1^-$. We show that the convergence may be arbitrarily slow.

Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so--called {\em tangency variety} of $f$ at $\bar{x},$ we first provide necessary and…

For a transcendental entire function $f$, the property that there exists $r>0$ such that $m^n(r)\to\infty$ as $n\to\infty$, where $m(r)=\min \{|f(z)|:|z|=r\}$, is related to conjectures of Eremenko and of Baker, for both of which order…

The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose $K$ is a compact set in the complex plane and 0 belongs to the boundary $\partial K$. Let ${\cal A}(K)$ denote the space of all functions $f$ on $K$…

There have been, over the last 8 years, a number of far reaching extensions of the famous original F. and M. Riesz's uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane $\Bbb C$ has the…

Let F be a class of functions with the uniqueness property: if a function f in F vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a…

For $\alpha > 0$, the $\alpha$-Lipschitz minorant of a function $f: \mathbb{R} \to \mathbb{R}$ is the greatest function $m : \mathbb{R} \to \mathbb{R}$ such that $m \leq f$ and $|m(s)-m(t)| \le \alpha |s-t|$ for all $s,t \in \mathbb{R}$,…

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…