Related papers: A Blaschke-type condition for analytic and subharm…
Given two compact sets, $E$ and $F$, on the unit circle, we study the class of subharmonic functions on the unit disk which can grow at the direction of $E$ and $F$ (sets of singularities) at different rate. The main result concerns the…
This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were…
We continue the study of analytic functions in the unit disk of finite order with arbitrary set of singular points on the unit circle, introduced in \cite{FG}. The main focus here is made upon the inverse problem: the existence of a…
We introduce a new geometric characteristic of compact sets on the plane called $r$-convexity, which fits nicely into the concept of generalized convexity and extends essentially the conventional convexity. For a class of subharmonic…
Extending the results of Borichev--Golinskii--Kupin [2009], we obtain refined Blaschke-type necessary conditions on the zero distribution of analytic functions on the unit disk and on the complex plane with a cut along the positive…
We obtain an upper bound for the derivative of a Blaschke product, whose zeros lie in a certain Stolz-type region. We show that the derivative belongs to the space of analytic functions in the unit disk, introduced recently in \cite{FG}. As…
We prove that resonances of the Schr\"{o}dinger operator with compactly supported potential can contain arbitrary subset of the angle $\{z: -\text{Im} z > C |\text{Re} z|\}$ that satisfies Blaschke condition. We also establish sufficient…
We give a survey of results on zero distribution and factorization of analytic functions in the unit disc in classes defined by the growth of $\log|f(re^{i\theta})|$ in the uniform and integral metrics. We restrict ourself by the case of…
We obtain a Blaschke-type necessary conditions on zeros of analytic functions on the unit disk with different types of exponential growth at the boundary. These conditions are used to prove Lieb-Thirring-type inequalities for the…
Answering a question of A.Zygmund in \cite{MR} G.MacLane and L.Rubel described boundedness of $L_2$-norm w.r.t. the argument of $\log |B|$, where $B$ is a Blaschke product. We generalize their results in several directions. We describe…
In this paper we will deal with problems in approximation theory of bounded analytic functions on the unit disc and their boundary behavior on the unit circle. We will attempt to unify two known such theorems to create a stronger theorem.…
Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$…
One of the purposes of this paper is to clarify the strong analogy between potential theory on the open unit disk and the homogeneous tree, to which we dedicate an introductory section. We then exemplify this analogy by a study of Riesz…
It is known that a subharmonic function of finite order $\rho$ can be approximated by the logarithm of the modulus of an entire function at the point $z$ outside an exceptional set up to $C\log|z|$. In this article we prove that if such an…
Let $f$ be an analytic function mapping the unit disk $\D$ to itself. We give necessary and sufficient conditions on the local behavior of $f$ near a finite set of boundary points that requires $f$ to be a finite Blaschke product.
We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is…
We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift $f(z) \mapsto \frac{f(z)-f(0)}{z}$ is a contraction on the space. We present a model for this operator and…
Consider the family of locally univalent analytic functions $h$ in the unit disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the condition $${\real} \left( \frac{z h''(z)}{\alpha h'(z)}\right) <\frac{1}{2} ~\mbox{ for…
Suppose $E$ is fully symmetric Banach function space on $(0,1)$ or $(0,\infty)$ or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on $f\in E$ so that its orbit $\Omega(f)$ is the closed convex hull of…
Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as…