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Bulat Khabibullin

Publications and source records attributed to Bulat Khabibullin.

11 recordsLinked to original sources
arXiv2019

A Small Intervals Theorem for Subharmonic Functions

Let $\mathbb C$ be the complex plane, $E$ be a measurable subset in a segment $[0, R]$ of the positive semiaxis $\mathbb R^+$, $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$. The main result of this article is an upper estimate of the integral of the module $|u|$ over a subset of $E$ through the maximum of the function $u$ on a circle of radius $R$ centered at zero and a linear Lebesgue measure of subset $E$. Our result develops one of the classical theorems of R. Nevanlinna in the case of $E=[0, R]$ and versions of so-called Small Arcs Lemma by Edrei-Fuchs for small intervals on $\mathbb R^+$ from the works of A.F. Grishin, M.L. Sodin, T.I. Malyutina. Our obtained estimate is uniform in the sense that the constants in the estimates are absolute and do not depend on the subharmonic function under the semi-normalization $u(0)\geq 0$.

math.CV↗
arXiv2018

Distribution of zeros and masses for holomorphic and subharmonic functions. I. Hadamard- and Blaschke-type conditions

Let $M$ be a subharmonic function on a domain $D$ in the complex plane $\mathbb C$ with the Riesz measure $ν_M$. Let $f$ be a non-zero holomorphic function on $D$ such that $\log |f|\leq M$ on $D$ and the function $f$ vanish on a sequence ${\tt Z}=\{{\tt z}_k\}_{k=1,2, \dots}\subset D$ {\large(}$u\not\equiv -\infty$ be a subharmonic function on $D$ with the Riesz measure or the mass distribution $ν_u$, and $u\leq M$ on $D$ resp.{\large)}. Then restrictions on the growth of the Riesz measure $ν_M$ of the function $M$ near the boundary of the domain $D$ entail certain restrictions on the distribution of points of the sequence $\tt Z$ (to the mass distribution $ν_u$ resp.). A quantitative form of research of this phenomenon is given immediately in the subharmonic framework. We also establish results in the inverse direction. We investigated in detail the cases when $D$ is $\mathbb C$, the unit disk, exterior of the unit disk, a concentric annulus, and $M$ is a radial function; $D$ is a regular domain and $M$ are constant on the level lines of Green's function of this domain $D$; $D$ is a domain of hyperbolic type, and $M$ are the superpositions of convex functions with functions that depend on the hyperbolic radius; $D$ is a regular domain and $M$ is the superposition of convex functions with a function dependent on the distance to some subset of the boundary of the domain $D$. All our main results and their implementation in more or less concrete situations are new not only for subharmonic functions $u$, and also for holomorphic functions $f$ even in the case when $D$ is $\mathbb C$, the unit disk, an annulus etc.

math.CV↗
arXiv2017

From integral estimates of functions to uniform. I. Exact versions

We give a general method to obtain from the integral restrictions of functions sharp pointwise and uniform estimates of these functions. This scheme is illustrated by the examples for Fock\,--\,Bargmann spaces of entire functions of several variables and Bergman spaces in balls and polydiscs.

math.CV↗
arXiv2016

Classic balayage of measures and subharmonic functions to a system of rays

We construct and apply the classic balayage (sweeping out) of measures and subharmonic functions on closed system of rays in the complex plane with vertex at the origin, including measures and subharmonic functions and infinite order. The need for such a procedure occurs in the study of the behavior of entire and subharmonic functions on systems of rays. The results apply to the complete regularity of the growth of subharmonic and entire functions on a system of rays, the study of the distribution of zeros of entire functions of exponential type of class A, touches upon the incompleteness of exponential systems. The paper contains a survey character sections and new results.

math.CV↗
arXiv2016

Holomorphic minorants of (pluri-)subharmonic functions

Let $u$ be a plurisubharmonic function. We prove the existence of a nonzero holomorphic function such that the logarithm of its modulus is not more than local averages of this function $u$. This is the abstract for scientific conference "Algebra, Analysis and Related Problems of Mathematical Modeling" (Vladikavkaz, June 26-27, 2015) dedicated to the 60th anniversary of Professor, Doctor of Physical and Mathematical Sciences Vladimir A. Koibaev

math.CV↗
arXiv2016

The order versions of the Hahn--Banach Theorem and envelopes. I. Homogeneous functions

We present here the General formulation of the problem of existence and construction of upper and lower envelope for an arbitrary function with values from the completion of the ordered set ${\rm S}$ for a certain class of functions with values in ${\rm S}$. The task is parsed only for the simplest case of model class of homogeneous functions. Consider only order-algebraic version without the involvement of the topology.

math.RA↗
arXiv2015

Zero (sub-)sequences of entire functions

We announce conditions under which a given sequence of points on the complex plane is a subsequence of zeros of an entire function with weight restrictions on growth.

math.CV↗
arXiv2010

Three equivalent conjectures on an estimate of integrals (in Russian)

We offer a conjecture on sharp estimation of a definite improper integral depend on a parameter $λ\in (0,+\infty)$ by means of given estimate of other definite integral depend on parameters $t\in [0,+\infty)$ and $λ$. Such sharp estimate is proved for $λ\leq 1$. Besides, an estimate is obtained for $λ>1$. The last estimate is not exact seemingly. We give also two conjectures that are equivalent to the original conjecture. Sources of our conjectures are extremal problems for entire, meromorphic, and plurisubharmonic functions of several variables.

math.CV↗