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Bulat N. Khabibullin

Publications and source records attributed to Bulat N. Khabibullin.

At least 19 recordsLinked to original sources
arXiv2021

Integrals of subharmonic functions and their differences with weight over small sets on a ray

Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv-\infty$ on the complex plane. An integral of the maximum on circles centered at zero of $U^+:=\sup\{0,U\} $ or $|u|$ over $E$ with a function-multiplier $g\in L^p(E)$ in the integrand is estimated, respectively, in terms of the characteristic function $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, and also in terms of the linear Lebesgue measure of $E$ and the $ L^p$-norm of $g$. Our main theorem develops the proof of one of the classical theorems of Rolf Nevanlinna in the case $E=[0,R]$, given in the classical monograph by Anatolii A. Gol'dberg and Iosif V. Ostrovskii, and also generalizes analogs of the Edrei-Fuchs Lemma on small arcs for small intervals from the works of A.F. Grishin, M.L. Sodin, T.I. Malyutina. Our estimates are uniform in the sense that the constants in these estimates do not depend on $U$ or $u$, provided that $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively.

math.CV↗
arXiv2020

Liouville-type theorems with constraints outside of small sets on circles or spheres for functions of finite order

We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside of such sets on spheres are constant. Our results and methods of proof are also new for functions of one complex variable.

math.CV↗
arXiv2020

Liouville-type theorems outside of small exceptional sets for functions of finite order

We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside some set of zero relative Lebesgue density are constant.

math.CV↗
arXiv2020

Preorders on Subharmonic Functions and Measures with Applications to the Distribution of Zeros of Holomorphic Functions

Let $X$ be a class of extended numerical functions on a domain $D$ of $d$-dimensional Euclidean space $\mathbb R^d$, $H\subset X$. Given $u,M\in X$, we write $u\prec_H M$ if there is a function $h\in H$ such that $u+h\leq M$ on $D$. We consider this special preorder $\prec_H$ for a pair of subharmonic unctions $u, M$ on $D$ in cases where $H$ is the space of all harmonic functions on $D$ or $H$ is the convex cone of all subharmonic functions $h \not\equiv -\infty$ on $D$. Main results are dual equivalent forms for this preorder $\prec_H$ in terms of balayage processes for Riesz measures of subharmonic functions $u$ and $M$, for Jensen and Arens-Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements $D\setminus S$ of non-empty precompact subsets $S\Subset D$. Applications to holomorphic functions $f$ on a domain $D\subset \mathbb C^n$ relate to the distribution of zero sets of functions $f$ under upper restrictions $|f|\leq \exp M$ on $D$. If a domain $D\subset \mathbb C$ is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of $D$, then our conditions for the distribution of zeros of $f\neq 0$ with $|f|\leq \exp M$ on $D$ are both necessary and sufficient.

math.CV↗
arXiv2020

Poisson-Jensen Formulas and Balayage of Measures

Our main results are certain developments of the classical Poisson--Jensen formula for subharmonic functions. The basis of the classical Poisson--Jensen formula is the natural duality between harmonic measures and Green's functions. Our generalizations use some duality between the balayage of measures and and their potentials.

math.CV↗
arXiv2020

The Logarithm of the Modulus of an Entire Function as a Minorant for a Subharmonic Function outside a Small Exceptional Set

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 functions ${\sf B}_u$ are integral averages of $u$ for rapidly shrinking disks as it approaches infinity. We give another equivalent version of this result with $\log |f|\leq u$ outside a very small exceptional set if $u$ is of finite order.

math.CV↗
arXiv2020

Affine Balayage of Measures and Distribution of Riesz Measures of Subharmonic Functions

We develop and use some key concepts of potential theory, such as balayage and duality between measures and their potentials, to study the distribution of masses of subharmonic functions while restrictions to their growth near the boundary of their domain of definition. They are formulated and proved in terms of the Riesz measure for a subharmonic function. In this article, applications to holomorphic functions concern the distribution of their zero sets under restrictions to growth of this functions. Auxiliary results of our article may also be of independent interest. This is, first of all, 1) theorems on gluing of subharmonic functions and on gluing a subharmonic function with Green's function; 2) an internal description of the duality between balayage of measures and their potentials; 3) the duality between the Jensen and Arens-Singer measures on the one hand and the Jensen and Arens-Singer potentials of on the other; 4) a generalization of classical Poisson-Jensen formula connecting subharmonic functions, their Riesz measures, harmonic measures, and Green's functions.

math.CV↗
arXiv2019

The distribution of zeros of entire functions of exponential type with restrictions on growth along the imaginary axis

Let $g$ be a entire function of exponential type on the complex plane $\mathbb C$, $Z=\{ z_k\}_{k=1,2,\dots}$ be a sequence of points in $\mathbb C$. We give a criterion for the existence of an entire function $f\neq 0$ of exponential type that vanishes on $ Z$ and satisfies the constraint $\ln |f(iy)|\leq \ln |g(iy)|+o(|y|)$, $y\to \pm\infty$. Our results generalize and develop a joint results of P. Malliavin and L. A. Rubel. Applications to multipliers for entire functions of exponential type, to analytic functionals and their convolutions on the complex plane, as well as to the completeness of exponential systems in the spaces of locally analytic functions on compacts in terms of the width of these compacts are given.

math.CV↗
arXiv2019

Integral averages and maximization of functions

Some properties of integral averages of functions on intervals and their asymptotic behavior are investigated. The results are aimed at applications to entire and subharmonic functions.

math.CV↗
arXiv2019

Variations on the proximate order

The concept of proximate order is widely used in the theories of entire, meromorphic, subharmonic and plurisubharmonic functions. We give a general interpretation of this concept as a proximate growth function relative to a model growth function. If a function is the proximate growth function with respect to the identity function on the positive semi-axis, then the logarithm of this function is the classical proximate order. Our definition uses only one condition. This form of definition is also new for the classical proximate order.

math.CV↗
arXiv2019

The growth of subharmonic functions along the imaginary axis

Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ are two subharmonic functions in the complex plane $\mathbb C$ with the Riesz measures $ν_u$ and $μ_M$ such that $u(z)\leq O(|z|)$ and $M(z)\leq O(|z|)$ as $z\to \infty$. If the growth of a function $M$ in some sense exceeds the growth of a function $u$ on some straight line, then we can expect measure $μ_M$ to dominate measure $ν_u$ in some sense. We give quantitative forms of such dominance. The main results are illustrated by a new uniqueness theorem for entire functions of exponential type.

math.CV↗
arXiv2019

Gluing Theorems for Subharmonic Functions

In our articles of recent years, the technique of gluing two subharmonic functions turned out to be very useful in studying the distribution of the roots or masses of holomorphic or subharmonic functions, respectively. Here we develop and improve this technique. Its applications will be given in our further works.

math.CV↗
arXiv2019

Classical Balayage of Charges and Measures

We investigate some properties of balayage of charges and measures for subclasses of subharmonic functions and their relationship to the geometry of domain or open set in finite-dimensional Euclidean space where this balayage is considered.

math.CV↗
arXiv2019

Balayage of measures and subharmonic functions on a system of rays. III. Growth of entire functions of exponential type

In the second part of this work was developed a technique of balayage of finite genus $q=0,1,2,\dots$ for measures (charges) and ($δ$-) subharmonic functions of finite order to an arbitrary closed system of rays $S$ with vertex at origin on the complex plane $\mathbb C$. In this third part of our work, we use only the case $q=1$ when $S$ is a pair of oppositely directed rays, i.e., $S$ is a straight line as the point set, and balayage is made from both sides of this line. We consider measures and subharmonic functions of finite type of order $1$. This bilateral balayage of genus $1$ will be applied to the non-triviality of weight classes of entire functions of exponential type $E$ that allocated only constraint on their growth along the line; for the full description of subsequences of zeros for classes $E$; the existence of entire functions-multipliers of $h$ for entire functions of exponential type $g$, limiting by the multiplication of their growth as $fh\in E$; the possibility of the representation of meromorphic functions in the form of a ratio of functions from $E$. The origins of the study lies in the classical Malliavin-Rubel Theorem on the conditions of existence of an entire function of exponential type vanishing on a given sequence of positive numbers. These studies also are parallel to the famous Beurling-Malliavin Theorems on the multiplier and on the radius of completeness.

math.CV↗
arXiv2018

Zeros of holomorphic functions in the unit disk and $ρ$-trigonometrically convex functions

Let $M$\/ be a subharmonic function with Riesz measure $μ_M$ on the unit disk $\mathbb D$ in the complex plane $\mathbb C$. Let $f$ be a nonzero holomorphic function on $\mathbb D$ such that $f$ vanishes on ${\sf Z}\subset \mathbb D$, and satisfies $|f| \leq \exp M$ on $\mathbb D$. Then restrictions on the growth of $μ_M$ near the boundary of $D$ imply certain restrictions on the distribution of $\sf Z$. We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using $ρ$-trigonometrically convex functions.

math.CV↗
arXiv2016

Subharmonic test functions and the distribution of zero sets of holomorphic functions

Let $m,n\geq 1$ are integers and $D$ be a domain in the $$ $\mathbb C^n$ or in the $m$-dimensional real space $\mathbb R^m$. We build positive subharmonic functions on $D$ vanishing on the boundary $\partial D$ of $D$. We use such (test) functions to study the distribution of zero sets of holomorphic functions $f$ on $D\subset \mathbb C^n$ with restrictions on the growth of $f$ near the boundary $\partial D$.

math.CV↗