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B. N. Khabibullin

Publications and source records attributed to B. N. Khabibullin.

At least 19 recordsLinked to original sources
arXiv2025

Decrease in growth of entire and meromorphic functions

We solve the following three problems. 1. How much can the radial growth of an entire function $f$ be reduced by multiplying it by some nonzero entire function? We give the answer in terms of the growth of the integral means of $\ln|f|$ over the circles centered at the origin. 2. We estimate the smallest possible radial growth of non zero entire functions that vanish on a given distribution of points $Z$. We solve this problem in terms of the growth of the radial integral counting function of $Z$. 3. Let $F=f/g$ be a meromorphic function with representations as the ratio of entire functions $f\neq 0$ and $g\neq 0$. How small can the radial growth of entire functions $f$ and $g$ be in such representations in relation to the growth of the Nevanlinna characteristic of $F$? All solutions have a non-asymptotic uniform character, and the obtained inequalities are sharp. All of them are based on some main theorem for subharmonic functions, which relies on the Govorov--Petrenko--Dahlberg--Essén inequality and uses our general results on the existence of subharmonic minorants.

math.CV↗
arXiv2023

Completeness of exponential systems in spaces of functions on a compact or domain and their area

We present new completeness conditions for exponential systems on the complex plane in Banach algebras of continuous functions on a compact with a connected complement that are simultaneously holomorphic in the interior of this compact if it is nonempty, as well as in spaces of holomorphic functions on a simply connected bounded domain with a topology of uniform convergence on compact subsets. These conditions are formulated in terms of the Euclidean area of the convex hull of a compact or a region on the one hand, and new characteristics of the distribution of exponential system indicators on the other.

math.CV↗
arXiv2023

Completeness of exponential systems and the perimeter of the convex hull

We give a new scale of completeness conditions for exponential systems in two types of functional spaces on subsets of the complex plane. The first is the Banach spaces of functions that are continuous on a compact and simultaneously holomorphic in the interior of this compact, if this interior is nonempty, with a uniform norm. The second is the spaces of holomorphic functions on a bounded open set with a topology of uniform convergence on compacts. These conditions are formulated in terms of the perimeter of the convex hull of the domain of determining of functions from space and new characteristics of distribution of exponents of exponential system.

math.CV↗
arXiv2023

Completeness of the exponential system in geometric terms of width, breadth and diameter

We establish a criterion for the completeness of an exponential system in the spaces of functions continuous on a convex compact set and holomorphic in the interior of this compact set, as well as in the spaces of holomorphic functions in the convex domain in terms of the breadth of the compact set or the domain in the direction. The main results are formulated exclusively through the relations between the breadth in the direction, width or diameter of the compact set or domain on the one hand and the logarithmic submeasures or logarithmic block densities of the distribution of exponents of exponential system on the other.

math.CV↗
arXiv2022

The Harnack distance and estimates of subharmonic functions from below on the unit disc

General estimates from below of holomorphic and subharmonic functions play one of the key roles in the theory of growth of holomorphic and subharmonic functions and in general in the theory of potential. At the same time, the most diverse technical apparatus was often used for this. In one of our recent papers, we showed in fairly general cases that in fact such a wide set of tools is not required. The classic Harnack distance is almost always enough. In this article, we will illustrate how this works in the cases of the unit disc in the complex plane.

math.CV↗
arXiv2022

Subharmonic addition to the Beurling-Malliavin multiplier theorem

We prove a version of the Beurling-Malliavin multiplier theorem. This version is formulated here in a simplified form. Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ be a pair of subharmonic functions on the complex plane $\mathbb C$ with positive parts $u^+:=\sup\{u,0\}$ and $M^+$ such that $$ \operatorname{type}[u]:=\limsup_{z\to \infty} \frac{u^+(z)}{|z|}<+\infty, \qquad \operatorname{type}[M]<+\infty, \qquad \int_{-\infty}^{+\infty}\frac{u^+(x)+M^+(x)}{1+x^2}\operatorname{d}x<+\infty. $$ If $\operatorname{type}[u]<a<+\infty$, $0<b<+\infty$, and $\operatorname{type}[M]<c<+\infty$, then there are an entire function $h\not\equiv 0$ with $\operatorname{type}[\log|h|]<c$ and a subset $iY$ in the imaginary axis $i\mathbb R$ of linear Lebesgue measure $<b$ such that the function $h$ is bounded on the real axis and $u(z)-M(z)+\log|h(z)|\leq a|\Im z|$ on each straight line parallel to the real axis and not intersecting $iY$.

math.CV↗
arXiv2022

The Malliavin-Rubel theorem on small entire functions of exponential type with given zeros: 60 years later

Let $Z$ and $W$ are distributions of points on the complex plane $\mathbb C$. The following problem goes back to the studies of F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane and others. For which $Z$ and $W$ for an entire function $g\neq 0$ of exponential type vanishing on $W$, there is an entire function $f\neq 0$ of exponential type vanishing on $Z$ such that $|f|\leq |g|$ on the imaginary axis? The classical Malliavin-Rubel theorem of the early 1960s completely solves this problem for "positive" $Z$ and $W$ lying only on the positive semiaxis. A number of generalizations of this criterion were established by us in the late 1980s for "complex" $Z\subset\mathbb C$ and $W\subset\mathbb C$ separated by angles from the imaginary axis, with some advances in the 2020s. In this paper, tougher problems are solved in a more general subharmonic framework for distributions of masses on $\mathbb C$. All the previously mentioned results can be obtained from the main results of the paper in a much stronger form even in the initial formulation for distributions of points $Z$ and $W$ and entire functions $f$ and $g$ of exponential type. Some of the results of the article are closely related to the famous Beurling-Malliavin theorems on the radius of completeness and the multiplier.

math.CV↗
arXiv2022

Balayage of charge distributions and subharmonic functions onto a strip

We consider two balayage constructions on the complex plane $\mathbb C$ with real axis $\mathbb R$ for $0\leq b\in \mathbb R$. Let $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$ of order $$\operatorname{ord}[u]:=\limsup_{z\to \infty} \frac{\ln \max\{1,u(z)\}}{\ln |z|}\leq 1,$$ $U=u-v$ be the difference of subharmonic functions $u$ and $v\not\equiv -\infty$ on $\mathbb C$ with $\operatorname{ord}[v]\leq 1$, i.e., $δ$-subharmonic function on $\mathbb C$ of order $\operatorname{ord}[U]\leq 1$. Then there is a $δ$-subharmonic function $V\not\equiv \pm\infty$ on $\mathbb C$ of order $\operatorname{ord}[V]\leq 1$ such that $V$ is harmonic on $\bigl\{ z \in \mathbb C\bigm| |\Re z|> b\bigr\}$ and $U(z)\equiv V(z)$ for all $z\in \bigl\{ z \in \mathbb C\bigm| |\Re z|\leq b\bigr\}\setminus E$ where $E\subset \mathbb C$ is polar. If $u$ is a subharmonic function of finite type under order $1$, i.e., $$\limsup_{z\to \infty} \frac{u(z)}{|z|}<+\infty,$$ then there exist subharmonic functions $u_{\mathbb R}$ and $u_b$ of finite type under order $1$ that are harmonic respectively on $\mathbb C\setminus \mathbb R$ and $\bigl\{ z \in \mathbb C\bigm| |\Re z|> b\bigr\}$ such that $$\begin{cases} u(z)\equiv u_{\mathbb R}(z)+u_b(z) \text{ for all $z\in {\mathbb R}\bigcup \bigl\{ z \in \mathbb C\bigm| |\Re z|\leq b\bigr\}$},\\ u(z)\leq u_{\mathbb R}(z) + u_b(z) \text{ for each $z\in \mathbb C$.}\end{cases}$$ At the same time, we trace special relationships between the various logarithmic characteristics of the Riesz mass and charge distributions of subharmonic and $δ$-subharmonic functions.

math.CV↗
arXiv2022

The Lindelöf Condition for Charge Distribution and Balayage

Let $ν$ be a charge distribution on the complex plane $\mathbb C$, i.e. the real Radon measure on $\mathbb C$ with total variation $|ν|$. The charge distribution $ν$ is of finite upper density under order of $1$ if $$ \limsup_{0<r\to+\infty} \frac{1}{t}|ν|\Bigl(\bigl\{z\in \mathbb C \bigm| |z|\leq r\bigr\}\Bigr)<+\infty. $$ The charge distribution $ν$ satisfies the Lindelöf condition of genus $1$ if $$ \limsup_{1\leq r\to +\infty}\biggl|\int_{1\leq |z|\leq r} \frac{1}{z}\operatorname{d}\!ν(z)\biggr|<+\infty. $$ These concepts play a key role in the study of entire functions of exponential type and subharmonic functions of finite type under order of $1$, as well as in their applications. In our previous works, a technique was developed for balayage of finite genus $q=0,1,\dots$ of the charge distribution $ν$ from the half-plane. We show that balayage of genus $q=1$ of the charge distribution $ν$ from the half-plane preserves this pair of properties under some condition on $ν$.

math.CV↗
arXiv2022

The restriction from below of the subharmonic function by the logarithm of the module of entire function

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. Then for any function $r\colon\mathbb C\to (0,1]$ satisfying the condition $$\inf_{z\in\mathbb C}\frac{\ln r(z)}{\ln(2+|z|)}>-\infty,$$ there is an entire function $f\not\equiv 0$ such that $$ \ln |f(z)|\leq \frac{1}{2π}\int_0^{2π}u(z+r(z)e^{iθ})\,{\mathrm d}θ\quad\text{for all $z\in\mathbb C$.}$$ A similar result is established for subharmonic functions of finite order with inequalities of the form $\ln|f(z)|\leq u(z)$ at all points $z\in\mathbb C\setminus E$, where the exceptional set $E$ is small in terms of $d$-dimensional Hausdorff content of $E$ with variable radius $r$.

math.CV↗
arXiv2021

Lower bounds for subharmonic functions in terms of the Harnack distance

Let $G$ be a nonempty bounded domain in a finite-dimensional Euclidean space. The main results are general estimates from below at points from $G$ for an arbitrary subharmonic function $u\not\equiv -\infty$ on the closure of the domain $G$ through the maximum of the function $u$ on the boundary of the domain $G$. These results are new for planar domains $G$, and for intervals of $G$ on the numerical line have also not been previously noted. They show that the Harnack distance plays a key role in these estimates. Further applications to subharmonic, convex, holomorphic functions, as well as to meromorphic functions and differences of subharmonic functions in domains of a particular type are supposed to be outlined in the continuation of this article.

math.CV↗
arXiv2021

On the Harnack distance

In this note, we consider two types of estimates for the Harnack distance in bounded domains of a finite-dimensional Euclidean space. The first type is based on the geometric concept of the entropy of arcwise connectedness. We used this concept earlier to study zero sets of holomorphic functions in weighted classes of holomorphic functions. The second type is based on the concept of separatedness for subsets in domain, which is introduced in this note. The latter concept, generally speaking, can be more subtle.

math.CV↗
arXiv2021

Integrals of differences of subharmonic functions. III. Hausdorff measure and content, integration over Lipschitz curves and surfaces

Additional integral inequalities are obtained for integrals of the differences of subharmonic functions by Borel measures on balls in a multidimensional Euclidean space. These integrals are still estimated from above through the Nevanlinna characteristic and various characteristics of the Borel measure and its support. All the results are also new for logarithms of modules of meromorphic functions on discs in the complex plane. Integration by Borel measures with a support on fractal sets is allowed, and estimates in these cases are given by the Hausdorff measure and the Hausdorff content for the support of the Borel measure. We consider separately important for applications special cases of functions in the entire complex plane and space, in the unit disc or ball, as well as cases of integration over length on subsets of Lipschitz curves and over area on subsets of Lipschitz hypersurfaces.

math.CV↗
arXiv2021

Integrals of differences of subharmonic functions. II. A criterion

We supplement the result of the first part of the work with estimates of the integrals of the difference of subharmonic functions in measure with some deterioration of the absolute constants, but these estimates have the form of a criterion, which is formulated exclusively in terms of the measure and its potential. This criterion is also new for meromorphic functions on the complex plane.

math.CV↗
arXiv2021

Integrals of differences of subharmonic functions. I. An integral inequality with Nevanlinna characteristic and modulus of continuity of measure

We obtain new integral inequalities for the integrals of the difference of subharmonic functions in measure through their Nevanlinna characteristic and some functional characteristic of the measure. These results are new also for meromorphic functions. Let us illustrate the main result for the case of a meromorphic function. We denote by $D_z(r)$ a closed disc of radius $r$ centered at $z$ in the complex plane $\mathbb C$. Let $μ\geq 0$ be a Borel measure on $D_0(r)$, and $M:=μ(D_0(r))<+\infty$. The modulus of continuity of measure $μ$ is the function $h(t):=\sup_{z\in \mathbb C}μ(D_z(t)), \quad t\geq 0.$ Suppose that $\int_0\frac{h(t)}{t} dt<+\infty$. Let $f\neq 0$ be a meromorphic function on a neighbourhood of $D_0(R)$ of radius $R>r$ with $f(0)\in \mathbb C$ and the Nevanlinna characteristic $T_f$ . Then there is $$\int \ln^+|f| \,dμ\leq 2\frac{R+r}{R-r}T_f(R)M\biggl(1+\int_0^{R+r}\frac{h(t)}{t} dt\biggr).$$

math.CV↗
arXiv2021

The growth of entire functions of exponential type and characteristics of distributions of points along a straight line

Let $Z$ and $W$ be a pair of point distributions of finite upper density on the complex plane $\mathbb C$ with the real axis $\mathbb R$. We give several variants of necessary and at the same time sufficient conditions for their arrangement, under which for every entire function of exponential type $g\neq 0$ vanishing on $W$, there is, respectively, either an entire function of exponential type $f\neq 0$ vanishing on $Z$ and satisfies one of the two variants of the constraints: 1) $|f(iy)|\leq |g(iy)|$ at each $y\in \mathbb R$, i.e. everywhere on the imaginary axis $i\mathbb R$, 2) $\ln |f(iy)|\leq \ln |g(iy)|+o(|y|)$ as $y\to \pm \infty$, or for any number $\varepsilon >0$ there is an entire function of exponential type $f\neq 0$ vanishing on $Z$ and satisfies the inequality $\ln \bigl|f(iy)\bigr|\leq \ln \bigl|g(iy)\bigr|+\varepsilon |y|$ for all $y\in \mathbb R\setminus E$, where $E\subset \mathbb R$ is a set of finite linear Lebesgue measure. Our study is carried out within the framework of generalization of the development of the classical theorem of P. Malliavin and L.A. Rubel of the 1960s, where the case of the location of $Z\subset \mathbb R^+$ and $W\subset \mathbb R^+$ on the positive semiaxis $\mathbb R^+\subset \mathbb R$ is considered. Our criteria are given in terms of special logarithmic characteristics and (sub)measures for $Z$ and $W$. At the same time, in the third variant, there are no additional requirements for $Z$ and $W$, but in the first and second variants, we require the asymptotic separation of $Z$ and $W$ by angles from the imaginary axis together with the Lindelöf-type condition for $W$ along the imaginary axis $iR$ on a certain symmetry of the imaginary parts of $1/w$ for $w\in W$ in the form $\Bigl|\sum_{1\leq |w|\leq r}\Im (1/w)\Bigr|=O(1)$ as $r\to +\infty$.

math.CV↗