The minimal growth of entire functions with given zeros along unbounded sets

Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition$$\varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0$$is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that$$\varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0.$$Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$.

Sufficient conditions for the improved regular growth of entire functions in terms of their averaging

Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is of improved regular growth (an entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that $\log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1})$, $U\not\ni z=re^{i\varphi}\to\infty$), then for some $\rho_3\in (0,\rho)$ the relation \begin{equation*} \int_1^r {\frac{\log |{f(te^{i\varphi})}|}{t}}\, dt=\frac{r^\rho}{\rho}h(\varphi)+o(r^{\rho_3}),\quad r\to +\infty, \end{equation*} holds uniformly in $\varphi\in [0,2\pi]$. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some $\rho_3\in (0,\rho)$ the last asymptotic relation holds uniformly in $\varphi\in [0,2\pi]$, then $f$ is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko.

The Growth of Entire Functions Defined by Laplace–Stieltjes Transforms Related to Proximate Order and Approximation

In this article, we discuss the growth of entire functions represented by Laplace–Stieltjes transform converges on the whole complex plane and obtain some equivalence conditions about proximate growth of Laplace–Stieltjes transforms with finite order and infinite order. In addition, we also investigate the approximation of Laplace–Stieltjes transform with the proximate order and obtain some results containing the proximate growth order, the error, An∗, and λn, which are the extension and improvement of the previous theorems given by Luo and Kong and Singhal and Srivastava.

Growth and Approximation of Laplace-Stieltjes Transform with p,q-Proximate Order Converges on the Whole Plane

One purpose of this paper is to study the growth of entire functions defined by Laplace-Stieltjes transform converges on the whole complex plane, by introducing the concept of p,q-proximate order, and one equivalence theorem of the p,q-proximate order of Laplace-Stieltjes transforms is obtained. Besides, the second purpose of this paper is to investigate the approximation of entire functions defined by Laplace-Stieltjes transforms with p,q-proximate order, and some results about the p,q-proximate order, the error, and the coefficients of Laplace-Stieltjes transforms are obtained, which are generalization and improvement of the previous theorems given by Luo and Kong, Singhal, and Srivastava.

Algebraic Values of Entire Functions with Extremal Growth Orders: An Extension of a Theorem of Boxall and Jones

Given an entire function $f$ of finite order $\unicode[STIX]{x1D70C}$ and positive lower order $\unicode[STIX]{x1D706}$, Boxall and Jones proved a bound of the form $C(\log H)^{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D70C})}$ for the density of algebraic points of bounded degree and height at most $H$ on the restrictions to compact sets of the graph of $f$. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ are effectively computable from certain data associated with the function. In this followup note, using different measures of the growth of entire functions, we obtain similar bounds for other classes of functions to which the original theorem does not apply.

On the growth of system of entire homogenous polynomials of several complex variables

In this paper we study the growth of entire functions represented by homogenous polynomials of two complex variables. The characterizations of their order and type have been obtained