ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS

In this paper, we shall prove Theorem 1 Let $f$ be nonconstant meromorphic in $\mathbb{C}$ with finite positive order $\lambda$, $\lambda(r)$ be a proximate order of $f$ and $U(r, f)=r^{\lambda(r)}$, then for each number $\alpha$,$0<\alpha<\pi/2$, there exists a number $\phi_0$ with $0\le \phi_0 < 2\pi$ such that the inequality \[ \limsup_{r\to\infty}\sum_{i=1}^3 n(r, \phi_0, \alpha, f=a_i(z))/U(r, f)>0,\] holds for any three distinct meromorphic function $a_i(z)(i=1, 2, 3)$ with $T(r,a_i)=o(U(r, f))$ as $r\to\infty$.

Asymptotics of $\delta$-subharmonic functions of finite order

For $\delta$-subharmonic in $\mathbb{R}^m$, $m\geq2$, function $u=u_1-u_2$ of finite positiveorder we found the asymptotical representation of the form\[u(x)=-I(x,u_1)+I(x,u_2) +O\left(V(|x|)\right),\ x\to\infty,\]where $I(x,u_i)=\int\limits_{|a-x|\leq|x|}K(x,a)d\mu_i(a)$, $K(x,a)=\ln\frac{|x|}{|x-a|}$ for $m=2$,$K(x,a)=|x-a|^{2-m}-|x|^{2-m}$ for $m\geq3,$$\mu_i$ is a Riesz measure of the subharmonic function $u_i,$ $V(r)=r^{\rho(r)},$ $\rho(r)$ is a proximate order of $u$.The obtained result generalizes one theorem of I.F. Krasichkov for entire functions.

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.

A Phragmén–Lindelöf Theorem via Proximate Orders, and the Propagation of Asymptotics

We prove that, for asymptotically bounded holomorphic functions in a sector in $$\mathbb {C},$$ C , an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by Fruchard and Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén–Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of Lindelöf and Valiron.

A Note on Relative $(p,q)$ th Proximate Order of Entire Functions

Relative order of functions measures specifically how different in growth two given functions are which helps to settle the exact physical state of a system. In this paper for any two positive integers $p$ and $q,$ we introduce the notion of relative $(p,q)$ th proximate order of an entire function with respect to another entire function and prove its existence.