The growth at infinity of a sequence of entire functions of bounded orders

TL;DR

This paper investigates the growth behavior at infinity of sequences of entire functions with bounded orders, extending previous results for genus zero functions and establishing near-optimal conditions based on zeros.

Contribution

It introduces new conditions on zeros of entire functions of bounded orders that determine their growth limits, extending prior work on genus zero functions.

Findings

Established nearly optimal zero-based conditions for growth limits

Extended growth results from genus zero to bounded order functions

Connected growth analysis to extremal functions similar to Siciak's extremal function

Abstract

In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in \cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of entire functions of bounded orders $P_n(z)$, we found a nearly optimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that we have \begin{eqnarray*} \limsup_{n\to\infty}|P_n(z)|^{1/k_n}\leq 1 \end{eqnarray*} for all $z\in \mathbb C$ (see Theorem \ref{theo5}). Exploring the growth of a sequence of entire functions of bounded orders lead naturally to an extremal function which is similar to the Siciak's extremal function (See Section 6).