Asymptotic estimates of entire functions of bounded $\mathbf{L}$-index in joint variables

TL;DR

This paper derives growth estimates for entire functions of bounded L-index in several complex variables, generalizing previous results and providing sharper bounds, which enhance understanding of their maximum modulus behavior.

Contribution

It extends known growth estimates for entire functions of bounded L-index to a broader class in multiple variables, improving existing bounds.

Findings

Derived growth estimates for entire functions in several complex variables.

Generalized previous results to wider classes of functions.

Provided sharper bounds than existing estimates for certain cases.

Abstract

In this paper, there are obtained growth estimates of entire in $\mathbb{C}^n$ function of bounded $\mathbf{L}$-index in joint variables. They describe the behaviour of maximum modulus of entire function on a skeleton in a polydisc by behaviour of the function $\mathbf{L}(z)=(l_1(z),\ldots,l_n(z)),$ where for every $j\in\{1,\ldots, n\}$ \ $l_j:\mathbb{C}^n\to \mathbb{R}_+$ is a continuous function. We generalised known results of W. K. Hayman, M. M. Sheremeta, A. D. Kuzyk, M. T. Borduyak, T. O. Banakh and V. O. Kushnir for a wider class of functions $\mathbf{L}.$ One of our estimates is sharper even for entire in $\mathbb{C}$ functions of bounded $l$-index than Sheremeta's estimate.