On Harmonic Entire mappings

TL;DR

This paper explores the properties of harmonic entire mappings, characterizing their order and type, and examining conditions under which certain derivatives are univalent, providing bounds and relationships among these properties.

Contribution

It offers new characterizations of the order and type of harmonic entire mappings and establishes conditions for their derivatives to be univalent, including bounds on their order.

Findings

Characterization of order and type for harmonic entire mappings

Necessary conditions for derivatives to be univalent

Upper bounds for the order of such mappings

Abstract

In this paper, we investigate properties of harmonic entire mappings. Firstly, we give the characterizations of the order and the type for a harmonic entire mapping $f=h+\overline{g}$, respectively, and also consider the relationship between the order and the type of $f$, $h$, and $g$. Secondly, we investigate the harmonic mappings $f=h+\overline{g}$ such that $f^{(n_p)}=h^{(n_p)}+\overline{g^{(n_p)}}$ are univalent in the unit disk, where $\{n_p\}_{p=1}^{\infty}$ be a strictly increasing sequence of nonnegative integers. In terms of the sequence $\{n_p\}_{p=1}^{\infty}$, we derive several necessary conditions for these mappings to be entire and also establish an upper bound for the order of these mappings.