Lower and upper order of harmonic mappings

TL;DR

This paper introduces the concepts of upper and lower order for sense-preserving harmonic mappings, generalizes known results from holomorphic functions, and explores implications of finite upper order with illustrative examples.

Contribution

It defines new order concepts for harmonic mappings, extends classical results, and improves existing distortion theorems in this context.

Findings

Defined upper and lower order for harmonic mappings

Generalized known results from holomorphic functions

Improved distortion theorem for finite upper order

Abstract

In this paper, we define both the upper and lower order of a sense-preserving harmonic mapping in $\mathbb{D}$. We generalize to the harmonic case some known results about holomorphic functions with positive lower order and we show some consequences of a function having finite upper order. In addition, we improve a related result in the case when there is equality in a known distortion theorem for harmonic mappings with finite upper order. Some examples are provided to illustrate the developed theory.