Harmonic spirallike functions and harmonic strongly starlike functions

TL;DR

This paper introduces two new classes of harmonic univalent functions, hereditarily $bblambda$-spirallike and hereditarily strongly starlike, generalizing existing classes and exploring their properties and relations.

Contribution

It defines and analyzes two novel classes of harmonic univalent functions, providing characterizations, relations, and coefficient conditions that extend prior work on harmonic starlike functions.

Findings

Established relations between hereditarily spirallike and strongly starlike functions.

Provided analytic characterizations of hereditarily spirallike functions.

Presented coefficient conditions for hereditary strong starlikeness and spirallikeness.

Abstract

Harmonic functions are natural generalizations of conformal mappings. In recent years, a lot of work have been done by some researchers who focus on harmonic starlike functions. In this paper, we aim to introduce two classes of harmonic univalent functions of the unit disk, called hereditarily $\lambda$-spirallike functions and hereditarily strongly starlike functions, which are the generalizations of $\lambda$-spirallike functions and strongly starlike functions, respectively. We note that a relation can be obtained between this two classes. We also investigate analytic characterization of hereditarily spirallike functions and uniform boundedness of hereditarily strongly starlike functions. Some coefficient conditions are given for hereditary strong starlikeness and hereditary spirallikeness. As a simple application, we consider a special form of harmonic functions.