Geometric properties of $\log$-polyharmonic mappings

TL;DR

This paper introduces a new class of log-polyharmonic mappings in the unit disk, explores their geometric properties like starlikeness and convexity, and proves the Goodman-Saff conjecture within this class.

Contribution

It defines the class of log-polyharmonic mappings and establishes key geometric properties, including the proof of the Goodman-Saff conjecture for a subclass.

Findings

The class $ ext{L}_p ext{H}$ is well-defined and studied.

Several geometric properties such as starlikeness, convexity, and univalence are characterized.

The Goodman-Saff conjecture is proven true within $ ext{L}_p ext{H}(G)$.

Abstract

In this paper, a class of $\log$-polyharmonic mappings $\mathcal{L}_p\mathcal{H}$ together with its subclass $\mathcal{L}_p\mathcal{H}(G)$ in the unit disk $\mathbb{D}=\{z: |z|<1\}$ is introduced, and several geometrical properties such as the starlikeness, convexity and univalence are investigated. In particular, we consider the Goodman-Saff conjecture and prove that the conjecture is true in $\mathcal{L}_p\mathcal{H}(G)$.