Pre-Schwarzian and Schwarzian norm estimates for harmonic functions with fixed analytic part

TL;DR

This paper provides improved estimates for the pre-Schwarzian and Schwarzian norms of harmonic functions with fixed analytic parts in the unit disk, introducing a new class and analyzing its properties.

Contribution

It rectifies previous results, introduces the class alF_0, and derives optimal norm estimates and distortion results for harmonic functions within this class.

Findings

Rectified earlier pre-Schwarzian norm estimates.

Established best possible estimates for alF_0 class.

Derived distortion and coefficient bounds for the co-analytic part.

Abstract

In the present article, we discuss about the estimate of the pre-Schwarzian and Schwarzian norms for locally univalent harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$. In this regard, we first rectify an earlier result of Kanas \emph{et al.} [J. Math. Anal. Appl., {\bf 474}(2) (2019), 931--943] and prove a general result for the pre-Schwarzian norm. We also consider a new class $\mathcal{F}_0$ consisting of all harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that ${\rm Re\,}\left(1+z\frac{h''(z)}{h'(z)}\right)>0$ for $z\in\mathbb{D}$ with dilatation $\omega_f(z)\in Aut(\mathbb{D})$ and obtain best possible estimates of the pre-Schwarzian and Schwarzian norms for functions in the class $\mathcal{F}_0$. Moreover, we obtain the distortion and coefficient estimates of the co-analytic function $g$ when $f=h+\overline{g}\in\mathcal{F}_0$.