Pre-Schwarzian and Schwarzian norm estimates for harmonic functions with fixed 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 $\mathrm{Re}\left(1+z\frac{h^{\prime \prime }(z)}{h^{\prime }(z)}\right)>0$ 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 $f=h+\overline{g}\in {\mathcal{F}}_0$.