2023/07/27 by Ali, Md Firoz, Pandit, Sushil · 1 citation
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2307.14793
In the present article, we discuss about the estimate of the pre-Schwarzian and Schwarzian norms for locally univalent harmonic functions f=h+g in the unit disk \mathbbD:=\z∈ℂ: |z|<1\. In this regard, we first rectify an earlier result of Kanas 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 F0 consisting of all harmonic functions f=h+g in the unit disk \mathbbD such that \rm Re (1+z(h''(z))/(h'(z)))>0 for z∈\mathbbD with dilatation ωf(z)∈ Aut(\mathbbD) and obtain best possible estimates of the pre-Schwarzian and Schwarzian norms for functions in the class F0. Moreover, we obtain the distortion and coefficient estimates of the co-analytic function g when f=h+g\inF0.