2020/09/18 by Allu, Vasudevarao, Halder, Himadri · 1 citation
#30C45 #30C50 #30C80 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2009.08683
Let ϕ be analytic and univalent (\it i.e., one-to-one) in \mathbbD:=\z∈ℂ: |z|<1\ such that ϕ(\mathbbD) has positive real part, is symmetric with respect to the real axis, starlike with respect to ϕ(0)=1, and ϕ' (0)>0. A function f ∈ C(ϕ) if 1+ zf''(z)/f'(z) \prec ϕ(z), and f∈ Cc(ϕ) if 2(zf'(z))'/(f(z)+f(z))' \prec ϕ(z) for z∈ \mathbbD. In this article, we consider the classes HC(ϕ) and HCc(ϕ) consisting of harmonic mappings f=h+g of the form h(z)=z+ ∑ n=2∞ anzn and g(z)=∑ n=2∞ bnzn in the unit disk \mathbbD, where h belongs to C(ϕ) and Cc(ϕ) respectively, with the dilation g'(z)=αz h'(z) and |α|<1. Using the Bohr phenomenon for subordination classes \cite[Lemma 1]bhowmik-2018, we find the radius Rf<1 such that Bohr inequality |z|+∑n=2∞ (|an|+|bn|)|z|n ≤ d(f(0),∂ f(\mathbbD)) holds for |z|=r≤ Rf for the classes HC(ϕ) and HCc(ϕ) . As a consequence of these results, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes.