The Bohr inequality for certain harmonic mappings

Abstract

Let HC(?) and HCc(?) denote the classes of sense-preserving harmonic mappings f=h+g� in D with the dilation g?(z)=?zh?(z) for |?|<1 such that h is Ma�Minda type convex function and Ma�Minda type convex function with respect to the conjugate points respectively. Here, the function ?:D??, called Ma�Minda function, is analytic and univalent in D such that ?(D) has positive real part, symmetric with respect to the real axis, starlike with respect to ?(0)=1 and ??(0)>0. The classes HC(?) and HCc(?) are derived from the work of Sun et al. (2016) on the class M(?,?). We study the growth theorems for the functions in these classes. For general ?, the sharp coefficient bounds of the functions in the classes HC(?) and HCc(?) are not yet known. For the function f=h+g� of the form h(z)=z+?n=2?anzn and g(z)=?n=2?bnzn in D, using the Bohr phenomenon for subordination classes, we find the radius Rf<1 such that Bohr inequality |z|+?n=2?(|an|+|bn|)|z|n?d(f(0),?f(D))holds for |z|=r?Rf for functions in the classes HC(?) and HCc(?). As a consequence, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes. In addition, we discuss the area theorem for functions in the class HC(?) and obtain an improved version of Bohr inequality for functions in HC(?). � 2021 Royal Dutch Mathematical Society (KWG)