The Bohr inequality for certain harmonic mappings (original) (raw)

A function f ∈ C(φ) if 1 + zf ′′ (z)/f ′ (z) ≺ φ(z) and f ∈ C c (φ) if 2(zf ′ (z)) ′ /(f (z) + f (z)) ′ ≺ φ(z) for z ∈ D := {z ∈ C : |z| < 1}. In this article, we consider the classes HC(φ) and HC c (φ) consisting of harmonic mappings f = h+ g of the form h(z) = z + ∞ n=2 a n z n and g(z) = ∞ n=2 b n z n in the unit disk D, where h belongs to C(φ) and C c (φ) respectively, with the dilation g ′ (z) = αzh ′ (z) and |α| < 1. Using Bohr phenomenon for subordination classes [13, Lemma 1], we find a radius R f < 1 such that Bohr inequality |z| + ∞ n=2 (|a n | + |b n |)|z| n ≤ d(f (0), ∂f (D)) holds for |z| = r ≤ R f for the classes HC(φ) and HC c (φ) .