Bohr Radius for Subordination and K-quasiconformal Harmonic Mappings

The present article concerns the Bohr radius for K-quasiconformal sense-preserving harmonic mappings f= h+ g¯ in the unit disk D for which the analytic part h is subordinated to some analytic function φ, and the purpose is to look into two cases: when φ is convex, or a general univalent function in D. The results state that if h(z)=∑n=0∞anzn and g(z)=∑n=1∞bnzn, then ∑n=1∞(|an|+|bn|)rn≤dist(φ(0),∂φ(D))forr≤r∗and give estimates for the largest possible r∗ depending only on the geometric property of φ(D) and the parameter K. Improved versions of the theorems are given for the case when b1= 0 and corollaries are drawn for the case when K→ ∞.