Bohr Radius for Some Classes of Harmonic Mappings

In this note, we introduce a general class of sense-preserving harmonic mappings defined as follows: B-H(0)(M) := {f = h + (g) over bar : Sigma(m=2)(gamma(m)vertical bar a(m)vertical bar + delta(m)vertical bar b(m)vertical bar) <= M, M > 0}, where h(z) = z + Sigma(infinity)(m=2) a(m)Z(m), g(z) = Sigma(infinity)(m=2) b(m)z(m) are analytic functions in D := {z is an element of C : vertical bar z vertical bar <= 1} and gamma(m), delta(m) >= alpha(2) := min{gamma(2), delta(2)} > 0, for all m >= 2. We obtain Growth Theorem, Covering Theorem, and derive the Bohr radius for the class B-H(0) (M). As an application of our results, we obtain the Bohr radius for many classes of harmonic univalent functions and some classes of univalent functions.