On harmonic bloch and normal functions

Abstract

Let f = h + (g) over bar be a harmonic univalent and sense preserving function on the unit disk, where h and g are analytic. We give the definition of a normal harmonic function. The necessary and sufficient conditions for f = h + g to be Bloch or normal are determined. The sharp upper bound for vertical bar f (z) vertical bar is obtained whenever f = h + (g) over bar is a Bloch function. We obtain sharp estimates for the normality order of f when f maps the unit disk onto the unit disk.