The logarithmic derivative of areally mean p-valent functions

Let U(p, k) denote the class of functions f which are areally mean p-valent and have k maximal growth directions in Δ = {z: |z| < 1}. In this paper we give an asymptotic formula of the mean square of the logarithmic derivative f /f for f ∈ U(p, k) with some restricted condition. This generalizes the corresponding result in [J. Anal. Math. 36 (1979) 36–43]. Also we construct a function f ∈ U(p, k) which does not satisfy this asymptotic formula.  2004 Elsevier Inc. All rights reserved.