The logarithmic derivative of areally mean \(p\)-valent functions (Q1877812)

This paper is concerned with areally mean \(p\)-valent analytic functions in the unit disk \(\Delta\) having \(k\) directions with maximal growth. As the author points out, \(p\) must be positive, but need not be an integer. Specifically, the class of functions considered here requires \(k\) distinct angles \(\theta_j\), \(j\) = \(1, 2, \dots, k\) and an infinite sequence \(r_n \rightarrow 1^-\) for which \(| f(r_n e^{i\theta_j})| = O((1-r_n)^{-2p/k}\) for each \(j\). If such a function satisfies a rather restrictive area growth requirement then it is proved that \((1-r)\int_0^{2\pi} | f'(re^{i\theta})/ f(re^{i\theta})| ^2 d\theta \rightarrow 4\pi p^2/k\) as \(r \rightarrow 1\). An example is given showing that without the area growth requirement this result need not hold. The paper also includes results about the coefficients of \(\log f(z)/h(z)\) where \(h(z) = b\prod_{j=1}^q (z - z_j)\) with the \(z_j\), \(j\) = \(1, 2, \dots, q\) being the finite number of zeros of \(f(z)\) in \(\Delta\) and \(b\) is chosen so \(f(0)/h(0) = 1\).