On the order of a starlike function

Abstract:

It is shown that if $f \in S$, the class of normalised starlike functions in the unit $\operatorname {disc} \Delta$, then \[ ({\text {i}})\quad \quad \lim \limits _{r \to 1 - } \frac {{\log {P_\lambda }(r)}}{{ - \log (1 - r)}} = \alpha \lambda {\text { for }}\lambda > 0;\] \[ ({\text {ii}})\quad \quad \lim \limits _{r \to 1 - } \frac {{\log ||{f_r}|{|_p}}}{{ - \log (1 - r)}} = \alpha p - 1{\text { for }}\alpha p > 1;\] and \[ ({\text {iii}})\quad \lim \limits _{r \to 1 - } \frac {{\log ||f’_r||_{p}}}{{ - \log (1 - r)}} = (1 + \alpha )p - 1\quad {\text {for (1 + }}\alpha )p > 1,\] where ${P_\lambda }(r) = \Sigma _{n = 1}^\infty {n^{\lambda - 1}}|{a_n}{|^\lambda }{r^n},({a_n})$ is the sequence of coefficients and $\alpha$ the order of $f$, and where \[ ||{f_r}|{|_p} = \frac {1}{{2\pi }}\int _0^{2\pi } {|f(r{e^{i\theta }})} {|^p}d\theta .\] The results extend work of Pommerenke. The methods of the paper yield various other results, one in particular being \[ \lim \sup \limits _{n \to \infty } \frac {{{{\log }^ + }n|{a_n}|}}{{\log n}} = \alpha \], a result which has an analogy in the theory of entire functions.