Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the order of a starlike function
HTML articles powered by AMS MathViewer

by F. Holland and D. K. Thomas
Trans. Amer. Math. Soc. 158 (1971), 189-201
DOI: https://doi.org/10.1090/S0002-9947-1971-0277705-5

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.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 30.42
  • Retrieve articles in all journals with MSC: 30.42
Bibliographic Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 158 (1971), 189-201
  • MSC: Primary 30.42
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0277705-5
  • MathSciNet review: 0277705