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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the order of a starlike function


Authors: F. Holland and D. K. Thomas
Journal: Trans. Amer. Math. Soc. 158 (1971), 189-201
MSC: Primary 30.42
MathSciNet review: 0277705
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Abstract: It is shown that if $ f \in S$, the class of normalised starlike functions in the unit $ \operatorname{disc} \Delta $, then

$\displaystyle ({\text{i}})\quad \quad \mathop {\lim }\limits_{r \to 1 - } \frac... ...P_\lambda }(r)}}{{ - \log (1 - r)}} = \alpha \lambda {\text{ for }}\lambda > 0;$

$\displaystyle ({\text{ii}})\quad \quad \mathop {\lim }\limits_{r \to 1 - } \fra... ...}\vert{\vert _p}}}{{ - \log (1 - r)}} = \alpha p - 1{\text{ for }}\alpha p > 1;$

and

$\displaystyle ({\text{iii}})\quad \mathop {\lim }\limits_{r \to 1 - } \frac{{\l... ...}{{ - \log (1 - r)}} = (1 + \alpha )p - 1\quad {\text{for (1 + }}\alpha )p > 1,$

where $ {P_\lambda }(r) = \Sigma _{n = 1}^\infty {n^{\lambda - 1}}\vert{a_n}{\vert^\lambda }{r^n},({a_n})$ is the sequence of coefficients and $ \alpha $ the order of $ f$, and where

$\displaystyle \vert\vert{f_r}\vert{\vert _p} = \frac{1}{{2\pi }}\int_0^{2\pi } {\vert f(r{e^{i\theta }})} {\vert^p}d\theta .$

The results extend work of Pommerenke.

The methods of the paper yield various other results, one in particular being

$\displaystyle \mathop {\lim \sup }\limits_{n \to \infty } \frac{{{{\log }^ + }n\vert{a_n}\vert}}{{\log n}} = \alpha $

, a result which has an analogy in the theory of entire functions.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0277705-5
PII: S 0002-9947(1971)0277705-5
Keywords: Starlike function, order of a starlike function, integral means, coefficient means
Article copyright: © Copyright 1971 American Mathematical Society