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Powers of p–valent functions

Published online by Cambridge University Press:  09 April 2009

J. W. Noonan
J. W. Noonan
Affiliation:
College of the Holy CrossWorcester MA 01610, USA
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Abstract

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If f is areally mean p–valent in the unit disc, if λ>0 and if fλ is defined as a single-valued analytic function on the unit disc with finitely many arcs removed, several results in the recent literature suggest that fλ might be areally mean pλ–valent. The purpose of this note is to determine the valence of fλ when f is areally mean p–valent, and also to characterize those functions for which fλ is pλ–valent for all λ>0. Analogous results are obtained for functions which are either s–dimensionally mean p–valent or logarithmically mean p–valent.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 1 , February 1978 , pp. 66 - 70
Copyright
Copyright © Australian Mathematical Society 1978

References

Eke, B. G. (1967), “The asymptotic behavior of areally mean valent functions”, J. Anal. Math. 20, 147–212.CrossRefGoogle Scholar
Hayman, W. K. (1967), Multivalent Functions (Cambridge University Press).Google Scholar
Jenkins, J. A. and Oikawa, K. (1971), “A remark on p-valent functions”, J. Austral. Math. Soc. 12, 397–404.CrossRefGoogle Scholar
Spencer, D. C. (1940), “On finitely mean valent functions, II”, Trans. Amer. Math. Soc. 48, 418–435.CrossRefGoogle Scholar