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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Coefficient differences and Hankel determinants of areally mean $ p$-valent functions

Author: James W. Noonan
Journal: Proc. Amer. Math. Soc. 46 (1974), 29-37
MSC: Primary 30A36
MathSciNet review: 0352440
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Abstract: With $ p > 0$, denote by $ {S_p}$ the class of functions analytic and areally mean $ p$-valent in the open unit disc. If $ f \in {S_p}$, it is well known that $ \alpha (f) = {\lim _{r \to 1}}{(1 - r)^{2p}}M(r,f)$ exists and is finite. If $ q \geq 1$ is an integer, denote the $ q$ Hankel determinant of $ f$ by $ {H_q}(n,f)$. In this paper the asymptotic behavior of $ {H_q}(n,f)$, as $ n \to \infty $, is related to $ \alpha (f)$. A typical result is: if $ \alpha (f) > 0$, and if $ p > q - 3/4$, then

$\displaystyle \vert{H_q}(n,f)\vert/{n^{2pq - {q^2}}} \sim \vert{Q_q}(p)\vert{(\alpha (f)/\Gamma (2p))^q},$

where $ {Q_q}$ is a polynomial of degree at most $ {q^2} - q$. In the course of the proof, asymptotic results are proved concerning certain coefficient differences, and in particular concerning $ \vert{a_n}\vert - \vert{a_{n - 1}}\vert$.

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Keywords: Areally mean $ p$-valent, Hankel determinants, coefficient differences
Article copyright: © Copyright 1974 American Mathematical Society

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