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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A new criterion for $ p$-valent functions


Authors: R. M. Goel and N. S. Sohi
Journal: Proc. Amer. Math. Soc. 78 (1980), 353-357
MSC: Primary 30C45
MathSciNet review: 553375
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Abstract: In this paper we consider the classes $ {K_{n + p - 1}}$ of functions $ f(z) = {z^p} + {a_{p + 1}}{z^{p + 1}} + \cdots $ which are regular in the unit disc $ E = \{ z:\vert z\vert < 1\} $ and satisfying the condition

$\displaystyle \operatorname{Re} \left( {{{({z^n}f)}^{(n + p)}}/{{({z^{n - 1}}f)}^{(n + p - 1)}}} \right) > (n + p)/2,$

where p is a positive integer and n is any integer greater than $ - p$. It is proved that $ {K_{n + p}} \subset {K_{n + p - 1}}$. Since $ {K_0}$ is the class of p-valent functions, consequently it follows that all functions in $ {K_{n + p - 1}}$ are p-valent. We also obtain some special elements of $ {K_{n + p - 1}}$ via the Hadamard product.

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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0553375-4
PII: S 0002-9939(1980)0553375-4
Keywords: p-valent functions, p-valent starlike functions, Hadamard product
Article copyright: © Copyright 1980 American Mathematical Society