A new criterion for 𝑝-valent functions

Goel, R. M.; Sohi, N. S.

doi:10.1090/S0002-9939-1980-0553375-4

A new criterion for $p$-valent functions
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by R. M. Goel and N. S. Sohi

Proc. Amer. Math. Soc. 78 (1980), 353-357

DOI: https://doi.org/10.1090/S0002-9939-1980-0553375-4

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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 $E = \{ z:|z| < 1\}$ and satisfying the condition \[ \operatorname {Re} \left ( {{{({z^n}f)}^{(n + p)}}/{{({z^{n - 1}}f)}^{(n + p - 1)}}} \right ) > (n + p)/2,\] 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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