ON CERTAIN CLASSES OF MEROMORPHICALLY P-VALENT STARLIKE FUNCTIONS

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YUANG HSIANG LIN

Abstract

Let $M_{n, p}(\alpha)$ denote the class of functions of the form


\[\int f(z) = \frac{1}{z^p}+\frac{a_0}{z^{p-1}}+\cdots+a_{k+p+1}z^k+\cdots, \quad p\in\mathbb{N}\]





that are regular in the punctured disk $E = \{z : 0 < |z| < 1\}$ and satisfy


\[\text{Re}\left\{\frac{z(D^{n+p-1}f(z))'}{D^{n+p-1}f(z)}\right\}<-\alpha\]





for $0\le \alpha<p$ and $z\in U= \{z:|z|<1\}$, where


\[D^{n+p-1}f(z)=\frac{1}{z^p}\left(\frac{z^{n+2p-1}f(z)}{(n+p-1)!}\right)^{n+p-1}\]





N. E. Cho and S. Owa [1] showed that $M_{n+1,p}(\alpha) \subset M_{n, p}(\alpha)$. In this paper, we use the Miller and Mocanu's lemma [3] to improve this property










Article Details

How to Cite
LIN, Y. H. (1995). ON CERTAIN CLASSES OF MEROMORPHICALLY P-VALENT STARLIKE FUNCTIONS. Tamkang Journal of Mathematics, 26(1), 59–64. https://doi.org/10.5556/j.tkjm.26.1995.4379
Section
Papers

References

Nak Eun Cho and Shigeyoshi Owa, "On Certain Class of Meromorphically p-valent Starlike Fune­ tions," Topics in Univalent Functions and Its Applications, Su ribaisekikenkyusho Kokyilroku, 821 (1993), 159-165.

I. S. Jack, "Functions starlike and convex of order a," ]. London Math. Soc. (2) 3 (1971), 469-474.

S. S. Miller and P. T. Mocanu, "Second order differential inequalities in the complex plane," J. Math. Appl. 65 (1978), 289-305.