Some extension results concerning analytic and meromorphic multivalent functions
Bull. Korean Math. Soc.
Published online 2019 Jan 14
Ali Ebadian, ٰVali Soltani Masih, and Shahram Najafzadeh
Department of Mathematics, Faculty of Science, Payame Noor University, P. O. Box 19395-3697, Tehran, Iran
Abstract : Let $\mathscr{B}_{p,n}^{\eta, \mu}\left(\alpha\right)$; $\left( \eta, \mu\in \mathbb{R}, n,p\in \mathbb{N}\right) $ denote all multivalent functions $f$ class in the unit disk $\mathbb{U}$ as
$f(z)=z^p+\sum_{k=n+p}^{\infty}a_kz^k$ which satisfy:
\[
\left| \left[ \frac{f'(z)}{pz^{p-1}}\right]^{\eta} \left[ \frac{z^p}{f(z)}\right] ^{\mu}-1\right| <1-\frac{\alpha}{p}; \quad \left( z\in \mathbb{U}, \: 0\leq \alpha<p\right),
\]
And $\mathscr{M}_{p,n}^{\eta,\mu}\left(\alpha\right)$ indicates all multivalent meromorphic functions $h$ in the punctured unit disk $\mathbb{U}^{\ast}$ as $h(z)=z^{-p}+\sum_{k=n-p}^{\infty}b_kz^k$ which satisfy:
\[
\left| \left[ \frac{h'(z)}{-pz^{-p-1}}\right]^{\eta} \left[ \frac{1}{z^p h(z)}\right]^{\mu}-1\right| <1-\frac{\alpha}{p}; \quad \left( z\in \mathbb{U}, \: 0\leq \alpha<p\right).
\]
In this paper, using Jack’s Lemma, sufficient conditions for these functions of the above-mentioned classes are investigated. Furthermore, sufficient conditions for strongly starlike $p\mspace{1mu}$-valent functions and convex of order $\gamma$ and type $\beta$, are also considered.
Keywords : Multivalent functions, Multivalent meromorphic functions, Punctured unit disk, Jack’s Lemma, $p\mspace{1mu}$\nobreakdash-\hspace{0pt}Valent strongly starlike and convex functions of order $\gamma$ and type $\beta$
MSC numbers : Primary: 30C45; Secondary: 30A10
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