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Zbl 1032.30007
Cho, Nak Eun; Kim, Tae Hwa
Multiplier transformations and strongly close-to-convex functions.
(English)
[J] Bull. Korean Math. Soc. 40, No.3, 399-410 (2003). ISSN 1015-8634

Let ${\cal A}$ be the class of functions $f(z)=z+\sum_{k=2}^\infty a_kz^k$ that are analytic in the unit disc ${\cal U}=\{z:|z|<1\}.$ For any integer $n,$ $\lambda\ge 0$ and $f\in {\cal A}$ a multiplier transformation is defined by $$I_n^\lambda f(z)=z+\sum_{k=2}^\infty k\left(\frac{1+\lambda}{k+\lambda}\right)^n a_kz^k.$$ Further, for $0\le\gamma<1$ and $0<\delta\le 1,$ the class ${\cal K}_n^\lambda(\gamma,\delta,\eta,A,B)$ consists of functions $f\in {\cal A}$ satisfying $$\left|\arg\left(\frac{z\left(I_n^\lambda f(z)\right)'}{I_n^\lambda g(z)}-\gamma\right)\right|<\delta\frac{\delta}{2},\quad z\in{\cal U},$$ for some $g\in {\cal S}_n^\lambda (\eta,A,B),$ where $${\cal S}_n^\lambda (\eta,A,B)=\left\{g\in{\cal A}: \frac{1}{1-\eta}\left(\frac{z\left(I_n^\lambda g(z)\right)'}{I_n^\lambda g(z)}-\eta\right)\prec \frac{1+Az}{1+Bz}\right\}$$ and $\prec$'' denotes the usual subordination. In this paper, the authors give the basic inclusion relationships among the classes ${\cal K}_n^\lambda(\gamma,\delta,\eta,A,B)$ and some integral preserving properties in connection with the operator $I_n^\lambda.$
[Nikola Tuneski (Skopje)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions

Keywords: subordinate; multiplier transformation; strongly close-to-convex; integral operator

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