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Zbl 1107.30012
Kim, Yong Chan; Sugawa, Toshiyuki
The Alexander transform of a spirallike function. (English)
[J] J. Math. Anal. Appl. 325, No. 1, 608-611 (2007). ISSN 0022-247X

Summary: Let ${\cal A}$ denote the class of analytic functions $f$ on the unit disk $\bbfD=\{z\in\bbfC:|z|<1\}$ normalized by $f(0)=0=f'(0)-1$ and let $\bbfS\subset {\cal A}$ consist of univalent functions. Let $$\bbfS_p (\beta)=\left\{f\in{\cal A}:\text{Re}\left(e^{i\beta}\frac{zf'(z)}{f(z)} \right)>0\text{ for }z\in\bbfD \right\},\quad\frac{-\pi}{2}<\beta<\frac {\pi}{2}$$ and $\bbfS_p=\cup_\beta \bbfS_p(\beta)$. Denote $$J[f](z): =\int^z_0\frac{f(\zeta)}{\zeta}d\zeta;\quad I_\alpha[f](z):=\int^z_0 \bigl[f'(\zeta)\bigr]^\alpha d\zeta,\quad \alpha\in \bbfC,\ 1^\alpha=1, \text{ for }f'(z)\ne 0.$$ The authors obtained for example: Theorem 1. The inclusion relation $J(\bbfS_p(\beta))\subset\bbfS$ holds precisely if either $\cos\beta\le\frac 12$ or $\beta=0$. Theorem 2. $A(J (\bbfS_p))= \{\alpha\in\bbfC:|\alpha|\le\frac 12\}$ where $A({\cal F}):=\{\alpha \in \bbfC:I_\alpha\le({\cal F})\subset\bbfS\}$.

MSC 2000:: *30C45 Special classes of univalent and multivalent functions

Keywords: univalent function; integral transform; spirallike function

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Zentralblatt MATH Berlin [Germany]