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On some classes of bi-univalent functions. (English) Zbl 0614.30017
Let S denote the class of all functions $$f(z)=z+a_ 2z^ 2+a_ 3z^ 3+..$$. which are analytic and univalent in the unit disc $$U=\{z:| z| <1\}$$. Next, let $$\sigma$$ denote the class of functions f which are analytic and bi-univalent in U, that is $$f\in S$$ and $$f^{-1}$$ has a univalent analytic continuation to $$\{| w| <1\}$$. The authors introduce the following subclasses of the class $$\sigma$$ : the class $$S^*_{\sigma}[\alpha]$$ of strongly bi-starlike functions of order $$\alpha$$, $$0<\alpha \leq 1$$, i.e. of the functions satisfyng the conditions $| \arg (zf'(z)/f(z))| <\alpha (\pi /2),\quad | \arg (wg'(w)/g(w))| <\alpha (\pi /2),\quad | z| <1,\quad | w| <1;$ the class $$S^*_{\sigma}(\beta)$$ of bi-starlike functions of order $$\beta$$, $$0\leq \beta <1$$, i.e. of the functions satisfying the conditions $Re(zf'(z)/f(z))>\beta,\quad Re(wg'(w)/g(w))>\beta,\quad | z| <1,\quad | w| <1;$ the class $$C_{\sigma}(\beta)$$ of bi-convex functions of order $$\beta$$, $$0\leq \beta <1$$, i.e. of the functions such that $Re\{1+(zf''(z)/f'(z))\}>\beta,\quad Re\{1+(wg''(w)/g'(w))\}>\beta,\quad | z| <1,\quad | w| <1,$ where $g(w)=w-a_ 2w^ 2+(2a^ 2_ 2-a_ 3)w^ 3+...$ is the extension of $$f^{-1}$$ to the whole of $$| w| <1$$. In this paper the authors obtain upper bounds for $$| a_ 2|$$ and $$| a_ 3|$$ when f belongs to $$S^*_{\sigma}[\alpha]$$, $$S^*_{\sigma}(\beta)$$ or $$C_{\sigma}(\beta)$$.
Reviewer: J.Kaczmarski

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)