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


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