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Starlikeness of functions with bounded mean modulus. (English. Russian original) Zbl 0622.30003
Sib. Math. J. 27, 154-161 (1986); translation from Sib. Mat. Zh. 27, No. 2(156), 14-22 (1986).
For $$\delta >0$$ let $$H^ m_{\delta}(c_ m)$$ be the class of functions f analytic in $$\{$$ $$z: | z| <1\}$$ and such that $\frac{1}{2\pi}\int^{2\pi}_{0}| f(re^{i\theta})|^{\delta} d\theta \leq 1,\quad r\leq 1,$ f(z)$$=c_ mz^ m+c_{m+1}z^{m+1}+..$$. and f(z)$$\neq 0$$ for $$z\neq 0$$. The author solves an extremal problem to minimize $\sup \{r: J(f)=Re(zf'(z)/f(z))\leq 0,\quad for\quad | z| =r\}$ on the class $$H^ m_{\delta}(c_ m)$$.
Reviewer: V.V.Peller
##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)