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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.)
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