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Some starlikeness conditions for analytic functions. (English) Zbl 0652.30004
Let A be the class of functions f, analytic in \[ U=\{u\in {\mathbb{C}}:| z| <1,\quad f(0)=f'(0)-1=0, \] and let \(S^*\) be the class of functions f, starlike univalent in U i.e. \[ S^*=(f\in A:Re zf'(z)/f(z)>0,\quad u\in U\}. \] Some new starlikeness conditions in terms of \(f'(z), f'(z)+\alpha zf''(z)\) and \((1-\alpha)f(z)/z+\alpha f'(z)\) are obtained.
Examples: 1. Suppose that for \(f\in A\), \(z\in U\) one of the following conditions hold: \[ (i)\quad | \arg f'(z)| <\alpha_ 0\frac{\pi}{2},\quad \alpha_ 0=0,6165..., \] \[ (ii)\quad | f'(z)- 1| <\frac{2}{\sqrt{5}}, \] \[ (iii)\quad | f'(z)+\alpha zf''(z)- 1| <1,\quad \alpha \geq (\sqrt{5}-2). \] Then \(f\in S^*\).
2. Let \(\gamma >-1\), \(0<M\leq 2(\gamma +2)/\sqrt{5}(\gamma +1).\)
If \(g\in A\) and \(| g'(z)-1| <M\), \((z\in U)\), then \[ (\gamma +1)z^{-\gamma}\int^{z}_{0}g(t)t^{\gamma -1}dt\in S^*. \]
Reviewer: J.Waniurski

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