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A sufficient condition for starlikeness. (English) Zbl 0834.30010
Let the function \(f\) be analytic in the unit disk \(U= \{z: |z|< 1\}\), \(f(0)= f'(0)- 1= 0\), satisfy for \(\alpha\geq 0\): \[ \text{Re}\Biggl(\alpha z^2 {f''(z)\over f'(z)}+ {zf'(z)\over f(z)}\Biggr)> 0,\qquad z\in U.\tag{\(*\)} \] Then \(f\) is starlike in \(U\).
If \(\alpha= 1\) then \((*)\) implies \(f\) is strongly starlike of order \({1\over 2}\), namely \(|\arg {zf'(z)\over f(z)}|< {1\over 2} {\pi\over 2}\) \((z\in U)\).

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