A sufficient condition for starlikeness. (English) Zbl 0834.30010

Ramesha, C.; Kumar, Sampath; Padmanabhan, K. S.

Chin. J. Math. 23, No. 2, 167-171 (1995).

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 \(|\arg {zf'(z)\over f(z)}|< {1\over 2} {\pi\over 2}\) \((z\in U)\).

Reviewer: J.Waniurski (Lublin)

MSC:

30C45

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

Keywords:

starlike