## 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.)

starlike