# zbMATH — the first resource for mathematics

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

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