## Some simple criteria of starlikeness and convexity for meromorphic functions.(English)Zbl 0884.30009

Let $$\Sigma$$ be the usual class of univalent functions in the unit disc $$U$$, with a pole at the origin. Denote by $$\Sigma_k$$ the functions in $$\Sigma$$ s.t. $f= {\textstyle \frac 1z}+ \sum a_kz^k, \qquad 0< |z|1, \quad k\geq 0.$ The authors recall the definition of the class $$\Sigma^*$$, and $$\Sigma^c$$ of starlike and convex functions, where $$\Sigma^*\subset \Sigma$$, $$\Sigma^c\subset \Sigma$$. Let $F(z)= \frac{c}{z^{c+1}} \int_0^z t^c f(t)dt.$ Denote $$F=I_c(f)$$. Earlier Aksent’ev proved that if $$f\in\Sigma$$ and $$|z^2 f'(z)+1|<1$$, then $$f$$ is univalent.
Theorem. $$f\in \Sigma_k$$, $$|z^2\lambda'(z)+1|< \frac{c+k+1}{c} \Rightarrow |z^2 F'(z)+1| < 1\Rightarrow F$$ is univalent. Other results are about sufficient conditions for $$f\in \Sigma^*$$ and $$f\in \Sigma^c$$.

### MSC:

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