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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 $f\in \Sigma^c$.

30C45Special classes of univalent and multivalent functions