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 \(f\in \Sigma^c\).


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