On univalence of certain integral operators (Q1585207)

Let \(A\) denote the class of functions \(f(z)\), \(f(z)= z+ a_2 z^2+\cdots\), \(|z|< 1\) that satisfy the condition \(|{z^2 f'(z)\over f^2(z)}- 1|< 1\). Such functions are known to be univalent in the unit disk. In this paper the author considers operators on \(A\) \[ f\to \int^z_0 \Biggl({f(t)\over t}\Biggr)^{\alpha+ 1} dt,\quad f\to\Biggl\{\alpha \int^z_0[f(t)]^{\alpha- 1} dt\Biggr\}^{{1\over\alpha}},\quad |f|< 1, \] and proves that, under certain conditions on the complex number \(\alpha\), they preserve univalence. Proofs are based on Ahlfors' type conditions of univalence.