Some integral operators and their univalence (Q1281337)

Given a normalized univalent function \(g \in S\) such that \(| g''(z)/g'(z)| \leq 1\), \(| z| <1\), and a complex number \(\alpha\), a sufficient condition is stated for the function \(G_{\alpha}(z) = \int_{0}^{z} (g'(u))^{\alpha} du\) to be univalent. The condition is extended to an \(n\)-fold symmetric counterpart of the integral operator. In the statememt of Theorem 2 it should be noted that \(n >1\), whereas for \(n >2\) the constant \(4\) can be easily replaced by \({1 \over 2}(n+2) (1+2/n)^{n/2}\).