Starlikeness of a certain integral (Q1175140)

Let \(A\) denote the class of functions of the form \(f(z)=z+\sum_{n=2}^ \infty a_ nz^ n\) which are analytic in the unit disk \(U\). Let \(R(\alpha)\) be the subclass of \(A\) consisting of all functions satisfying \(\text{Re}\{f'(z)\}>\alpha\) (\(z\in U\)) for some \(\alpha\) (\(\alpha<1\)). Denoting by \(S^*\) the subclass of \(A\) consisting of all starlike functions, the author proves: Theorem. If \(f(z)\in R(\alpha)\) with \(\beta\leq\alpha<1\), then the function \(g(z)\) defined by \(g(z)=\int_ 0^ z{f(t)\over t}dt\) is in the class \(S^*\), where \[ \beta=-{{8\log 2- 3-4(\log 2)^ 2} \over {8\log 2-6-4(\log 2)^ 2}}=-0.262\dots\;. \] The result is sharp.