Improvement of a criterion for starlikeness (Q422071)

Let \(\mathbb{U}=\{z \in C:|z|<1\}\) denote the open unit disc in \(\mathbb{C}\) and let \(A\) be the class of analytic functions \(f\) defined on \(\mathbb{U}\) of the form: \(f(z)=z+a_{2}z^{2}+a_{3}z^{3}+ \dots \). The Alexander operator is defined to be NEWLINE\[NEWLINE\displaystyle{A(f)(z)=\int^{z}_{0} \frac{f(t)}{t} dt}.NEWLINE\]NEWLINE It has been proved that the Alexander operator does not map the class of close-to-convex functions to the class of starlike functions. The main result of the authors is the following:NEWLINENEWLINETheorem. If \(f,g \in A\) and NEWLINE\[NEWLINE\displaystyle{\mathrm{Re} \frac{g(z)}{z}> \frac{100}{83} \left|\mathrm{Im} \frac{g(z)}{z} \right| }, \quad z \in \mathbb{U},NEWLINE\]NEWLINE then the condition NEWLINE\[NEWLINE\displaystyle{\mathrm{Re} \frac{zf^{\prime}(z)}{g(z)}>0, \quad z \in \mathbb{U}}NEWLINE\]NEWLINE implies that \(F=A(f)\in S^{*}\).