Subclasses of uniformly starlike functions (Q1196597)

Recently, \textit{A. W. Goodman} [On uniformly starlike functions, J. Math. Anal. (to appear)] introduced the class UST of uniformly starlike functions. A normalized (\(f(0)=0\) and \(f'(0)=1\)) univalent function in the unit disk \(\mathbb{D}\) belongs to UST if for any circle \(\gamma\) with center \(\zeta\) in \(\mathbb{D}\), the image curve \(f(\gamma\cap\mathbb{D})\) is starlike with respect to the point \(f(\zeta)\). At present, few properties of the class UST are known. The authors investigate subclasses of UST in order to generate information about UST itself. We state some of their results. \textit{A. W. Goodman} gave an analytic characterization for UST; the authors recast this characterization in terms of Hadamard products. As a consequence, they show that \((1/\rho)f(\rho z)\in \text{UST}\) for \(0<\rho\leq 1/\sqrt{2}\) whenever \(f\) is a normalized convex univalent function. The radius \(\rho=\sqrt{2}\) is best possible. This shows how to generate functions in UST from convex univalent functions. An analogous result is obtained for prestarlike functions.