Images of disks under convex and starlike functions
For \(0\leq \alpha <1\), let K(\(\alpha)\) denote the class of functions in S which are convex of order \(\alpha\). \(K=K(0)\) and \(S^*\) denote the familiar convex and starlike functions in S. It is known that a convex function or a starlike function will map each disk \(| z| <r<1\) onto a convex region or starlike region, respectively. An old result of E. Study and M. S. Robertson asserts that any convex function f maps every disk in \(| z| <1\) onto a convex region. A simple proof of this result and an extension to the classes K(\(\alpha)\) are given. It is shown that the corresponding result for \(S^*\) is false. That is, it is not necessarily true that each \(f\in S^*\) maps each disk \[ | z- z_ 0| <r<1-| z_ 0|,\quad | z_ 0| <1, \] onto a region starlike with respect to \(f(z_ 0)\). In a positive direction, however, it is proved more generally if \(f\in S\) then f maps each disk \(| z-z_ 0| <\rho^*\), \(| z_ 0| <1\), onto a region starlike with respect to \(f(z_ 0)\). The constant \(\rho^*\) is sharp for the full class S and is the best estimate known for \(S^*\).
scientific article; zbMATH DE number 4081898