Estimate of \(d_ 0/d^ *\) for starlike functions
Let \(S^*\) be the class of starlike functions in the unit disk \(\Delta =\{z\in {\mathbb{C}}:| z| <1\}\). By \(D_ f\), \(r_ 0\), \(d_ 0\) and \(d^*\) let us denote: \(D_ f=f(\Delta)\), \(r_ 0\) the radius of convexity of f, \[ d_ 0=\min_{| z| =r_ 0}| f(z)|,\quad d^*=\inf_{\beta \not\in D_ f}| \beta |,\text{ where } f\in S^*. \] In 1953 A. Schild formulated the problem of finding the exact greatest lower bound for \(d_ 0/d^*\) in the class \(S^*\). In this note, the author proves that \(d_ 0/d^*\geq 0.4101492\).
scientific article; zbMATH DE number 3991841