The derivative of a holomorphic function and estimates of the Poincaré density
The author proves several inequalities for the derivative of a function holomorphic in the unit disc \(D\), generalizing classical inequalities, and gives applications to the Poincaré density of the image domain \(G=f(D)\) and the Schwarzian derivative of \(f\). From these results we quote Theorem 3. Let \(f\) be holomorphic in \(D\). Then \(\delta(z,f)\leq{4\rho(z,f)F(z,f)\over \{\rho(z,f)+1\}^ 2}\). Equality for some \(z=z_ 0\) with \(f'(z_ 0)\neq 0\) is possible only for one of the functions \(f(z)=p{z-z_ 0\over 1-\overline{z_ 0}z}+q\), \(p\neq 0\) and \(q\) constants. Here, \(F(z,f)=(1-| z|^ 2)| f'(z)|\), \(\delta(z,f)\) is the radius of the largest schlicht disc contained in the image domain (or Riemann surface) \(f(D)\) centred at \(f(z)\), and \(\rho(z,f)\) is the radius of the largest non-euclidean disc \(\left|{\zeta-z\over 1-\overline z\zeta}\right|<r\) centred at \(z\) where \(f\) is univalent.
scientific article; zbMATH DE number 92935