Estimates of spherical derivative of meromorphic functions
Let \(f\) be a meromorphic function on the unit disk \(D\). The author considers various analytic-geometric quantities related to \(f\). These quantities include: (i) The spherical derivative \(f^\sharp (z)= |f^\prime(z)|/(1+|f(z)|^2)\). (ii) The maximum \(\rho_a(z,f)\) of all \(r\in (0,1]\) such that \(f(w)\neq -1/\overline{f(z)}\), for all \(w\) in the hyperbolic disk \(\Delta(z,r)\) of center \(z\) and hyperbolic radius \(r\). (iii) \(\rho_a(f):= \inf\{\rho_a(z,f):z\in D\}\). (iv) The maximum \(\rho(z,f)\) of all \(r\in (0,1]\) such that \(f\) is univalent in \(\Delta(z,r)\). (v) The maximum \(\rho_\omega(z,f)\) of all \(r\in (0,1]\) such that \(e^{f(w)}+e^{f(z)}\neq 0\) for all \(w\in \Delta(z,r)\). (vi) \(\rho_\omega (f):= \inf\{\rho_\omega (z,f):z\in D\}\). (vii) The Schwarzian derivative of \(f\), and several other quantities. The author proves various sharp estimates involving the above quantities. Of particular interest is the investigation of the equality cases and the identification of the extremal functions. Among the main results of the paper are the following: (a) \(f\) is normal (that is, the function \((1-|z|^2) f^\sharp(z)\) is bounded in \(D\)) if and only if \(\rho_a(f)>0\). (b) A holomorphic function \(f\) on \(D\) is Bloch (that is, \(\sup_{z\in D}(1-|z|^2) |f^\prime(z)|<\infty\)) if and only if \(\rho_\omega(f)>0\).
scientific article; zbMATH DE number 1751377