Spherical derivative of meromorphic function with image of finite spherical area. (Q1974158)

Let \(\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\) be the Riemann sphere. For a Borel set \(E\subset\overline{\mathbb{C}}\) \(A(E)\) stands for the spherical area of \(E\). Let \(\Omega\subset\mathbb{C}\) be a domain and \(f\) be a meromorphic function in \(\Omega\). The spherical derivative is defined as \[ f^\# (z)=\frac{| f (z)| }{1+| f(z)| ^2}\,. \] The author introces the family \(F(\Omega,\beta)\) as follow: \(f\in F(\Omega,\beta)\) if \(A(f(\Omega))\leq\beta\). Note that \[ A(f(\Omega))\leq\int\!\!\!\int_{\Omega}(f^\#(z))^2\,dx\,dy\,; \] we have the equality if \(f\) is univalent. Such definition of \(F(\Omega,\beta)\) is the feature of the article. The author proves that if \(\Omega\) is a hyperbolic domain, then the family \(F(\Omega,\beta)\) is normal and \[ f^\# (z)\leq\sqrt{\frac\beta{\pi-\beta}}\,P_\Omega(z),\quad f\in F(\Omega,\beta)\,, \] where \( P_\Omega(z)\,| dz| \) is the Poincaré metric. The case of equality is investigated as well.