Goluzin's extension of the Schwarz-Pick inequality (Q1372380)

\nopagenumbers \noindent Let \(f(z)\) be holomorphic and bounded in the unit disk, \(|f(z)|< 1\), with the expansion \(f(z) = a_{0} + a_{n}z^{n} + \cdots\). Set \(\Gamma(z,f) = (1-|z|^{2})|f'(z)|/(1-|f(z)|^{2})\), \(A = |a_{n}|/(1-|a_{0}|^{2})\) and \(\Upsilon(z) = z^{n}(z+A)/(1+Az)\). An extension of the Schwarz-Pick lemma due to G.M. Goluzin is \(\Gamma(z,f) \leq \Gamma(|z|,\Upsilon)\). The author presents an improvement of this inequality in which the right hand side is sharpened. In this form a characterization of equality is possible. A similar result in terms of the Poincaré metric is given in the case of a holomorphic map between hyperbolic domains.