The punishing factors for convex pairs are \(2^{n-1}\)
Let \(\Omega\) and \(\Pi\) be simply connected proper subdomains of the complex plane, \(f:\Omega\rightarrow\Pi\) be a holomorphic function, \(\lambda_\Omega(z)\) and \(\lambda_\Pi(z)\) be the density of the Poincaré metric with curvature \(K=-4\) at \(z\in\Omega\) and \(w\in\Pi,\) respectively. The authors prove that if \(\Omega\) and \(\Pi\) are convex then \[ \frac{|f^{(n)}(z)|}{n!}\leq 2^{n-1}\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))} \] and if \(\Omega\) is convex and \(\Pi\) linearly accessible then \[ \frac{|f^{(n)}(z)|}{(n+1)!}\leq 2^{n-2}\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))} \] for \(n=2,3,\ldots\) and \(z\in\Omega.\) Even more \(2^{n-1}\) and \(2^{n-2}\) are the best possible for any admissible pair \((\Omega,\Pi).\) The title of the paper is motivated by the fact that the quotient \(\frac{(\lambda_\Omega(z))^n}{\lambda_\Pi(f(z))}\) reflects the influence of the position of the points \(z\) and \(f(z)\) in \(\Omega\) and \(\Pi\) on the \(f^{(n)}(z)\) and the coefficients \(2^{n-1}\) and \(2^{n-2}\) are factors that ``punish'' bad behavior of \(\Omega\) and \(\Pi\) at the boundary.
scientific article; zbMATH DE number 5281233
scientific article