The punishing factors for convex pairs are \(2^{n-1}\). (English) Zbl 1147.30014

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.


30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30D50 Blaschke products, etc. (MSC2000)
Full Text: DOI Euclid EuDML


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