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Schwarz-Pick inequalities for derivatives of arbitrary order. (English) Zbl 1018.30018

Let Ω and Π be two simply connected domains in the complex plane which are not equal to . We denote by λ Ω and λ Π the densities of the hyperbolic metrics in Ω and Π respectively. Let A(Ω,Π) denote the set of functions f:ΩΠ analytic in Ω. The authors consider the quantities

M n (z,Ω,Π):=sup|f (n) (z)| n!λ Π (f(z)) (λ Ω (z)) n :fA(Ω,Π),n,zΩ,
C n (Ω,Π):=sup{M n (z,Ω,Π):zΩ}·

It follows by the Schwarz-Pick lemma that C 1 (Ω,Π)=1 for any pair of simply connected domains. The quantity C n (Ω,Π) has been computed in some special cases by St. Ruscheweyh (for example, when Ω is the unit disk Δ and Π is a half-plane).

The authors of the article under review show that for any convex domain Π,

M n (z,Δ,Π)=(1+|z|) n-1 ·

It follows that C n (Δ,Π)=2 n-1 for convex domain Π. Furthermore, they show that C n (Ω,Π)4 n-1 holds for arbitrary simply connected domains whereas the inequality 2 n-1 C n (Ω,Π) is proved only under some technical restriction upon Ω and Π.

MSC:
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30C55General theory of univalent and multivalent functions