Schwarz-pick inequalities for derivatives of arbitrary order

Abstract

Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λΩ and λΠ denote the densities of the Poincaré metric in Ω and Π, respectively. For f : Ω → Π analytic in Ω, inequalities of the type |f(n)(z)|/n! ≤ Mn (z, Ω, Π) (λΩ(z))n/λΠ(f(z)), z ∈ Ω, are considered where Mn(z, Ω, Π) does not depend on f and represents the smallest value possible at this place. We prove that Mn(z, Δ, Π) = (1 + |z|)n-1 if Δ is the unit disk and Π is a convex domain. This generalizes a result of St. Ruscheweyh. Furthermore, we show that Cn(Ω, Π) = sup{Mn(z, Ω, Π) | z ∈ Ω} ≤ 4n-1 holds for arbitrary simply connected domains whereas the inequality 2n-1 ≤ Cn(Ω, Π) is proved only under some technical restrictions upon Ω and Π.