Schwarz-Pick inequalities for convex domains

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 A (Ω, Π) denote the set of functions f : Ω → Π analytic in Ω. Define the quantities C_n (Ω, Π) by
C_n (Ω, Π) := $¥sup¥limits_{f¥in A(¥Omega,¥Pi)}¥sup¥limits_{z¥in ¥Omega} ¥frac{|f^{(n)}(z)|¥lambda_{¥Pi}(f(z))}{n!(¥lambda_{¥Omega}(z))^{n}}$, n ∈ N
where λ_Ω and λ_Π are the densities of the Poincaré metric in Ω and Π, respectively. We derive sharp upper bounds for |f⁽ⁿ⁾(z)| (z ∈ Ω) and C_n(Ω,Π) if 2 ≤ n ≤ 8 and Ω is a convex domain. The detailed equality condition of the estimate on |f⁽ⁿ⁾(z)| is also given.