Estimates for derivatives of holomorphic functions in a hyperbolic domain

Let f (z) be a holomorphic function in a hyperbolic domain Ω. For 2 n 8, the sharp estimate of |f (n)(z)/f (z)| associated with the Poincaré density λΩ(z) and the radius of convexity ρΩc(z) at z ∈ Ω is established for f (z) univalent or convex in each Δc(z) and z ∈ Ω. The detailed equality condition of the estimate is given. Further application of the results to the Avkhadiev–Wirths conjecture is also discussed. © 2006 Elsevier Inc. All rights reserved.