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.