Estimates for derivatives of holomorphic functions in a hyperbolic domain (Q868806)

Let \(f(z)\) be a holomorphic function in a hyperbolic domain \(\Omega\). For \(2\leq n\leq 8\), the sharp estimate of \(| f(n)(z)/f'(z)| \) associated with the Poincaré density \(\lambda_\Omega(z)\) and the radius of convexity \(\rho_{\Omega_c}(z)\) at \(z\in \Omega\) is established for \(f(z)\) univalent or convex in each \(\Delta_c(z) \) and \(z\in\Omega\). The detailed equality condition of the estimate is given. Further application of the results to the Avkhadiev-Wirths conjecture is also discussed.