Localization of the coefficient theorem

Abstract

Let f be holomorphic and univalent in D={|z|<1} and set K(z)=z/(1−z)². We prove |f⁽ⁿ⁾(z)/f'(z)|≤K⁽ⁿ⁾(|z|)/K'(|z|) at each z∈D and for each n≥2. This inequality at z=0 is just the coefficient theorem of de Branges, the very solution of the Bieberbach conjecture. The equality condition is given in detail. In the specified case where f(D) is convex we have again a similar and sharp result. We also consider |f⁽ⁿ⁾(z)/f'(z)| for f univalent in a hyperbolic domain Ω with the Poincaré density P_Ω(z) and the radius of univalency ρ_Ω(z) at z∈Ω. We obtain the estimate (ρ_Ω(z)/P_Ω(z))ⁿ⁻¹|f⁽ⁿ⁾(z)/f'(z)|≤n!4ⁿ⁻¹ at z∈Ω for n≥2, together with the detailed equality condition on f, Ω, and z.