A General Coefficient Theorem for Univalent Functions: Generalization of the Bieberbach and Zalcman Conjectures

The main result of this paper is that any rotationally homogeneous polynomial functional J(f)=∑|α|=n0NCm1,…,msam1αm1…amsαms with |α|=αm1+⋯+αms≥3,2<am1<⋯<ams<∞, depending on the distinguished finitely many coefficients amj, is maximized on S by Koebe’s function κθ(z)=z/(1−eiθz)2 with θ∈[−π,π]. This includes, in particular, the well-known Bieberbach and Zalcman conjectures and covers many other coefficient estimates for univalent functions. As an application, the main theorem provides the solution of the generalized Zalcman conjecture posed by Ma.