Qualitative Results in the Bombieri Problem for Conformal Mappings

Abstract

Bombieri’s numbers σ_mn characterize a behavior of the coefficient body for the class S of all holomorphic and univalent functions f in the unit disk normalized by f(z) = z + a₂z² + …. The number σ_mn is the limit of ratio for Re(n − a_n) and Re (m − a_m) as f tends to the Koebe function K(z) = z(1 − z)⁻². It is showed in the paper that Bombieri’s conjecture about explicit values of σ_mn implies a sliding regime in an associated control theory problem generated by the Loewner differential equation. We develop also an asymptotical approach in verification of necessary criteria for Bombieri’s conjecture.