Analogy of Bombieri’s number for bounded univalent functions

Abstract

Bombieri’s numbers σ_mn characterize the 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. In particular, σ₂₃=0. We define analogous numbers σ_mn(M) for the class S(M) ⊂ S of bounded functions |f(z)|< M, |z| < 1, M >1, with the limit of ratio for Re(p_n(M) − a_n) and Re(p_m(M) − a_m) as f tends to the Pick function P_M(z) = MK⁻¹(K(z)/M) = z + Σ _n=2^∞p_n(M)zⁿ. We prove that σ₂₃(M) = −4/M, M > 1.