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 + a ₂ z ² +.... 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, σ ₂₃=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)z ⁿ. We prove that σ ₂₃(M) = −4/M, M > 1.