Two Coefficient Conjectures for Nonvanishing Hardy Functions, I

Abstract

There are two eminent still open conjectures for nonvanishing holomorphic Hardy functions f(z) on the unit disk. The Hummel–Scheinberg–Zalcman conjecture posed in 1977 extends Krzyz’s conjecture of 1968 to Hp spaces with finite p > 1 and states that Taylor’s coefficients of nonvanishing holomorphic functions f ∈ Hp with norm ‖f‖p ≤ 1 are sharply estimated by |cn| ≤ (2/e)1 − 1/p, with appropriate extremal functions. Both conjectures have been investigated by many authors; however still remain open. The desired Krzyz’s estimate |cn| ≤ 2/e for f ∈ H∞ with ‖f‖∞ ≤ 1 was established only for the initial coefficients cn with n ≤ 5 The only known results for the Hummel–Scheinberg–Zalcman conjecture are that it is true for n = 1 and n = 2 as well as some results for special subclasses of Hp. We prove here that the Hummel–Scheinberg–Zalcman conjecture is true for all spaces H2m with m ∈ N. In the limit as m → ∞, this also provides the proof of Krzyz’s conjecture. Our approach involves deep results from Teichmüller space theory, especially the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces, and special quasiconformal deformations of H2m functions.