A new look at Krzyz's conjecture

Recently the author has presented a new approach to solving extremal problems of geometric function theory. It involves the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. We show here that this approach, combined with quasiconformal theory, can be also applied to nonvanishing holomorphic functions from $H^{\mathrm{\infty }}$. In particular this gives a proof of an old open Krzyz conjecture for such functions and of its generalizations. The unit ball $H_1^{\mathrm{\infty }}$ of $H^{\mathrm{\infty }}$ is naturally embedded into the universal Teichmuller space, and the functions $f\in H_1^{\mathrm{\infty }}$ are regarded as the Schwarzian derivatives of univalent functions in the unit disk.