  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 $H^\infty$. In
particular this gives a proof of an old open Krzyz conjecture for such
functions and of its generalizations.
  The unit ball $H_1^\infty$ of $H^\infty$ is naturally embedded into the
universal Teichmuller space, and the functions $f \in H_1^\infty$ are regarded
as the Schwarzian derivatives of univalent functions in the unit disk.
