  At the end of 1960's, Lawrence Zalcman posed a conjecture that the
coefficients of univalent functions $f(z) = z + \sum\limits_2^\infty a_n z^n$
on the unit disk satisfy the sharp inequality $|a_n^2 - a_{2n-1}| \le (n-1)^2$,
with equality only for the Koebe function. This remarkable conjecture implies
the Bieberbach conjecture, investigated by many mathematicians, and still
remains a very difficult open problem for all n > 3; it was proved only in
certain special cases.
  We provide a proof of Zalcman's conjecture based on results concerning the
plurisubharmonic functionals and metrics on the universal Teichm\"uller space.
As a corollary, this implies a new proof of the Bieberbach conjecture. Our
method gives also other new sharp estimates for large coefficients.
