  We show that any compact surface of genus zero in Euclidean 3-space that
satisfies a quasiconformal inequality between its principal curvatures is a
round sphere. This solves an old open problem by H. Hopf, and gives a spherical
version of Simon's quasiconformal Bernstein theorem. The result generalizes,
among others, Hopf's theorem for constant mean curvature spheres, the
classification of round spheres as the only compact elliptic Weingarten
surfaces of genus zero, and the uniqueness theorem for ovaloids by Han,
Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of
solutions to linear elliptic equations with discontinuous coefficients.
