  We study the ground-state properties of weakly frustrated Heisenberg
ferrimagnetic chains with nearest and next-nearest neighbor antiferromagnetic
exchange interactions and two types of alternating sublattice spins S_1 > S_2,
using 1/S spin-wave expansions, density-matrix renormalization group, and
exact- diagonalization techniques. It is argued that the zero-point spin
fluctuations completely destroy the classical commensurate- incommensurate
continuous transition. Instead, the long-range ferrimagnetic state disappears
through a discontinuous transition to a singlet state at a larger value of the
frustration parameter. In the ferrimagnetic phase we find a disorder point
marking the onset of incommensurate real-space short-range spin-spin
correlations.
