  Motivated by the magnetic properties of the spin-chain compounds
LiCuSbO$_4$$\equiv$LiSbCuO$_4$ and Rb$_2$Cu$_2$Mo$_3$O$_{12}$, we study the
ground state of the Heisenberg chain with dimerized nearest-neighbor
ferromagnetic (FM)($J_1, J_1^\prime<0$) and next-nearest-neighbor
antiferromagnetic ($J_2>0$) couplings. Using the density-matrix renormalization
group technique and spin-wave theory we find a first-order transition between a
fully-polarized FM and an incommensurate spiral state at
$2\alpha=\beta/(1+\beta)$, where $\alpha$ is the frustration ratio $J_2/|J_1|$
and $\beta$ the degree of dimerization $J_1^\prime/J_1$.In the singlet spiral
state the spin-gap is vanishingly small in the vicinity of the FM transition,
corresponding to a situation of LiCuSbO$_4$. For larger $\alpha$, corresponding
to Rb$_2$Cu$_2$Mo$_3$O$_{12}$, and smaller $\beta$ there is a crossover from
this frustration induced incommensurate state to an
Affleck-Lieb-Kennedy-Tasaki-type valence bond solid state with substantial
spin-gaps.
