  We use the coupled cluster method for infinite chains complemented by exact
diagonalization of finite periodic chains to discuss the influence of a
third-neighbor exchange J3 on the ground state of the spin-1/2 Heisenberg chain
with ferromagnetic nearest-neighbor interaction J1 and frustrating
antiferromagnetic next-nearest-neighbor interaction J2. A third-neighbor
exchange J3 might be relevant to describe the magnetic properties of the
quasi-one-dimensional edge-shared cuprates, such as LiVCuO4 or LiCu2O2. In
particular, we calculate the critical point J2^c as a function of J3, where the
ferromagnetic ground state gives way for a ground state with incommensurate
spiral correlations. For antiferromagnetic J3 the ferro-spiral transition is
always continuous and the critical values J2^c of the classical and the quantum
model coincide. On the other hand, for ferromagnetic J3 \lesssim
-(0.01...0.02)|J1| the critical value J2^c of the quantum model is smaller than
that of the classical model. Moreover, the transition becomes discontinuous,
i.e. the model exhibits a quantum tricritical point. We also calculate the
height of the jump of the spiral pitch angle at the discontinuous ferro-spiral
transition.
