Higher order quantum effects on the magnetic phase diagram induced by four-spin ring exchange on plaquettes are investigated for a two-dimensional quantum antiferromagnet with S = 1/2. Spatial anisotropy and frustration are allowed for. Using a perturbative spin-wave expansion up to second order in 1/S we obtain the spin-wave energy dispersion, sublattice magnetization, and the magnetic phase diagram. We find that for substantial four-spin ring exchange the quantum fluctuations are stronger than in the standard Heisenberg model. A moderate amount of four-spin ring exchange couplings stabilizes the ordered antiferromagnetic Néel state while a large amount renders it unstable. Comparison with inelastic neutron scattering data points toward a moderate ring exchange coupling of 27% to 29% of the nearest-neighbor exchange coupling.