Spectral observations below Lyman-α are now obtained with the Cosmic Origin Spectrograph (COS) on the Hubble Space Telescope. It is therefore necessary to provide an accurate treatment of the blue wing of the Lyman-α line that enables correct calculations of radiative transport in DA and DBA white dwarf stars. On the theoretical front, we very recently developed very accurate H-He potential energies for the hydrogen 1s, 2s, and 2p states. Nevertheless, an uncertainty remained about the asymptotic correlation of the Σ states and the electronic dipole transition moments. A similar difficulty occurred in our first calculations for the resonance broadening of hydrogen perturbed by collisions with neutral H atoms. The aim of this paper is twofold. First, we clarify the question of the asymptotic correlation of the Σ states, and we show that relativistic contributions, even very tiny, may need to be accounted for a correct long-range and asymptotic description of the states because of the specific 2s 2p Coulomb degeneracy in hydrogen. This effect of relativistic corrections, inducing small splitting of the 2s and 2p states of H, is shown to be important for the Σ-Σ transition dipole moments in H-He and is also discussed in H-H. Second, we use existent (H-H) and newly determined (H-He) accurate potentials and properties to provide a theoretical investigation of the collisional effects on the blue wing of the Lyman-α line of H perturbed by He and H. We study the relative contributions in the blue wing of the H and He atoms according to their relative densities. We finally achieve a comparison with recent COS observations and propose an assignment for a feature centered at 1190 Å.