Under the assumption of lattice preferred orientation of iron hcp crystals, the seismic anisotropy of the inner core is related to single crystal elastic properties by a fabric model. This model is able to explain a new seismological constraint: the constant behaviour of the ratio k of two anisotropy coefficients, despite radial and lateral variations of the anisotropy level in the inner core. The relevance of the theoretical calculations of iron single crystal elastic properties at inner core conditions are tested by a comparison between seismic estimates and computed values of this ratio k. Taking into account the error bar of the seismic estimate, iron hcp structures presenting fast and slow c axis could fit the data equally well.