The linear stability of three-layer plane Poiseuille flow is studied in the longwave limit and for moderate wavelengths. The fluids are assumed to follow Oldroyd-B constitutive equations with constant viscosities and elasticities. We find that the jumps of the Poiseuille shear rate at both interfaces which give the convexity of the Poiseuille velocity profile, allow us to determine the longwave stability for Newtonian fluids. On the other hand, the stability of viscoelastic fluids is analyzed by using the additive character of the longwave eigenvalues with respect to viscous and elastic terms. The stability with respect to moderate wavelength disturbances has to deal with two different modes called ＇shortwave＇ (SW) and ＇longwave＇ (LW), according to their values at zero wavenumber. The SW eigenvalues can become the most dangerous modes for large Weissenberg numbers and their influences can be studied by means of shortwave analysis. Moreover, we point out that the longwave stability analysis and convexity of the Poiseuille velocity profile allow us to determine the LW eigenvalues which are stable with respect to order one wavelength disturbances.