We show that for any rotationally symmetric apodizer on the full aperture, there is a family of apodizers on the annular aperture with the same functional Strehl ratio vs defocus (W(20)) and vs spherical aberration (W(40)). However, in the latter case, the coefficients W(20) and W(40) are reduced by the factors (1 - epsilon(2)) and (1 - epsilon(2))(2) respectively, where is the central obscuration ratio. We indicate that the best focal plane is shifted from W(20) = - W(40) to W(20) = -(1 + epsilon(2)) W(40). These general results allow us to design and to compare novel apodizers on annular apertures which reduce the influence of W(20) and W(40). The Strehl ratios of a novel family of apodizers are discussed to illustrate our general results.