The Dallenbach layer is composed of an absorbing magnetic-dielectric layer attached to a perfect electric conductor (PEC) sheet. Under linearity and time invariance (LTI) assumptions, Rozanov has established analytically a sum-rule tradeoff between the absorption efficacy over a predefined bandwidth and the thickness of the layer, which is the so called Rozanov bound. In recent years, several proposals have been introduced to bypass this bound by using non-LTI absorbers. However, in practice, their implementation may be challenging. Here, we expose additional hidden assumptions in Rozanov's derivation, and thus we introduce several sum rules for LTI layer absorbers that are not covered by the original Rozanov criterion, and they give rise to more relaxed constraints on the absorption limit. We demonstrate practical LTI designs of absorbing thin layers that provide absorption beyond Rozanov's bound. These designs are based on the replacement of the original PEC boundary by various types of penetrable impedance sheets.