A unified approach is presented for establishing a broad class of Cramér–Rao inequalities for the location parameter, including, as special cases, the original inequality of Cramér and Rao, as well as an$L^{p}$version recently established by the authors. The new approach allows for generalized moments and Fisher information measures to be defined by convex functions that are not necessarily homogeneous. In particular, it is shown that associated with any log-concave random variable whose density satisfies certain boundary conditions is a Cramér–Rao inequality for which the given log-concave random variable is the extremal. Applications to specific instances are also provided.