In this note we study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. We introduce and study the class of relatively operator Lipschitz functions. We obtain a trace formula in the case of relatively trace class perturbations and show that this class of functions is the maximal class of functions for which the trace formula holds. Our method also gives us a new approach to the inequality R |ξ(t )|(1 + |t |)−1 dt < ∞ for the spectral shift function ξ in the case of relatively trace class perturbations. © 2025 Academie des sciences. All rights reserved.