In this paper, we consider a birth–death process with generator $\mathcal{L}$ and reversible invariant probability π. Given an increasing function ρ and the associated Lipschitz norm ‖⋅‖Lip(ρ), we find an explicit formula for $\|(-\mathcal{L})^{-1}\|_{\operatorname {Lip}(\rho)}$ . As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of $\mathcal{L}$ in L2(π). Moreover, by Lyons–Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth–death processes.