Addendum to: Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics

In this addendum we extend Theorem 4.6 on the negative binomial distribution in `Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics’ (Journal of Mathematical Biology 92:40, 2026; arXiv:2503.21403). We prove that the fractional linear lower bound to the negative binomial generating function derived there is indeed valid for every $x\in [0,1]$, and not only for $x\in [0,P_{\mathrm{N}\mathrm{B}}^{\mathrm{\infty }}]$, where $P_{\mathrm{N}\mathrm{B}}^{\mathrm{\infty }}$ is the extinction probability of the associated Galton-Watson process.