On the maximum $f$-divergence of probability distributions given the value of their coupling

The paper is a supplement to the author's paper [1]. Here we present explicit upper bounds (which are optimal in some cases) on the maximum value of the $f$-divergence $D_f(P \| Q)$ of discrete probability distributions $P$ and $Q$ provided that the distribution $Q$ (or its minimal component $q_{\min}$) and the value of the coupling of $P$ and $Q$ are fixed. We also obtain an explicit expression for the maximum value of the divergence $D_f(P \| Q)$ provided that only the value of the coupling of $P$ and $Q$ is given. Results of [1] concerning the Kullback–Leibler divergence and $\chi^2$-divergence are particular cases of the results proved in the present paper.