On some optimization problems for the Rényi divergence

We consider the problem of determining the maximum and minimum of the Rényi divergence $D_{\lambda}(P\parallel Q)$ and $D_{\lambda}(Q\parallel P)$ for two probability distribution $P$ and $Q$ of discrete random variables $X$ and $Y$ provided that the probability distribution $P$ and the parameter $\alpha$ of $\alpha$-coupling between $X$ and $Y$ are fixed, i.e., provided that $\mathrm{Pr}\{X=Y\}=\alpha$.