Given two random variables $X,Y$ which are both $\mathbb{N}$-valued and have the same expected value, and denote their probability mass functions by ${\bf p} = (p_0,p_1,\ldots)$ and ${\bf q} = (q_0,q_1,\ldots)$, respectively. We also assume that $q_n > 0$ for all $n \in \mathbb N$. I am pretty sure one can upper bound the Wasserstein distance (of order 1) $W_1({\bf p},{\bf q})$ by the so-called $\chi^2$ distance defined via $$\chi^2({\bf p},{\bf q}) = \sum\limits_{n\geq 0} \frac{|p_n-q_n|^2}{q_n}.$$ However, I fail to found any specific reference containing a statement like $$W_1({\bf p},{\bf q}) \leq f\left(\chi^2({\bf p},{\bf q})\right)$$ for some appropriate non-negative function $f$ which vanishes at the origin. I will appreciate any help in locating a suitable reference for the advertised bound...

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Remark: I forgot to mention that ${\bf p}$ and ${\bf q}$ have the same mean value in my original post.