Let $X$ and $Y$ be two random variables with first order moments, i.e. $E[|X|]$, $E[|Y|]<+\infty$. Assume further that

$$E\left[|X-Y|\right]<\varepsilon.$$

Set $Law(X)=\mu$ and $Law(Y)=\nu$, it is clear that $\mu$ and $\nu$ are close in the Prokhorov metric, see

https://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric

for definition. Denote by $\rho(\cdot,\cdot)$ the Prokhorov metric. My question is how to estimate $\rho(\mu,\nu)$. For example, could we show that $\rho(\mu,\nu)<\varepsilon$ or $\rho(\mu,\nu)<\sqrt{\varepsilon}$? Thanks for the reply!