Upper Bound for the Difference Between Expectations

Let us consider two random variables, $X_0 \sim \mathbb{P}_0$ and $X_1 \sim \mathbb{P}_1$. Is it possible to establish an upper bound to the difference between the expectations of $X_0$ and $X_1$ based on the mean difference between $X_0$ and $X_1$, i.e.

$$\bigg|\mathbb{E}_{\mathbb{P}_0}[X_0] - \mathbb{E}_{\mathbb{P}_1}[X_1] \bigg| < \mathbb{E}[d(x_0, x_1)]?$$If not, is there any upper bound to $\bigg|\mathbb{E}_{\mathbb{P}_0}[X_0] - \mathbb{E}_{\mathbb{P}_1}[X_1] \bigg|$ at all?