On an arithmetic inequality on ${\mathbb{P}}_{\mathbb{Q}}^1$

We establish an inequality comparing the height and the $\chi$-arithmetic volume of toric metrized divisors on ${\mathbb{P}}_{\mathbb{Q}}^1$. This gives a partial answer to a question of Burgos, Moriwaki, Philippon and Sombra ([5, remark 5.13]).