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