Upper bound for the critical density of biased Activated Random Walk

We provide a new upper bound for the critical density of Activated Random Walk in case of biased distribution of jumps. With finite sleeping rate the bound is strictly less than one in one dimension for all initial particles distri-bution and in higher dimension for some initial particles distributions. This answers a question from Dickman, Rolla and Sidoravicius (2010) in case of bias