Sharp Reverse Hölder property for A∞A∞ weights on spaces of homogeneous type

In this article we present a new proof of a sharp Reverse Hölder Inequality for A∞A∞ weights. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy–Littlewood maximal function involving A∞A∞ constants:‖M‖Lp(w)⩽c(1p−1[w]Ap[σ]A∞)1/p, where 1