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

In this article we present a new proof of a sharp Reverse Hölder Inequality for 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∞ constants: M Lp(w) c 1 p −1 [w]Ap [σ]A∞ 1/p , where 1