Sharp reverse H\"older inequality for $C_p$ weights and applications

We prove an appropriate sharp quantitative reverse H\"older inequality for the $C_p$ class of weights from which we obtain as a limiting case the sharp reverse H\"older inequality for the $A_\infty$ class of weights. We use this result to provide a quantitative weighted norm inequality between Calder\'on-Zygmund operators and the Hardy-Littlewood maximal function, precisely $$\|Tf\|_{L^p(w)} \lesssim_{T,n,p,q} [w]_{C_q}(1+\log^+[w]_{C_q})\|Mf\|_{L^p(w)},$$ for $w\in C_q$ and $q>p>1$, quantifying Sawyer's theorem.