  We prove the sharp mixed $A_{p}-A_{\infty}$ weighted estimate for the
Hardy-Littlewood maximal function in the context of weighted Lorentz spaces,
namely \[ \|M\|_{L^{p,q}(w)} \lesssim_{p,q,n}
[w]^{\frac1p}_{A_p}[\sigma]^{\frac1{\min(p,q)}}_{A_{\infty}}, \] where
$\sigma=w^{\frac{1}{1-p}}$. Our method is rearrangement free and can also be
used to bound similar operators, even in the two-weight setting. We use this to
also obtain new quantitative bounds for the strong maximal operator and for $M$
in a dual setting.
