A second alternative approach for the study of the Muckenhoupt class $A_1(\mathbb{R})$
We find the exact best possible range of those $p > 1$ for which any function which belongs to $A_1(\mathbb{R})$, with $A_1$-constant equal to $c$, must also belong to $L^p$. In this way we provide alternative proofs of the results in [2] and [10].