In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore’s inequality for the moduli of smoothness and a logarithmic variant of Bennett–DeVore–Sharpley’s inequality for rearrangements. As a consequence, we improve the classical Stein–Zygmund embedding deriving B ˙ ∞ d / p L p , ∞ ( R d ) ↪ BMO ( R d ) \dot {B}^{d/p}_\infty L_{p,\infty }(\mathbb {R}^d) \hookrightarrow \text {BMO}(\mathbb {R}^d) for 1 > p > ∞ 1 > p > \infty . Moreover, these results are also applied to establish new Fefferman–Stein inequalities, Calderón–Scott type inequalities, and extrapolation estimates. Our approach is based on the limiting interpolation techniques.