  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 $\dot{B}^{d/p}_\infty L_{p,\infty}(\mathbb{R}^d)
\hookrightarrow \text{BMO}(\mathbb{R}^d)$ for $1 < p < \infty$. Moreover, these
results are also applied to establish new Fefferman--Stein inequalities,
Calder\'on--Scott type inequalities, and extrapolation estimates. Our approach
is based on the limiting interpolation techniques.
