  We establish two new estimates which control a function in $L^1$ by only the
$L^1$ norm of its radial derivative. While the interior estimate holds for all
superharmonic functions, the boundary version is much more delicate. It
requires the function to be a stable solution of a semilinear elliptic equation
with a nonnegative, nondecreasing, and convex nonlinearity. As an application,
our estimates provide quantitative proofs of two results established in
[Cabr\'e, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] by
contradiction-compactness proofs. We recall that this work proved the H\"older
regularity of stable solutions to semilinear elliptic equations in the optimal
range of dimensions $n\leq 9$.
