Harnack inequality for harmonic functions
= Harnack inequality for harmonic functions
{c}
If $u\geq0$ is a <harmonic function> on a neighbourhood of $\overline{B_{4r}(y)}$, then
$$
\sup_{B_r(y)}u\leq3^n\inf_{B_r(y)}u.
$$
For $x,z\in B_r(y)$, use $B_r(x)\subset B_{3r}(z)\subset B_{4r}(y)$ and the <mean value property for harmonic functions>. Nonnegativity lets us enlarge the integral, and the ratio of volumes is $3^n$. This compares values without requiring derivative bounds.