Minimal harmonic function

Remarks

Some authors also normalise $h(x_0) = 1$ at a fixed reference point $x_0$. Example: on the unit disc $\mathbb{D}$, let $h(z) = \frac{1 - \abs{z}^2}{\abs{1 - z}^2}$, the Poisson kernel with pole $1$. If $0 \le v \le h$ is harmonic, the Herglotz representation theorem writes $v$ and $h - v$ as Poisson integrals of positive measures $\mu$ and $\nu$ with $\mu + \nu = \delta_1$, the measure of $h$. Then $\mu$ is a multiple of $\delta_1$, so $v$ is a multiple of $h$, and $h$ is minimal. Non-example: $h = 1$ on the unit disc is not minimal. The function $v(z) = \frac{1 + \operatorname{Re} z}{2}$ is harmonic with $0 \le v \le 1$ on $\mathbb{D}$, but it is not constant. On $\R^n$ every positive harmonic function is constant, so $h = 1$ is minimal there. The minimal functions with $h(x_0) = 1$ are the extreme points of the convex set of positive harmonic functions with value $1$ at $x_0$.