Assume $f, g: \mathbb{R}^d \to \mathbb{R}$ are harmonic functions.
- Assume that there exist $C < \infty$ and $\alpha < 1$ such that for all $x$,$$|f(x)| \le C|x|^\alpha.$$What is the easiest way to see that $f$ is constant?
- Assume that $g(x) > 0$ for all $x \in \mathbb{R}^d$. What is the easiest way to see that $g$ is constant?
The Poisson integral formula gives an analogue of "Cauchy's Estimates" in complex analysis: If $f$ is harmonic and $|f(x)|\le M$ for $|x-x_0|=r$ then $$|\nabla f(x_0)|\le c\frac Mr.$$You can use this to show that $\nabla f=0$ in the first question.
The second question is the same, except with a slightly different version of that estimate: The same result holds assuming just that the average of $|f(x)|$ on the sphere $|x-x_0|=r$ is no larger than $M$.
Now say $g>0$ is harmonic. The average of $|g(x)|$ for $|x|=r$ is the same as the average of $g(x)$, which is $g(0)$. This shows that $\nabla g(0)=0$. Since any translate of $g$ is also a positive harmonic function, in fact $\nabla g=0$ everywhere.