# Proving a function $f(m,n)$ which satisfies two conditions is a constant

I found the following question in a book only with one sentence. "This question can be solved by an elementary way. Note that the following two are false: (1) If a function is bounded from below, then it has minimum value. (2) A monotone decreasing sequence reaches a negative value."

Question: Let $m,n$ be integers. Supposing that a function $f(m,n)$ defined by $m,n$ satisfies the following two conditions, then prove that $f(m,n)$ is a constant.

1. $f(m,n)\ge0$.

2. $4f(m,n)=f(m-1,n)+f(m+1,n)+f(m,n-1)+f(m,n+1)$.

I suspect this question can be solved by a geometric aspect. I've tried to prove this, but I'm facing difficulty. Could you show me how to prove this?

• It is a discrete version of Laplace equation $\nabla^{2}\Phi = 0$ which leads to $\int\Psi\nabla^{2}\Psi\,{\rm d}V = 0$ and to $\int\left\vert\nabla\Psi\right\vert^{2} = 0$ which involves an integration by parts. The discrete version requires "Abel Summations by part" identity ( en.wikipedia.org/wiki/Summation_by_parts ). – Felix Marin Sep 6 '13 at 7:29
• Where did you get this problem? It feels familiar to me for some reason. – davidlowryduda Sep 6 '13 at 7:47
• @mixedmath: I found this in the book which a friend of mine has, but I don't know its name. I'll ask him. – mathlove Sep 6 '13 at 7:53
• @FelixMarin: Thank you for nice information. – mathlove Sep 6 '13 at 7:57
• @mathlove You're welcome. – Felix Marin Sep 6 '13 at 23:44