Sum of subharmonic functions attaining a maximum in an open, connected set.

Question

I want to prove the following statement: If $p_1,..,p_n$ are subharmonic functions on an open, connected subset $V$ of the complex plane and if $p_1+…+p_n$ attains a maximum on $V$, then each $p_i$ is a harmonic function.

My attempt: The sum $p_1+…+p_n$ is subharmonic. And so by the maximum principle for subharmonic functions, it is a constant. Therefore, $\Delta(p_1+…+p_n)=\Delta(p_1)+…+\Delta(p_n)=0$, where $\Delta$ is the Laplacian operator. I don't see how this forces $\Delta(p_i)=0$, for each $i$.

Also at my disposal are the mean value properties of harmonic and subharmonic functions, but I can't see a way to employ them. Any help is appreciated. Thanks

Answer 1

You already demonstrated that $$ p_1 + p_2 + \cdots + p_n = C $$ is constant. Then both $p_1$ and $$ -p_1 = p_2 + \cdots + p_n - C $$ are subharmonic, which means that $p_1$ is harmonic.

The same arguments works for $p_2, \ldots, p_n$.