Real parts of two analytic functions are equal?

[bmanbs2]

Homework Statement

Suppose $$f$$ and $$g$$ are analytic on a bounded domain $$D$$ and continuous on the domain's boundary $$B$$.
Also, $$Re\left(f\right) = Re\left(g\right)$$ on $$B$$.
Show that $$f = g + ia$$, where $$a$$ is a real number.

Homework Equations

The maximum modulus principle states that $$Re\left(f\right)$$ and $$Re\left(g\right)$$ have no local minima or maxima on $$D$$, and that the absolute values of $$Re\left(f\right)$$, $$Re\left(g\right)$$, $$f$$, and $$g$$ have maximums on $$B$$.

The Attempt at a Solution

I'm not sure how to show $$Re\left(f\right) = Re\left(g\right)$$ across the entire domain.

 

[bmanbs2]

Actually just found the answer, so I'll post it here.

Consider the function $$h = g - f$$. As $$f$$ and $$g$$ are analytic, $$h$$ is also analytic. But as $$Re\left(f\right) = \left(g\right)$$ on $$B$$, $$Re\left(h\right) = 0$$ on $$B$$. The Maximum Modulus Principle states that $$|Re\left(h\right)|$$ reaches its maximum on $$B$$, so $$Re\left(h\right) = 0$$ on D as well.