Real parts of two analytic functions are equal?

[bmanbs2]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

 

[bmanbs2]

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

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