Let f(x,y)=u+iv be a complex harmonic function such that (f(x,y))2 is also harmonic (both on a domain D). Show that f(x,y) must be a constant I've attempted to show this through brute force, but perhaps there is something more elegant? I know that the Laplace equations must hold, so what I have done (I'll spare the readers the details) is to calculate out f(x,y)2, separate the real and imaginary parts in terms of u and v, and calculate the second order partial derivatives for the Laplace equations. After some calculation and cancellation (in particular of the second partials using the fact that f itself is harmonic), I get: ux2+uy2-vx2-vy2=0 and uxvy+uxvy=0 I need to somehow get from these two equations to the fact that f is constant. Looking at this, it seems like I'm just a step away, but I am missing something. Any suggestions on what I might be overlooking?