If mu and v are constants then solve the equation mudeltaki

u,vu,v are harmonic conjugate with each other in some domain , then we need to show

u,vu,v must be constant.

as vv is harmonic conjugate of uu so f=u+ivf=u+iv is analytic.

as uu is harmonic conjugate of vv so g=v+iug=v+iu is analytic.

fig=2ufig=2u and f+ig=2ivf+ig=2iv are analytic, but from here how to conclude that u,vu,v are constant? well I know they are real valued function, so by open mapping theorem they are constant?

1. vv is a conjugate of uu if and only if uu is a conjugate of vv (since u+ivu+iv and viuviu are constant multiples of each other)
2. Since the harmonic conjugate is unique up to additive constant, the assumption that uu is a conjugate of vv implies (because of 1) that u=u+constu=u+const, and conclusion follows.
3. Related to 1: the Hilbert transform HH satisfies HH=idHH=id.