Homework Help: Set of Holomorphic functions s.t. |f1|^2 + + |fn|^2 = K

1. Sep 18, 2011

deluks917

1. The problem statement, all variables and given/known data

Given holomorphic functions f1, ...,fn defined on some open connected set s.t. |f1|2 + ... + |fn|2 = K. K is some constant. Prove all the fj's are constant.

2. Relevant equations

Cauchy Riemann:
fk = u +iv

ux = vy
uy = -vx

3. The attempt at a solution

For n=1

|f|^2 = k.
we have taking derivatives

u*ux + v*vx = 0
u*uy + v*vy = 0

and hence by CR:
u*ux + v*vx = 0
v*ux + -u*vx = 0

if u^2 + v^2 = 0 we are done

if not the determinant = -(u^2 + v^2) =/= 0. so the only solution is ux = vx = 0.

(f\bar = complex conjugate of f)

also f*f\bar = K (again assume K =/=0)

f\bar = K/f but since f =/= 0 this implies f\bar is holomorphic which is not possible unless f is constant.

I can neither get the linear algebra nor the solving for f\bar to work with more than one function. If anyone could even give me a hint for n=2 I'd appreciate it. This is not for a course (though I am taking Complex) I got the problem from Narasimhan complex analysis in one variable prob 48.6. Thanks in advance.

Last edited: Sep 18, 2011