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.