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Let $f$ and $g$ be two holomorphic functions on a complex region. If $\left | f\right |^2 + \left | g \right |^2$ is constant, then show that both of them are constants. I have tried to use Cauchy Riemann eq. and Laplace eq., however I did not manage to solve the problem.

Zhang Yifeng
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    A more general case is considered here: http://math.stackexchange.com/questions/289114/show-that-holomorphic-f-1-f-n-are-constant-if-sum-k-1n-left-f. – Martin R Mar 10 '16 at 12:44
  • A answer is given here.:http://math.stackexchange.com/questions/1410696/prove-that-if-h-f-12-cdots-f-n2-is-constant-then-f-i-is-constant?lq=1 – Zhang Yifeng Mar 10 '16 at 14:06

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