query: Holomorphic functions are constant given that the sum of their moduli is constant.
positive_passages.docid: 10194
positive_passages.text: If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
negative_passages.docid: 96050
negative_passages.text: $f$ an analytic function on $\mathbb{C}$ which takes values in $\mathbb{C}\backslash(-\infty,0]$ implies constant
negative_passages.docid: 74891
negative_passages.text: sum of holomorphic functions
negative_passages.docid: 120533
negative_passages.text: A harmonic function with sublinear growth at infinity is constant
negative_passages.docid: 92056
negative_passages.text: An inequality on holomorphic functions
negative_passages.docid: 63446
negative_passages.text: The existence of analytical branch of the logarithm of a holomorphic function
negative_passages.docid: 19399
negative_passages.text: $f^2$ and $f^3$ are holomorphic implies $f$ is holomorphic.
negative_passages.docid: 19750
negative_passages.text: If a function $f$ is holomorphic on the closed unit disk centered at the origin and is real valued whenever $|z| = 1$, then $f$ is constant.
negative_passages.docid: 39386
negative_passages.text: Harmonic non-surjective functions are constant
negative_passages.docid: 91352
negative_passages.text: If $\Re(f)$ is bounded then f is constant.
negative_passages.docid: 137971
negative_passages.text: Existence of an holomorphic function
negative_passages.docid: 74622
negative_passages.text: Composition of holomorphic functions is holomorphic
negative_passages.docid: 23842
negative_passages.text: $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant?
negative_passages.docid: 3818
negative_passages.text: Proving two entire functions are constant.
negative_passages.docid: 27772
negative_passages.text: Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.
negative_passages.docid: 71799
negative_passages.text: A complex polynomial with partial derivatives equal to zero is constant.
negative_passages.docid: 146654
negative_passages.text: Real valued analytic function defined on a connected set is constant