query: Show that if $f$ is analytic on a domain $D$, and if $|f|$ is constant, then $f$ is constant.
positive_passages.docid: 153411
positive_passages.text: If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.
negative_passages.docid: 10194
negative_passages.text: If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
negative_passages.docid: 132663
negative_passages.text: Show $f$ is analytic if $f^8$ is analytic
negative_passages.docid: 72856
negative_passages.text: Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
negative_passages.docid: 133386
negative_passages.text: If $f$ is entire and $\lim_\limits{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant
negative_passages.docid: 154742
negative_passages.text: If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
negative_passages.docid: 116092
negative_passages.text: Non constant analytic function from $\{z\in\mathbb{C}:z\neq 0\}$ to $\{z\in\mathbb{C}:|z|>1\}.$
negative_passages.docid: 51934
negative_passages.text: Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
negative_passages.docid: 80650
negative_passages.text: Complex differentiability of $f(z)=|z|$
negative_passages.docid: 32789
negative_passages.text: Show bounded harmonic function on $\mathbb{C}$ is constant.
negative_passages.docid: 158647
negative_passages.text: If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
negative_passages.docid: 65343
negative_passages.text: If $f:D\to D’$ is analytic and $u: D'\to R$ is harmonic then the composition of $u$ and $f$ is harmonic in $D$
negative_passages.docid: 64650
negative_passages.text: If $a\operatorname{Re}(f(z))+b\operatorname{Im}(f(z))=c$ then f is constant
negative_passages.docid: 106290
negative_passages.text: Prove a function is constant. (complex analysis)
negative_passages.docid: 71094
negative_passages.text: Proof that 1-1 analytic functions have nonzero derivative
negative_passages.docid: 84510
negative_passages.text: If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
negative_passages.docid: 47748
negative_passages.text: Analytic function on an open disc.