query: Let $f$ be nonconstant analytic in a domain $D$ and continuous on its closure. If $|f|$ is constant on $\partial D$, then $f$ has a zero in $D$.
positive_passages.docid: 142079
positive_passages.text: If $f$ is a non-constant analytic function on $B$ such that $|f|$ is a constant on $\partial B$, then $f$ must have a zero in $B$
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: 75390
negative_passages.text: How does one prove that if function's partial derivative respect to every variable is zero, function is constant?
negative_passages.docid: 20955
negative_passages.text: Let $f(z)$ be analytic on $\mathbb{D}$ = {${z\in\mathbb{C}:|z-1|<1}$} such that $f(1) = 1$, if $f(z) = f(z^2)$ for all $z\in\mathbb{D}$,
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: 110219
negative_passages.text: Bound on the derivative of a holomorphic function at $0$
negative_passages.docid: 103125
negative_passages.text: Show that a non-constant holomorphic function has zeroes in a compact set
negative_passages.docid: 68071
negative_passages.text: If $f$ is a nonconstant entire function such that $|f(z)|\geq M|z|^n$ for $|z|\geq R$, then $f$ is a polynomial of degree atleast $n$.
negative_passages.docid: 84313
negative_passages.text: $|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
negative_passages.docid: 43440
negative_passages.text: Complex Analysis: Showing analytic function is zero
negative_passages.docid: 147516
negative_passages.text: Continuous function whose right hand derivative equals 0 is constant?
negative_passages.docid: 75638
negative_passages.text: Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.
negative_passages.docid: 84510
negative_passages.text: If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
negative_passages.docid: 141115
negative_passages.text: $f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$
negative_passages.docid: 138809
negative_passages.text: Consequences of a fixed point of a bounded, complex analytic function
negative_passages.docid: 131000
negative_passages.text: A proof of a criterion of vanishing of a function.
negative_passages.docid: 114826
negative_passages.text: Zero Partials imply Constant Function Theorem or Proof