query: Proving a holomorphic function is constant on an open connected set under certain conditions.
positive_passages.docid: 84103
positive_passages.text: Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem
negative_passages.docid: 64172
negative_passages.text: Show that if $f$ is analytic in a domain $D$ and $|f(z)|$ is constant in $D$, then the function $f(z)$ is constant in $D$
negative_passages.docid: 153191
negative_passages.text: A function satisfy $x\frac{\partial f(x,y)}{\partial x}+y\frac{\partial f(x,y)}{\partial y}=0$ in a convex domain implies it is a constant
negative_passages.docid: 47748
negative_passages.text: Analytic function on an open disc.
negative_passages.docid: 76147
negative_passages.text: Bounding holomorphic function at 0 given other bounds
negative_passages.docid: 14117
negative_passages.text: Holomorphic function has at most countably zeros
negative_passages.docid: 62640
negative_passages.text: Show a given analytic function is constant
negative_passages.docid: 109943
negative_passages.text: Estimates of the value of a holomorphic function and its derivative
negative_passages.docid: 105172
negative_passages.text: Prove that the entire function is constant over certain conditions.
negative_passages.docid: 89050
negative_passages.text: If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
negative_passages.docid: 40208
negative_passages.text: Locally Constant Functions on Connected Spaces are Constant
negative_passages.docid: 103125
negative_passages.text: Show that a non-constant holomorphic function has zeroes in a compact set
negative_passages.docid: 153411
negative_passages.text: If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.
negative_passages.docid: 32789
negative_passages.text: Show bounded harmonic function on $\mathbb{C}$ is constant.
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: 115863
negative_passages.text: Every harmonic function is the real part of a holomorphic function
negative_passages.docid: 115142
negative_passages.text: Holomorphic functions on a complex compact manifold are only constants