query: Entire Function is Constant with $Re(f(z))^2 = Im(f(z)) \forall z \in C$
positive_passages.docid: 27772
positive_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: 82745
negative_passages.text: Showing Entire Function is Bounded
negative_passages.docid: 105172
negative_passages.text: Prove that the entire function is constant over certain conditions.
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: 107518
negative_passages.text: Entire function $f$ such that $|f(z)| ≤ K |z|^3$ for $|z|\ge1$ and $f(z) = f(iz)$ for every $z∈C$
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: 27251
negative_passages.text: Holomorphic function with $f\circ f=f$
negative_passages.docid: 98399
negative_passages.text: Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.
negative_passages.docid: 158647
negative_passages.text: If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
negative_passages.docid: 148981
negative_passages.text: Entire $f$ and $g$ constant if $e^{f(z)}+e^{g(z)}=1$
negative_passages.docid: 135577
negative_passages.text: Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
negative_passages.docid: 133837
negative_passages.text: Entire function dominated by another entire function is a constant multiple
negative_passages.docid: 86517
negative_passages.text: If $|f(z)|\le |g(z)|$, can we say $f$ is constant?
negative_passages.docid: 2527
negative_passages.text: Entire function bounded by a polynomial is a polynomial
negative_passages.docid: 1043
negative_passages.text: Image of $L=\{z\in\Bbb C \mid a\operatorname{Re}(z)+b\operatorname{Im}(z)+c=0; a^2+b^2\neq0; a,b,c\in \Bbb R \}$ under the function $e^z$
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: 65054
negative_passages.text: How is $e^z$ the only function that is entire with the property $f'(z) = f(z)$ and $f(0) = 1$