query: Suppose that $f$ is an entire function with $Re[f(z)]\neq 0$ $\forall z\in \mathbb{C}$Then $f$ is constant.
positive_passages.docid: 141863
positive_passages.text: If the imaginary part of an entire function is never zero, the function is constant
negative_passages.docid: 91352
negative_passages.text: If $\Re(f)$ is bounded then f is constant.
negative_passages.docid: 58178
negative_passages.text: Property of Entire Functions
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: 48275
negative_passages.text: $f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
negative_passages.docid: 64905
negative_passages.text: Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$.
negative_passages.docid: 86517
negative_passages.text: If $|f(z)|\le |g(z)|$, can we say $f$ is constant?
negative_passages.docid: 158519
negative_passages.text: Hints for a complex limit: Prove if $\lim_{z \to \infty} f(z)/z = 0$ then $f(z)$ is constant.
negative_passages.docid: 63255
negative_passages.text: Proving an entire, complex function with below bounded modulus is constant.
negative_passages.docid: 2527
negative_passages.text: Entire function bounded by a polynomial is a polynomial
negative_passages.docid: 65548
negative_passages.text: Prove if f(z) is entire function s.t $|f(z)|\leq M|z|$ then $f(z)$ Is polynomial of degree $1$
negative_passages.docid: 7878
negative_passages.text: Proving that a doubly-periodic entire function $f$ is constant.
negative_passages.docid: 25520
negative_passages.text: Why: A holomorphic function with constant magnitude must be constant.
negative_passages.docid: 114245
negative_passages.text: What might I use to show that an entire function with positive real parts is constant?
negative_passages.docid: 23585
negative_passages.text: Show that a holomorphic function satisfying $|f(z)|\leq c|z|^n$ must be $f(z)=a\cdot z^n$ for some a.
negative_passages.docid: 140602
negative_passages.text: If the absolute value of an analytic function $f$ is a constant, must $f$ be a constant?
negative_passages.docid: 97371
negative_passages.text: Proving a function is constant, under certain conditions?