Prove that $f$ is constant if $f$ is real when $|z|=1$
84103
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
154034
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
31522
Proving $f$ has at least one zero inside unit disk
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
86517
If $|f(z)|\le |g(z)|$, can we say $f$ is constant?
114488
If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
56127
Prove a function is constant under certain conditions
82367
Prove that a holomorphic function with postive real part is constant
81758
Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero.
54059
Showing that $\frac{z^n}{1+z^{2n}}$ is real for $|z|=1$.
64650
If $a\operatorname{Re}(f(z))+b\operatorname{Im}(f(z))=c$ then f is constant
56967
Show that $ f^2 − g^2 = C $ for some real constant C
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
128477
Entire function constant, where $f(z)=f(z+1)$ and $|f(z)|< e^{|z|}$.
61640
Prove c exists for f:[0,1] -> R