Let $f$ be entire in $\Bbb C$. If $Re(z)>0$ $\forall z \in \Bbb C$. Then prove that $f(z)$ is constant.
75638
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.
26388
proof that function is constant
154742
If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
100765
If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
63255
Proving an entire, complex function with below bounded modulus is constant.
36379
Suppose that $ f $ is entire and that for each $ z $, either $ |f(z)| \leq 1 $ or $ |f^\prime (z) |\leq 1 $. Prove that $ f $ is a linear polynomial.
68071
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 $n$.
89050
If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
158647
If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
28620
Function $f(z)$ such that $|f(z)| > |z|$ for every $z\in\mathbb{C}, 1/f(z)$
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
77267
Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
34223
Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.
97371
Proving a function is constant, under certain conditions?
114245
What might I use to show that an entire function with positive real parts is constant?
34719
Show this holomorphic function is constant
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$