If $f$ satisfies $|f(z)|\le |f(z^2)|$ then $f$ is constant
158647
If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
102553
How to show that $f(z) = f(i)$
132113
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
28399
Showing $f$ is constant
48275
$f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
48619
Let $f$ be a non-constant analytic function on $\mathbb{D}$ satisfying $|f(z^2)| ≤ |f(z)|$ for $|z| < 1.$
89050
If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
141863
If the imaginary part of an entire function is never zero, the function is constant
114488
If $|f|+|g|$ is constant on $D,$ prove that holomorphic functions $f,~g$ are constant on $D$.
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 $n$.
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
19750
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.
69804
Prove that if $f$ is entire and $\vert f(z^2) \vert \leq 2 \vert f(z) \vert$, then $f$ is constant
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.
93400
Prove that an analytic function with a constant imaginary part is constant itself
23663
Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.
144589
Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$