Suppose that $f$ is an entire function satisfying $f(2z)=\frac{f(z)+f(z+1)}{2}$. Show that $f$ is constant.
129605
$ 2g(z)=g(\frac{z}{2}) + g(\frac{z+1}{2})$
39921
show that f is constant on I
98399
Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.
7878
Proving that a doubly-periodic entire function $f$ is constant.
82229
Show if complex function $f(z)$ contains all real coefficients then $f(\bar z)$ = $\bar {f(z)}$
73547
Prove that is $\bar{f}g$ is analytic then $f$ is constant or...?
65548
Prove if f(z) is entire function s.t $|f(z)|\leq M|z|$ then $f(z)$ Is polynomial of degree $1$
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
153855
Complex analysis, showing a function is constant
10194
If $f,g$ are analytic in the unit disk, and $|f|^2+|g|^2=1$, then $f,g$ constant.
148981
Entire $f$ and $g$ constant if $e^{f(z)}+e^{g(z)}=1$
105700
Proving $\,f$ 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
42075
If $f$ is analytic then $|f|$ is not constant unless $f$ is constant
132113
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
28399
Showing $f$ is constant
65054
How is $e^z$ the only function that is entire with the property $f'(z) = f(z)$ and $f(0) = 1$