if $Ref(z)$>0 for every $z$ in the complex plane, then the function is constant
114245
What might I use to show that an entire function with positive real parts is constant?
158519
Hints for a complex limit: Prove if $\lim_{z \to \infty} f(z)/z = 0$ then $f(z)$ is constant.
113343
$f$ is constant if derivative equals zero
56978
How to Prove that if $f(z)$ is entire, and $f(z+i) = f(z), f(z+1) = f(z)$, then $f(z)$ is constant?
153411
If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.
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
20955
Let $f(z)$ be analytic on $\mathbb{D}$ = {${z\in\mathbb{C}:|z-1|<1}$} such that $f(1) = 1$, if $f(z) = f(z^2)$ for all $z\in\mathbb{D}$,
64172
Show that if $f$ is analytic in a domain $D$ and $|f(z)|$ is constant in $D$, then the function $f(z)$ is constant in $D$
20047
At $z=0$ the function $f(z)=\exp({z\over 1-\cos z})$ has
114826
Zero Partials imply Constant Function Theorem or Proof
47474
Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
148981
Entire $f$ and $g$ constant if $e^{f(z)}+e^{g(z)}=1$
142138
Show that $f(x)$ is a constant function if $\lim\limits_{h\to 0}\frac{1}{h^3}\int_{-h}^{h}f(x+t)\cdot t\,dt=0$ for all $x$
89050
If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
72856
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
145919
If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.