Prove $f(z)=c$ when $\operatorname{Re}(f)$ has an upper bound ($f$ is analytic)
17408
An entire function whose real part is bounded must be constant.
158647
If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
125814
The function $f(z)=z+\frac{1}{z}$ maps $\Omega$ biholomorphically on the upper plane
138809
Consequences of a fixed point of a bounded, complex analytic function
62901
Bound for Analytic Function on Unit Disk
111635
Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$.
106290
Prove a function is constant. (complex analysis)
41149
Radius of convergence of $f(z)?$
28620
Function $f(z)$ such that $|f(z)| > |z|$ for every $z\in\mathbb{C}, 1/f(z)$
55138
An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$
158561
Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, then$f (i) $ is
125566
The Identity Theorem for real analytic functions
65901
Solution verification to complex analysis problem, prove $f$ is a polynomial.
76285
Suppose $f$ is entire and $|f(z)| \leq 1/|Re z|^2$ for all $z$. Show that $f $ is identically $0$.
111076
If $f$ is non-constant and entire, prove that there exists a $z_0 \in C$ such that $f(z_0)$ is a positive real number
18877
$f(z)$ and $\overline{f(\overline{z})}$ simultaneously holomorphic
76971
Entire functions bounded along $\vert z\vert = R$