If we have a non-constant entire function $f(z)$ show that $u(z)$ is not bounded.
17408
An entire function whose real part is bounded must be constant.
141863
If the imaginary part of an entire function is never zero, the function is constant
90533
Show that if $p(x)$ is reducible in $F[x]$, then $F[x]/(p(x))$ is not an integral domain.
67485
The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets
63255
Proving an entire, complex function with below bounded modulus is constant.
116092
Non constant analytic function from $\{z\in\mathbb{C}:z\neq 0\}$ to $\{z\in\mathbb{C}:|z|>1\}.$
138809
Consequences of a fixed point of a bounded, complex analytic function
2527
Entire function bounded by a polynomial is a polynomial
74501
$\delta : C([0,1])\to\mathbb{R}$, $f\mapsto f(0)$, with sup-norm is bounded but unbounded with one-norm.
101700
Liouvilles theorem question - show $f$ is constant
82371
$f$ is an entire function with Im $f\geq 0$
39583
ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$
138559
How can I show that $\frac{1}{1+|z|^2}$ isn't analytic at $z=1$?
28399
Showing $f$ is constant
63418
Show that $f_n(x) = \frac{x^2}{n}$ for $x \in \mathbb{R}$ does not converge uniformly towards $0$ for $n \rightarrow \infty$
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
81927
Example of closed and bounded in $\mathbb Q$ set that is not compact