$|f(z)-1| < 1$ implies $ \int_{\gamma} \frac{f'(z)}{f(z)}dz = 0$
11479
Integral yields log of holomorphic function
46844
Show that if $\int_0^x f(y)dy \sim Ax^\alpha$ then $f(x)\sim \alpha Ax^{\alpha -1}$
96023
Prove: $\int_\gamma \bar z \, dz=2i\operatorname{Area}(G)$
14759
Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.
15886
Proof that $\frac {1} {2\pi i} \oint \frac {{\rm d}z} {P(z)} $ over a closed curve is zero.
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
94962
For complex $z$, $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right)  = 0$
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
153107
Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?
154264
$f \in C[-1,1]$, Prove ${\lim_{h \to 0^+}}{\int_{-1}^1 \frac{h}{h^2+x^2}f(x)\,dx} = \pi f(0)$
101778
Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
136966
Riemann integral proof $\int^b_a f(x) \, dx>0$
53726
Prove that if $f$ is a real continuous function such that $|f|\le 1$ then $|\int_{|z|=1} f(z)dz| \le 4$
31522
Proving $f$ has at least one zero inside unit disk
132431
Proving $|f'(z)|\leq\frac{\text{Re}(f(z))}{\text{Re}(z)}$
138054
Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$
55138
An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$