Prove that $|Im\{f(z)\}|≤ $$\frac{2}{\pi} \log \frac{1+|z|}{1 - |z|},$ $z \in \mathbb{D}$.
137045
The estimate of real and imaginary part of a holomorphic function
105641
Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$
11062
$f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}d\zeta$ if $\lim_{z\to\infty}f(z)=0$ and $f$ is analytic in $|z|\ge1$
37871
Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
95266
Show $|t|\leq \pi \Rightarrow |e^{it}-1|\geq 2|t|/ \pi$
66707
How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $
30897
$\pi(x)\geqslant\frac{\log x}{2\log2}$ for all $x\geqslant2.$
125680
Show that $(1+u) \log (1+u) - u \ge \frac{u^2}{2(1+u/3)} $
147986
Prove: if $|z|=1$ than $\frac{z+1}{z-1}$ is an imaginary number
145152
How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in[1,2]^3$
102553
How to show that $f(z) = f(i)$
28218
Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
15237
Prove an inequality using complex analysis
136855
Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID
71299
Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$
127397
$z\overline{w}\ne 1 $ Prove that $|\frac{w-z}{1-\overline{w}z}|\le 1|$ if $|z|\le 1$ and $|w|\le 1 $
106401
Prove: $p-mq | f(m)$ where $m$ is any integer