$|f(z)| \leq \frac{1}{1-|z|}$ implies $|f'(z)| \leq \frac{4}{(1 - |z|)^2}$
100780
Deriving upper bound for derivative of analytical function
97643
Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$
61799
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
4498
Proof that $\|fx\| \leq \|f\|\cdot\|x\|$
89050
If $|f(z)|\leq |f(z^2)|$ on the unit disk, show that $f$ is a constant function.
54380
Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$
15441
Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$
152053
Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
106767
$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
4065
What are the entire functions $f$ such that $|f'(z)| \leq |f(z)| \ ? $
76285
Suppose $f$ is entire and $|f(z)| \leq 1/|Re z|^2$ for all $z$. Show that $f $ is identically $0$.
149634
$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$
80826
To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
73002
For $n \geq 2$, prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
28125
Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$
84313
$|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$