Show that $|f(0)| \leq \frac{1}{4}$ for every holomorphic function $f:\Bbb D \to \Bbb D$ such that $|f(\frac{1}{2})|+|f(-\frac{1}{2})|=0$
2414
If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$
103125
Show that a non-constant holomorphic function has zeroes in a compact set
64172
Show that if $f$ is analytic in a domain $D$ and $|f(z)|$ is constant in $D$, then the function $f(z)$ is constant in $D$
14759
Prove that $ f(z) = 0 $ for all $ z \in \overline{D}(0,1) $.
61799
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
74051
No analytic function in $\{|z|<1\}$ such that $f(0)=1$ and $|f(z)|\ge 1+|z|^2$. Proof strategy
63446
The existence of analytical branch of the logarithm of a holomorphic function
75653
Prove that $|f ''|\ge 4$
149634
$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$
63526
Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
28620
Function $f(z)$ such that $|f(z)| > |z|$ for every $z\in\mathbb{C}, 1/f(z)$
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
87723
Show a holomorphic function on unit disc with with $|f'(z) - f'(0)| < |f'(0)|$ is injective
5365
Bound on $|f'(0)|$ of a holomorphic funtion in the unit disk
81974
Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ (Riemann integral)
76147
Bounding holomorphic function at 0 given other bounds
100608
Prove that $\|v\|_{H^1(\Omega )}\leq C(\|f\|_{L^2(\Omega )}+\|v\|_{H^{1/2}(\partial \Omega )}+\|\partial _\nu v\|_{H^{1/2}(\partial \Omega )}))$