$f: \mathbb{D} \to \mathbb{D}$ holomorphic, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$. Prove $|f(0)| \leq \frac{1}{4}$
2414
If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$
92056
An inequality on holomorphic functions
148494
How does one prove that $\frac{1}{2}\cdot\frac{3}{4}\cdots \frac{2n-1}{2n}\leq \frac{1}{\sqrt{3n+1}}?$
142299
Proving theorem connecting the inverse of a holomorphic function to a contour integral of the function.
78319
Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
34719
Show this holomorphic function is constant
23663
Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant.
74392
Show that a function satisfies the Cauchy-Riemann function if and only if $\frac{df}{d\overline{z}}=0$
103125
Show that a non-constant holomorphic function has zeroes in a compact set
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
19896
Prove $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$
149328
A function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$ satisfies $f(z)+f(-z)=0$.(MADHAVA-2020)
100608
Prove that $\|v\|_{H^1(\Omega )}\leq C(\|f\|_{L^2(\Omega )}+\|v\|_{H^{1/2}(\partial \Omega )}+\|\partial _\nu v\|_{H^{1/2}(\partial \Omega )}))$
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
107968
holomorphic function on punctured disk satisfying $\left|f\left(\frac{1}{n}\right)\right|\leq\frac{1}{n!}$ has an essential singularity at $0$
5365
Bound on $|f'(0)|$ of a holomorphic funtion in the unit disk
15797
Holomorphic function having finitely many zeros in the open unit disc