$f(z)\ne 0$ if $|z|<1/2$ in this situation?
123249
An analytic function $f$ on $|z|<1$ with $\sup_{|z|<1} |f(z)|=1$ and $|f(0)|=\frac{1}{2}$ has no zeros in $|z|<\frac{1}{2}$
34308
Given $|z|<1$ prove $\lim_{n \to \infty} z^n=0$.
85739
Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $
138544
For which values of $x$, will $f (x) . f '(x) < 0$?
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
130322
Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
145573
How to show that $f$ is a zero function?
133386
If $f$ is entire and $\lim_\limits{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant
63418
Show that $f_n(x) = \frac{x^2}{n}$ for $x \in \mathbb{R}$ does not converge uniformly towards $0$ for $n \rightarrow \infty$
28011
Does there exists an entire function such that for each $|z|>C$ for some constant $C$, $|f(z)|=|z|+1$?
76593
Show that $f_n(z)=z^n$ converges locally uniformly on $|z|<1$ but not uniformly(?)
48553
$f$ has an essential singularity in $z_0$. What about $1/f$?
62239
For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
107254
Real analysis question $e^{-1/x^2}$
120406
$f$ strictly increasing does not imply $f'>0$