$f \in C^2(0,\infty)$ , then how to show that $\Big(\sup_{x>0} |f'(x)|\Big)^2\le 4\big(\sup _{x>0}|f(x)|\big)\big(\sup_{x>0}|f''(x)|\big)$ ?
89429
Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
42108
Prove that $c_0$ is closed in $\ell^\infty$
158647
If $|f(z)|\le |f(z^2)|$ then prove that $f$ is constant
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
145984
if $f: (0,\infty) \to (0,\infty)$ is a strictly decreasing then $f \circ f$ is decreasing?
966
Show that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$
98836
Showing $\int_x^{x+1}f(t)\,dt \xrightarrow{x\to\infty}0$ for $f\in L^2 (\mathbb{R})$
111205
Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$
20192
Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
19986
How can this be proved $\lim_{x\to\infty}(f(x)+f'(x))=l$
71225
$f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
101286
How show that $|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$
41173
If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?
19072
$\int_{\mathbb R} |f(x)| dx < \infty \implies \int^{\infty}_{n} |f(x)| dx \to 0$ as $n \to \infty$?
96848
Prove: $\limsup_{n\to\infty} (a_n)\leq \sup(a_n)_{n=1}^\infty$
5365
Bound on $|f'(0)|$ of a holomorphic funtion in the unit disk
34136
$f$ is differentiable in $[0,1]$ ,$\sup_{x\in[0,1]}|f'(x)| \le M\lt+\infty $