query: Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
positive_passages.docid: 107894
positive_passages.text: Inequality involving the sup of a function and its first and second derivatives
negative_passages.docid: 153272
negative_passages.text: Basic question about $\sup_{x\neq 0}{} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1}{\|Ax\|} $, $x \in\mathbb{R}^n$
negative_passages.docid: 71857
negative_passages.text: Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
negative_passages.docid: 15232
negative_passages.text: Proving an inequality involving real powers
negative_passages.docid: 7974
negative_passages.text: lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $
negative_passages.docid: 257
negative_passages.text: Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$
negative_passages.docid: 149739
negative_passages.text: Prove: $f(t)\leq t+1$
negative_passages.docid: 143130
negative_passages.text: Proving that the supremum of $(a,b)$ is $b$.
negative_passages.docid: 35564
negative_passages.text: Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$
negative_passages.docid: 148494
negative_passages.text: How does one prove that $\frac{1}{2}\cdot\frac{3}{4}\cdots \frac{2n-1}{2n}\leq \frac{1}{\sqrt{3n+1}}?$
negative_passages.docid: 115495
negative_passages.text: Prove $\int _0^\infty f^2 dx \leq \cdots $ for $f$ convex
negative_passages.docid: 90343
negative_passages.text: Proving the inequality $|e^z-1|\leq e^{|z|}-1$
negative_passages.docid: 78463
negative_passages.text: Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$
negative_passages.docid: 138819
negative_passages.text: Prove that $\phi(n) \geq \sqrt{n}/2$
negative_passages.docid: 87239
negative_passages.text: Prove functions defined by sup and inf are continuous
negative_passages.docid: 58943
negative_passages.text: If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
negative_passages.docid: 75653
negative_passages.text: Prove that $|f ''|\ge 4$