query: Prove $(1+x)\ln(1+x) + (1-x)\ln(1-x) \leq 2x^2$
positive_passages.docid: 27570
positive_passages.text: Prove $(1-x)\ln(1-x)+(1+x)\ln(1+x)\leq 2x^2$ for $0<x<1$
negative_passages.docid: 3110
negative_passages.text: Prove $\frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1}$
negative_passages.docid: 145423
negative_passages.text: prove that if $1 \le m \le n$ then $\left(1+\frac1n\right)^m < 1+\frac mn + \left(\frac mn\right)^2$
negative_passages.docid: 48503
negative_passages.text: Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
negative_passages.docid: 32149
negative_passages.text: Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$
negative_passages.docid: 145909
negative_passages.text: Prove that $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{2n}\geq \frac{2}{3}$
negative_passages.docid: 54513
negative_passages.text: Prove $\frac 1{x+1}<\ln\left(\frac{x+1}x\right)<0.5\left(\frac 1x+\frac 1{x+1}\right)$
negative_passages.docid: 104688
negative_passages.text: How can I prove that $x-{x^2\over2}<\ln(1+x)$
negative_passages.docid: 66387
negative_passages.text: Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
negative_passages.docid: 142360
negative_passages.text: How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
negative_passages.docid: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 82591
negative_passages.text: Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
negative_passages.docid: 44328
negative_passages.text: Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$
negative_passages.docid: 80556
negative_passages.text: Demonstrate by induction the inequality: $\ln(1+n)\leq\sum_{i=1}^{n}\frac{1}{i}\leq1+\ln(n)$
negative_passages.docid: 121548
negative_passages.text: proof of an inequality $ \left( 1+ \frac{x}{p} \right) ^{p}$
negative_passages.docid: 101792
negative_passages.text: By expanding $e^x$ into a series prove the following inequality
negative_passages.docid: 16524
negative_passages.text: Proof of an inequality by induction: $(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + x_n$