query: How to show $ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$
positive_passages.docid: 145884
positive_passages.text: Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
negative_passages.docid: 47323
negative_passages.text: Show that $\left(1+\frac xk\right)^k$ is monotonically increasing
negative_passages.docid: 157068
negative_passages.text: Prove that $\left (\frac{1}{a}+1 \right)\left (\frac{1}{b}+1 \right)\left (\frac{1}{c}+1 \right) \geq 64.$
negative_passages.docid: 51864
negative_passages.text: $\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$
negative_passages.docid: 44403
negative_passages.text: Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
negative_passages.docid: 68858
negative_passages.text: Prove that $\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$
negative_passages.docid: 24311
negative_passages.text: Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$
negative_passages.docid: 35611
negative_passages.text: How can I show that $n! \leqslant \left(\frac{n+1}{2}\right)^n$?
negative_passages.docid: 82961
negative_passages.text: Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$.
negative_passages.docid: 158863
negative_passages.text: How to prove $\sum_p p^{-2} < \frac{1}{2}$?
negative_passages.docid: 49217
negative_passages.text: How do I show that $a^p + b^p > (a + b)^p$ for $a,b>0$ and $0<p<1$?
negative_passages.docid: 18914
negative_passages.text: How prove this $A=\left(B-\frac{1}{110}I\right)^T\left(B+\frac{1}{110}I\right)$
negative_passages.docid: 30386
negative_passages.text: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
negative_passages.docid: 128790
negative_passages.text: Prove an integral inequality: $ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $
negative_passages.docid: 2455
negative_passages.text: Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
negative_passages.docid: 67650
negative_passages.text: Short and intuitive proof that $\left(\frac{n}{k}\right)^k \leq \binom{n}{k}$
negative_passages.docid: 105145
negative_passages.text: Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$