query: $(x + y)^s \le x^s + y^s$ for $0 < s \le 1$
positive_passages.docid: 26741
positive_passages.text: How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
negative_passages.docid: 41364
negative_passages.text: How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
negative_passages.docid: 118559
negative_passages.text: Prove the inequality $\sum_{1\le i<j\le n}\sqrt{a_ia_j}<\frac{n-1}2\sum_{i=1}^n a_i$
negative_passages.docid: 25203
negative_passages.text: with inequality $\frac{y}{xy+2y+1}+\frac{z}{yz+2z+1}+\frac{x}{zx+2x+1}\le\frac{3}{4}$
negative_passages.docid: 60761
negative_passages.text: Proof: $\cos^p (\theta) \le \cos(p\theta)$
negative_passages.docid: 61472
negative_passages.text: If $0<x<y$, then prove that $\sqrt{x} <\sqrt{y}$ and $x <\sqrt{xy} <y$
negative_passages.docid: 7448
negative_passages.text: Prove $S \doteq \sum_{n=1}^{\infty} p_n  0$ assuming $0 \leq p_n < 1$.
negative_passages.docid: 104447
negative_passages.text: Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$
negative_passages.docid: 129331
negative_passages.text: Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
negative_passages.docid: 135747
negative_passages.text: Prove $x^{rs}-1=(x^s-1)(x^{s(r-1)}+...+x^s+1)$
negative_passages.docid: 64108
negative_passages.text: how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?
negative_passages.docid: 25419
negative_passages.text: If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$
negative_passages.docid: 10859
negative_passages.text: Prove $\sup(f+g) \le \sup f + \sup g$
negative_passages.docid: 27901
negative_passages.text: Proving $\sum\limits_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot \|y\|$
negative_passages.docid: 158551
negative_passages.text: If $X \perp Y$ and $X+Y \in L^1$ then $X,Y \in L^1$
negative_passages.docid: 79040
negative_passages.text: Graph of $|x| + |y| = 1$
negative_passages.docid: 27911
negative_passages.text: How can I prove that $xy\leq x^2+y^2$?