Prove that for $x,y \geq 0$ and $p>1$ , $xy \leq \frac {x^p} {p} +\frac {y^q} {q}$ whenever $ \frac {1} {p} + \frac {1} {q} = 1$.
14221
How can we prove it? ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$
90343
Proving the inequality $|e^z-1|\leq e^{|z|}-1$
66247
Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$
108704
Help with proof of $|x+y| \geq |x|-|y|$
110103
Prove that there are $0<c_1 < c_2<1$ s.t. $\frac{1}{g'(c_1)}+\frac{1}{g'(c_2)}=2$ . The same for $g'(c_1)g'(c_2)=1$.
137614
$\left|\frac{x}{|x|}-\frac{y}{|y|}\right|\leq |x-y|$, for $|x|, |y|\geq 1$?
3109
Prove that $(p+q)^m \leq p^m+q^m$
103030
Proving $\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}$ for $x> 0$
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
122326
Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$
83291
Prove that $p(x)=(x-a)q(x)$ for some $q(x)\in R[x]$ of degree= (degree of $p)-1$.
349
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
129331
Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
82591
Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
145909
Prove that $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{2n}\geq \frac{2}{3}$
103851
Show that $\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} \geq x+y+z $ by considering homogeneity