$(a+b)^\beta \leq a^\beta +b^\beta$ for $a,b\geq0$ and $0\leq\beta\leq1$
3109
Prove that $(p+q)^m \leq p^m+q^m$
136403
Prove that $\sqrt{a}\leq\frac{1+a}{2}$
5369
Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
72081
$\forall x,y>0, x^x+y^y \geq x^y + y^x$
33903
Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
108704
Help with proof of $|x+y| \geq |x|-|y|$
38448
Show that a simple inequality holds: $a<b$ implies $\frac ab < \frac{a+1}{b+1}$
91617
Prove this inequality $a \sqrt{1-b^2}+b\sqrt{1-a^2}\le1 $
81775
Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
23638
$a^2b^2+a^2c^2+b^2c^2\leq 3$
80095
Show $|a|+|b|+|c|+|a+b+c| \geq  |a+b|+|b+c|+|c+a|$ for complex $a$, $b$, $c$
154085
$\sum a_i \ln(b_i) \leq \sum a_i \ln(a_i)$ with $\sum a_i = \sum b_i = 1$
139503
How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
5465
This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$
130281
Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
76925
Algebraic proof that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $
41300
Show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$