$(x+y)^r \le x^r+y^r$ when $r \in (0,1)$ and $x,y$ are real positive numbers.
26741
How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
158844
If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
90896
proof of inequality $e^x\le x+e^{x^2}$
149889
Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$
138931
Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
50296
Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers
63651
If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax+by=1$.
43478
Prove that: $ m_a + m_b + m_c \le 4R + r $
13572
$x^y+y^x>1$ for all $(x, y)\in \mathbb{R_+^2}$
15232
Proving an inequality involving real powers
154945
$|px^2+qx+r|\le1$ for all $x$ in $[-1; 1]$, show that $|rx^2+qx+p|\le2$ for all $x$ in $[-1; 1]$
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 $x, y \ge 0$
25784
Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.
131596
Symmetric Inequality in $\mathbb{R}$
125209
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
77493
Classic $c_r$-inequality in $l_r$ space: $E(|X+Y|^r)\leq c_r[E(|X|^r)+E(|Y|^r)]$
25788
Prove that $i^i$ is a real number