To show $(x+y)^p\leq x^p+y^p$, where $0\leq p\leq1, x>0,y>0$?
3109
Prove that $(p+q)^m \leq p^m+q^m$
134348
When is $(p - 1)! + 1$ a power of $p$?
133228
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
98921
Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?
27911
How can I prove that $xy\leq x^2+y^2$?
20626
Prove that $x^y < y^x$
41364
How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
77026
Proof with inequalities
94119
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
25419
If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$
76863
How to show $ \sup \inf g(x,y) \leq \inf \sup g(x,y)$?
7775
Show that $|x-y|\leq |x|+|y|$.
44403
Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
88687
Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
5043
Show that $2^n>1+n\sqrt{2^{n-1}}$
61472
If $0<x<y$, then prove that $\sqrt{x} <\sqrt{y}$ and $x <\sqrt{xy} <y$