Prove $(x+y)^a\leq x^a+y^a$ if $0<a\leq1$ and $x,y\geq0$
3109
Prove that $(p+q)^m \leq p^m+q^m$
29462
Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$
125707
A simple proof (hopefully) for $2^x \neq 0$
29935
If $a, b, c, d>0$ and $abcd=1$ prove that an inequality holds true
147785
Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
86211
If $x+y+z \geq xyz$ Prove that $x^2 + y^2 + z^2 \geq xyz$.
138212
Please help me prove: $v(a+b)\leq v(a)+v(b)$, and $v(ab)\leq v(a)v(b)$ where $v(x)=\inf{\{\vert x^n \vert}^{1/n}: n\in\mathbb{N}\}$
111316
PROVE if $x \ge-1 $then $ (1+x)^n \ge 1+nx $ , Every $n \ge 1$
96327
Show that $a^x+b^x+c^x>(a+b+c)^x$ for all $a,b,c>0$ and $0<x<1$
41383
Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$
105316
Prove : $|x - y| \leqslant \epsilon$ $\forall \epsilon > 0$ iff $x = y$
90896
proof of inequality $e^x\le x+e^{x^2}$
155232
Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
156555
Inequality $\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$
78144
How to prove the inequality $a_1^{a_1}a_2^{a_2}\cdots a_n^{a_n} > \left(\frac{a_1+\dots+a_n}{n}\right)^{a_1+\dots+a_n}$?
8207
Can I prove this inequality algebraically?
5369
Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$