Prove $(a^p + b^p)^{1/p} \leq (a^2 + b^2)^{1/2}$
15232
Proving an inequality involving real powers
148451
Polynomial inequality proof
65297
Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$.
66387
Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
59513
For $a, b \geq 0$, $0 < x < 1$, show $(a+b)^x \leq a^x + b^x$
111295
Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$
45742
Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
6621
Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$
257
Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
25458
Prove inequalities $2\sqrt{n+1}-2 \leq 1 + 1/\sqrt{2} + 1/\sqrt{3} + .... + 1/\sqrt{n}\leq 2\sqrt{n}-1$
11693
If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$
23638
$a^2b^2+a^2c^2+b^2c^2\leq 3$
158863
How to prove $\sum_p p^{-2} < \frac{1}{2}$?
152053
Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
81818
Prove or disprove : $\det(A^k + B^k) \geq 0$
66261
Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$