Algebra
# Power Mean Inequalities

Given positive $a_1, a_2, \ldots, a_j,$ let $f$ be defined as

$f(n) = \left(\frac{a_1^n + a_2^n + \cdots + a_j^n}{j}\right)^{1/n}.$

For $m > n$, which of the following is necessarily true?

What is the smallest $n$ for which

$\left(a^3 + b^3\right)^2 \leq n (a^6 + b^6)$

holds for all positive $a$ and $b$?

What is the largest $n$ for which

$(x + y) (y + z) (x + y + z) \geq \frac{n x^2 y^3 z^2}{(x + y)(y + z)(xy + xz + yz)}$

holds for all positive $x, y, z$?