Power Mean Inequalities



Given positive a1,a2,,aj, a_1, a_2, \ldots, a_j, let f f be defined as

f(n)=(a1n+a2n++ajnj)1/n. f(n) = \left(\frac{a_1^n + a_2^n + \cdots + a_j^n}{j}\right)^{1/n}.

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

If the arithmetic mean and quadratic mean (root mean square) of a set of positive numbers are equal, which of the following means must also equal that same value?

What is the smallest n n for which

(a3+b3)2n(a6+b6) \left(a^3 + b^3\right)^2 \leq n (a^6 + b^6)

holds for all positive a a and b b ?

What is the largest n n for which

(x+y)(y+z)(x+y+z)nx2y3z2(x+y)(y+z)(xy+xz+yz) (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 x, y, z ?

Suppose the harmonic mean of 5 5 positive numbers is equal to 2. 2. What is the minimum possible sum of the numbers?


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