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Power Mean Inequalities

This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.

QM-AM-GM-HM

         

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?

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 \( n \) 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 \( x, y, z \)?

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

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