# arrow_back What is the generalized mean inequality ?

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What is the generalized mean inequality ?

The geometric-arithmetic mean inequality and all other mean inequalities are a special case of the Generalised Mean Inequality, a monster that compares an infinite number of means in a compact form. Equality holds when all numbers are the same, for any p and q. #Algebra

Let $$M_{p}\left(x_{1}, \ldots, x_{n}\right)= \begin{cases}\left(\frac{1}{n} \sum_{i=1}^{n} x_{i}^{p}\right)^{1 / 0} & \text { if } p \neq 0 \\ \sqrt[n]{\prod_{i=1}^{n} x_{i}} & \text { if } p=0\end{cases}$$ be the generalized mean of $x_{1}, \ldots, x_{n} .$ Then if $p<q$

$$M_{p}\left(x_{1}, \ldots, x_{n}\right) \leq M_{q}\left(x_{1}, \ldots, x_{n}\right)$$

Harmonic Mean $(p=-1)$
$M_{-1}=\frac{n}{\frac{1}{x_{1}}+\ldots+\frac{1}{x_{n}}} \leq \quad$ Geometric Mean $(p=0) M_{0}=\sqrt[n]{x_{1} \cdot \ldots \cdot x_{n}} \leq M_{1}=\frac{x_{1}+\ldots+x_{n}}{n} \leq M_{2}=\sqrt{\frac{x_{1}^{2}+\ldots+x_{n}^{2}}{n}}$
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