The Hölder mean with exponential $$p$$ is defined as

$M_p(x_1, x_2, \cdots, x_n) = \left\{\begin{array}{ll} \left(\frac{1}{n}\sum_{i=1}^{n}x_i^p\right)^{1/p} & \textrm{for }p\neq 0 \\ \sqrt[n]{\prod_{i=1}^{n}x_i} & \textrm{for }p=0 \end{array}\right.$

or a weighted version with $$w=w_1+w_2+\cdots+w_n$$

$M_p(x_1, x_2, \cdots, x_n) = \left\{\begin{array}{ll} \left(\frac{1}{w}\sum_{i=1}^{n}w_ix_i^p\right)^{1/p} & \textrm{for }p\neq 0 \\ \sqrt[w]{\prod_{i=1}^{n}x_i^{w_i}} & \textrm{for }p=0 \end{array}\right.$

which for $$p=\infty$$ and $$p=-\infty$$, then

\begin{aligned} M_\infty(x_1, x_2, \cdots, x_n) &=\max(x_1, x_2, \cdots, x_n) \\ M_{-\infty}(x_1, x_2, \cdots, x_n) &=\min(x_1, x_2, \cdots, x_n) \end{aligned}

Hölder mean is a generalization of means:

• $$p=-\infty$$: minimum
• $$p=-1$$: Harmonic mean
• $$p=0$$: Geometric mean (obtained by taking limit on $$p\to 0$$)
• $$p=1$$: Arithmetic mean
• $$p=2$$: Root-mean-square
• $$p=\infty$$: maximum

The Hölder mean has the property that, $$M_p \le M_q$$ for $$p<q$$ and the equality holds only when $$x_1=x_2=\cdots=x_n$$, or, for the weighted version, $$w_1=w_2=\cdots=w_n$$ and $$x_1=x_2=\cdots=x_n$$. This is the generalization of the mean inequality of $$\textrm{A.M.} \ge \textrm{G.M.} \ge \textrm{H.M.}$$