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 \(\textrm{A.M.} \ge \textrm{G.M.} \ge \textrm{H.M.}\)