The Hölder mean with exponential is defined as

or a weighted version with

which for and , then

Hölder mean is a generalization of means:

  • : minimum
  • : Harmonic mean
  • : Geometric mean (obtained by taking limit on )
  • : Arithmetic mean
  • : Root-mean-square
  • : maximum

The Hölder mean has the property that, for and the equality holds only when , or, for the weighted version, and . This is the generalization of the mean inequality of