Definition
If f is a function which maps an interval of the real line to the real numbers, and is both continuous and injective then we can define the f-mean of two numbers
as
For numbers
- ,
the f-mean is
We require f to be injective in order for the inverse function to exist. Since is defined over an interval, lies within the domain of .
Since f is injective and continuous, it follows that f is a strictly monotonic function, and therefore that the f-mean is neither larger than the largest number of the tuple nor smaller than the smallest number in .
Read more about this topic: Quasi-arithmetic Mean
Famous quotes containing the word definition:
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)