Quasi-arithmetic Mean - Definition

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:

    ... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.
    Adrienne Rich (b. 1929)

    One definition of man is “an intelligence served by organs.”
    Ralph Waldo Emerson (1803–1882)

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)