Quasi-arithmetic mean

In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function . It is also called Kolmogorov mean after Soviet mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean. If f is a function which maps an interval of the real line to the real numbers, and is both continuous and injective, the f-mean of numbers is defined as , which can also be written 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 . If = R, the real line, and , (or indeed any linear function , not equal to 0) then the f-mean corresponds to the arithmetic mean. If = R+, the positive real numbers and , then the f-mean corresponds to the geometric mean. According to the f-mean properties, the result does not depend on the base of the logarithm as long as it is positive and not 1. If = R+ and , then the f-mean corresponds to the harmonic mean. If = R+ and , then the f-mean corresponds to the power mean with exponent . If = R and , then the f-mean is the mean in the log semiring, which is a constant shifted version of the LogSumExp (LSE) function (which is the logarithmic sum), . The corresponds to dividing by n, since logarithmic division is linear subtraction. The LogSumExp function is a smooth maximum: a smooth approximation to the maximum function. The following properties hold for for any single function : Symmetry: The value of is unchanged if its arguments are permuted. Idempotency: for all x, . Monotonicity: is monotonic in each of its arguments (since is monotonic). Continuity: is continuous in each of its arguments (since is continuous).