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.
Examples
If = ℝ, the real line, and, 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 function,. The corresponds to dividing by, since logarithmic division is linear subtraction. The LogSumExp function is a smooth maximum: a smooth approximation to the maximum function.
Properties
The following properties hold for for any single function : Symmetry: The value of is unchanged if its arguments are permuted. Fixed point: for all x, . Monotonicity: is monotonic in each of its arguments. Continuity: is continuous in each of its arguments . Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With it holds: Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks: Self-distributivity: For any quasi-arithmetic mean of two variables:. Mediality: For any quasi-arithmetic mean of two variables:. Balancing: For any quasi-arithmetic mean of two variables:. Central limit theorem : Under regularity conditions, for a sufficiently large sample, is approximately normal. Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of : .
Characterization
There are several different sets of properties that characterize the quasi-arithmetic mean.
Mediality is essentially sufficient to characterize quasi-arithmetic means.
Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.
Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.
Balancing: An interesting problem is whether this condition implies that the mean is quasi-arithmetic. Georg Aumann showed in the 1930s that the answer is no in general, but that if one additionally assumes to be an analytic function then the answer is positive.
Homogeneity
s are usually homogeneous, but for most functions, the f-mean is not. Indeed, the only homogeneous quasi-arithmetic means are the power means ; see Hardy-Littlewood-Pólya, page 68. The homogeneity property can be achieved by normalizing the input values by some mean. However this modification may violate monotonicity and the partitioning property of the mean.
