x_{1} (li; number/quantity) is the weighted quasi-arithmetic mean/generalized f-mean of/on data x_{2} (completely specified ordered multiset/list) using function x_{3} (defaults according to the notes; if it is an extended-real number, then it has a particular interpretation according to the Notes) with weights x_{4} (completely specified ordered multiset/list with same cardinality/length as x_{2}; defaults according to Notes).

Potentially dimensionful. Make sure to convert x_{3} from an operator to a sumti; x_{3} is the 'f' in "f-mean" and must be a complex-valued, single-valued function which is defined and continuous on x_{2} and which is injective; it defaults to the pth-power function (z^{p}) for some nonzero p (note that it need not be positive or an integer) and indeterminate/variable/input z, or log, or exp (as functions); culture or context can further constrain the default. If x_{2} is set to "z^{+ \infty .}" (the exponent is positive infinity, given by "ma'uci'i") for indeterminate/variable z (the function is the functional limit of the monic, single-term polynomial as the degree increases without bound), then the result (x_{1}) is the weight-sum-scaled maximum of the products of the data (terms of x_{2}) with their corresponding weights (terms of x_{4}) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering; likewise, if x_{2} is set to "z^{- \infty .}" (the exponent is negative infinity, given by "ni'uci'i") for indeterminate/variable z (the function is the functional limit of the monic, single-term reciprocal-polynomial as the reciprocal-degree increases without bound (or the degree decreases without bound)), then the result (x_{1}) is the weight-sum-scaled minimum of the products of the data (terms of x_{2}) with their corresponding weights (terms of x_{4}) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering. The default of x_{4} is the ordered set of n terms with each term equal identically to 1/n, where the cardinality of x_{2} is n. Let "f" denote the sumti in x_{3}, "y_{i}" denote the ith term in x_{2} for all i, "n" denote the cardinality of x_{2} (thus also x_{4}), and "w_{i}" denote the ith term in x_{4} for any i; then the result x_{1} is equal to: f^{(-1)(}Sum(w_{i f(y} in Set(1,...,n)) / Sum(w_{i, i} in Set(1,...,n))). Note that if the weights are all 1 and x_{2} is set equal to not the pth-power function, but instead the pth-power function left-composed with the absolute value function (or the forward difference function), then the result is the p-norm on x_{2} scaled by n^{(-1/p)} for integer n being the cardinality of x_{2}. This should typically not refer to the mean of a function ( https://en.wikipedia.org/wiki/Mean_of_a_function ), although it generalizes easily; alternatively, with appropriate weighting, allow x_{2} to be the image (set) of the function whose average is desired over the entire relevant subset of its domain - notice that the weights will have to themselves be functions of the data or the domain of the function; in this context, the function is not necessarily x_{3}. If x_{3} is a single extended-real number p (not a function), then this word refers to the weighted power-mean and it is equivalent to letting x_{2} equal the pth-power function as before iff p is nonzero real, the max or min as before if p is infinite (according to its signum as before), and log if p=0 (thus making the overall mean refer to the geometric mean); this overloading is for convenience of usage and will not cause confusion because constant functions are very much so not injective.

- cnanlagau
- x
_{1}is the generalized arithmetic-geometric mean of the elements of the 2-element set x_{2}(set; cardinality must be 2) of order x_{3}(either single extended-real number xor an unordered pair/2-element set of extended-real numbers).