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
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—William James (1842–1910)
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