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Looking for a Few Good Means

Bruce Ebanks
The American Mathematical Monthly
Vol. 119, No. 8 (October 2012), pp. 658-669
DOI: 10.4169/amer.math.monthly.119.08.658
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Page Count: 12
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Looking for a Few Good Means
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Abstract The mean value theorem of integral calculus states that for any continuous realvalued function f on an interval I, and for any two distinct real numbers a, b∈ I, there exists a value V(a, b) in the open interval between a and b for which \documentclass{article} \pagestyle{empty}\begin{document} $$f(V(a,b))=\frac{1}{b-a}\int_{a}^{b}f(x)\,dx.$$ \end{document} If in addition f is strictly monotonic, then the function Vf/ defined by \documentclass{article} \pagestyle{empty}\begin{document} $$V_{f}(s,t)=f^{-1}\left(\frac{1}{t-s}\int_{s}^{t}f(x)\,dx\right)$$ \end{document} can be viewed as a two-variable mean on the interval I. We determine which of these means are homogeneous. The preparatory remarks preceding the proof provide a brief introduction to the theory of functional equations. Several questions for further exploration that the theorem suggests are indicated in the final sections.

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