Summary
Motivated by different mean value properties, the functional equationsf(x) − f(y)/x−y=φ[η(x, y)], (i)xf(y) − yf(x)/x−y=φ[ζ(x, y)] (ii) (x ≠ y) are completely solved whenζ, ŋ are arithmetic, geometric or harmonic means andx, y elements of proper real intervals. In view of a duality between (i) and (ii), three of the results are consequences of other three.
The equation (ii) is also solved whenζ is a (strictly monotonic) quasiarithmetic mean while the real interval contains 0 and whenζ is the arithmetic mean while the domain is a field of characteristic different from 2 and 3. (A result similar to the latter has been proved previously for (i).)
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References
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Aczél, J., Kuczma, M. On two mean value properties and functional equations associated with them. Aeq. Math. 38, 216–235 (1989). https://doi.org/10.1007/BF01840007
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DOI: https://doi.org/10.1007/BF01840007