Bull. Korean Math. Soc. 2013; 50(1): 263-273
Printed January 31, 2013
https://doi.org/10.4134/BKMS.2013.50.1.263
Copyright © The Korean Mathematical Society.
Janusz Matkowski
University of Zielona G\'{o}ra
A mean-value result, saying that the difference quotient of a differentiable function in a real interval is a mean value of its derivatives at the endpoints of the interval, leads to the functional equation \[ \frac{f(x)-F(y)}{x-y}=M\left( g(x),G(y)\right) ,\text{ \ \ \ \ }x\neq y, \] where $M$ is a given mean and $f,F,g,G$ are the unknown functions. Solving this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homographic and square-root functions. A new criterion of the monotonicity of a real function is presented.
Keywords: mean-value theorem, classical means, monotonic functions, quadratic function, homographic function, square root function, functional equation
MSC numbers: Primary 39B22, 26A24, 26A48
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