[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2013.50.1.263

MEAN-VALUE PROPERTY AND CHARACTERIZATIONS OF SOME ELEMENTARY FUNCTIONS

Matkowski, Janusz (Faculty of Mathematics Computer Science and Econometrics University of Zielona Gora)

Publication Information

Bulletin of the Korean Mathematical Society / v.50, no.1, 2013 , pp. 263-273 More about this Journal

Abstract

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 $\frac{f(x)-F(y)}{x-y}=M(g(x),\;G(y)),\;x{\neq}y$ , 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;

Citations & Related Records

Reference

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