MEAN-VALUE PROPERTY AND CHARACTERIZATIONS OF SOME ELEMENTARY FUNCTIONS

Title & Authors
MEAN-VALUE PROPERTY AND CHARACTERIZATIONS OF SOME ELEMENTARY FUNCTIONS
Matkowski, Janusz;

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 $\small{\frac{f(x)-F(y)}{x-y}=M(g(x),\;G(y)),\;x{\neq}y}$, where M is a given mean and $\small{f}$, F, $\small{g}$, 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;
Language
English
Cited by
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