MEAN-VALUE PROPERTY AND CHARACTERIZATIONS OF SOME ELEMENTARY FUNCTIONS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 50, Issue 1, 2013, pp.263-273
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2013.50.1.263

Title & Authors

MEAN-VALUE PROPERTY AND CHARACTERIZATIONS OF SOME ELEMENTARY FUNCTIONS

Matkowski, Janusz;

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 , where M is a given mean and , F, , 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

References

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J. Aczel, A mean-value property of the derivative of quadratic polynomials - without mean values and derivatives, Math. Mag. 58 (1985), no. 1, 42-45.

2.

J. Aczel and M. Kuczma, On two mean value properties and functional equations associated with them, Aequationes Math. 38 (1989), no. 2-3, 216-235.

3.

M. Kuczma, On the quasiarithmetic mean in a mean value property and the associated functional equation, Aequationes Math. 41 (1991), no. 1, 33-54.