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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 [수식] where M is a given mean and f, F, g,G are the unknown functions. Solv- ing this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homo- graphic and square-root functions. A new criterion of the monotonicity of a real function is presented.
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참고문헌 (5)
참고문헌 신청
J. Aczel / 1985 / A mean-value property of the derivative of quadratic polynomials - without mean values and derivatives / Math. Mag 58 (1) : 42 ~ 45
J. Aczel / 1989 / On two mean value properties and functional equations asso- ciated with them / Aequationes Math 38 (2) : 216 ~ 235
J. Matkowski / 2011 / A mean-value theorem and its applications / J. Math. Anal. Appl 373 (1) : 227 ~ 234
J. Matkowski / 2011 / Power means generated by some mean-value theorems / Proc. Amer. Math. Soc 139 (10) : 3601 ~ 3610
M. Kuczma / 1991 / On the quasiarithmetic mean in a mean value property and the associated functional equation / Aequationes Math 41 (1) : 33 ~ 54
J. Aczel / 1989 / On two mean value properties and functional equations asso- ciated with them / Aequationes Math 38 (2) : 216 ~ 235
J. Matkowski / 2011 / A mean-value theorem and its applications / J. Math. Anal. Appl 373 (1) : 227 ~ 234
J. Matkowski / 2011 / Power means generated by some mean-value theorems / Proc. Amer. Math. Soc 139 (10) : 3601 ~ 3610
M. Kuczma / 1991 / On the quasiarithmetic mean in a mean value property and the associated functional equation / Aequationes Math 41 (1) : 33 ~ 54
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