On a characterization of the logarithm by a mean value property

• Autorzy: Ger R.
• Języki publikacji: EN

Abstrakty

EN

Any real polynomial f(x) = ax² + bx + c, x ∈ IR, has the property that f (x)-f (y) x-y for every (x, y) ∈ IR, x ꞊ y. It turns out that that particular form of the Lagrange mean value theorem characterizes polynomials of at most second degree. Much more can be proved: J. Aczél [1] has shown that, with no regularity assumptions, a triple (/, g, h) of functions mapping IR into itself satisfies the equation f(x)-g(y) x-y= h(x + y) for all (x, y) ∈ IR, x ≠ y, if and only if there exist real constants a, 6, c such that f (x) = g(x) = ax² + b, x + c, x ∈ IR, and h(x) = ax + b, x ∈ IR. Generalizations involving weighted arithmetic means were also considered (see e.g. M. Falkowitz [3] and the references therein) and characterizations of polynomials of higher degrees (in the same spirit) were obtained (see [4] and [5], for instance). In what follows we are going to characterize the logarithm in a similar way. To this end, denote by D the open first quadrant of the real plane IR² with the diagonal removed, i.e. D := (O, ∞)² \ {(x, x) e IR² : x ∈ (0, ∞) }.Applying the classical Lagrange mean value theorem to the logaritmic function we derive the existence of a function D 3 (x, y) -> £(x,y) € intcony {x, y} such that the equality log a:-log y x-y £(z,y)

Słowa kluczowe

EN

logarithm; real polynomials; Lagrange theorem

PL

logarytmy; wielomiany rzeczywiste; twierdzenie Lagrange'a

• Wydawca: Wydawnictwo Uniwersytetu Humanistyczno-Przyrodniczego im. Jana Długosza w Częstochowie
• Czasopismo: Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
• Rocznik: 1998
• Tom: Vol. 4
• Strony: 1--7
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Ger R.

Bibliografia

• [1] J. Aczél, A mean value property of the derivative of quadratic polynomials - without mean values and derivatives. Math. Magazine 58 (1958), 42-45.
• [2] B. C. Carlson, The logarithmic mean. Math. Monthly 79 (1972), 615-618.
• [3] M. Falkowitz, A characterization of low degree polynomials. (Solution of problem E 3338 [1989, 641] proposed by Walter Rudin), Math. Monthly 98 (1991), 268-269.
• [4] R. Ger, Remark. The 29-th International Symposium on Functional Equations, June 3-10, 1991, Wolfville, N.S. (Canada); Report of Meeting, Aequationes Math. 43 (1992), 305.
• [5] R. Ger, A mean value property of the derivatives of polynomials. (submitted)
• [6] M. Kuczma, An introduction to the theory of functional equations and inequalities. Państwowe Wydawnictwo Naukowe & Uniwersytet Śląski, Warszawa-Kraków-Katowice, 1985.
• [7] A. W. Marshall & L Olkin, Inequalities: Theory of majorization and its applications. Academic Press, New York-London, 1979.
• [8] J. Matkowski, Mean value property and associated functional equations. (submitted).