Euler's Beta function diagonalized and a related functional equation

• Autorzy: Choczewski B. , Wach-Michalik A.
• Języki publikacji: EN

Abstrakty

EN

Euler's Gamma function is the unique logarithmically convex solution of the functional equation (1), cf. the Proposition. In this paper we deal with the function beta: R+ → R+, beta(x) := B(x, x), where B(x, y) is the Euler Beta function. We prove that, whenever a function h is asymptotically comparable at the origin with the function a log +b, a > 0, if varphi: R+ → R+ satisfies equation (5) and the function h o varphi is continuous and ultimately convex, then varphi = beta.

Słowa kluczowe

EN

Euler's Beta function; diagonalization; functional equations; convex functions

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2004
• Tom: Vol. 24, no. 1
• Strony: 35--41
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Choczewski B.

• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland

autor

Wach-Michalik A.

• AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland

Bibliografia

• [1] Artin E.: Einfuhrung in die Theorie der Gammafunktion. Hamburger Math. Einzelschr. Heft 1, B. G. Teubner, Leipzig-Berlin, 1931.
• [2] Bohr H., Mollerup J.: Lectures on mathematical analysis. III. Limit processes. K0benhavn 1922 (Danish).
• [3] Choczewski B., Wach-Michalik A.: A difference equation for q-gamma functions. Folia FNS Universitatis Masarykianae Brunensis, Mathematica 13 (2002), 71-75.
• [4] Kairies H.-H.: Convexity in the theory of the Gamma function, [in:] General Inequalities 1, ed. E. F. Beckenbach, ISNM, vol. 41, Birkhauser Verlag, Basel-Stuttgart, 1978, 49-62.
• [5] Kuczma M.: Functional equations in a single variable. Warszawa, PWN 1968.
• [6] Kuczma M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality. Warszawa-Kraków-Katowice, PWN, Uniwersytet Śląski 1985.
• [7] Kuczma M., Choczewski B., Ger R.: Iterative Functional Equations. Encyclopedia ol Math, and Its Appl., vol. 32, Cambridge, New York, Port Chester, Melbourne, Sydney, Cambridge University Press 1990.
• [8] Liebiediew N.N.: Special functions and their applications. Warszawa, PWN 1957 (Polish).
• [9] Wach-Michalik A.: Uniqueness conditions for solutions of functional equations characterizing some special functions. Thesis, Kraków, Pedagogical University 2001 (Polish).
• [10] Wach-Michalik A.: On a problem of H.-H. Kairies concerning Euler's Gamma function. Ann. Acad. Paedag. Cracoviensis, Studia Math. 19 (1) (2001), 151-161.