Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T01:55:36.416Z Has data issue: false hasContentIssue false
  • English
  • Français

Inequalities for the beta function of n variables

Published online by Cambridge University Press:  17 February 2009

Horst Alzer
Horst Alzer
Affiliation:
Morsbacher Str. 10, 51545 Waldbröl, Germany; e-mail: alzerhorst@freenet.de.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present various inequalities for Euler's beta function of n variables. One of our theorems states that the inequalities

hold for all xi ≥ (i = 1,… n; n ≥ 3) with the best possible constants an = 0 and bn = 1 − 1/(n − 1)!. This extends a recently published result of Dragomir et al., who investigated (*) for the special case n = 2.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 4 , April 2003 , pp. 609 - 623
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of mathematical functions with formulas, graphs and mathematical tables (Dover, New York, 1965).Google Scholar
[2]Alzer, H., “On some inequalities for the gamma and psi functions”, Math. Comp. 66 (1997) 373–389.CrossRefGoogle Scholar
[3]Alzer, H., “Sharp inequalities for the beta function”, Indag. Math. (N.S.) 12 (2001) 15–21.CrossRefGoogle Scholar
[4]Anderson, G. D., Vamanmurthy, M. K. and Vuorinen, M. K., Conformal invariants, inequalities, and quasiconformal maps (Wiley, New York, 1997).Google Scholar
[5]Andrews, G. E., Askey, R. and Roy, R., Special functions (Cambridge Univ. Press, Cambridge, 1999).CrossRefGoogle Scholar
[6]Beckenbach, E. F., “Superadditivity inequalities”, Pacific J. Math. 14 (1964) 421–438.CrossRefGoogle Scholar
[7]Bruckner, A. M. and Ostrow, E., “Some function classes related to the class of convex functions”, Pacific J. Math. 12 (1962) 1203–1215.CrossRefGoogle Scholar
[8]Carlson, B. C., Special functions of applied mathematics (Academic Press, New York, 1977).Google Scholar
[9]Dragomir, S. S., Agarwal, R. P. and Barnett, N. S., “Inequalities for beta and gamma functions via some classical and new integral inequalities”, J. Inequal. Appl. 5 (2000) 103–165.Google Scholar
[10]Gordon, L., “A stochastic approach to the gamma function”, Amer. Math. Monthly 101 (1994) 858–865.CrossRefGoogle Scholar
[11]Hille, E. and Phillips, R. S., Functional analysis and semi-groups, Amer. Math. Soc. Coll. Publ. 31, revised ed. (AMS, Providence, RI, 1957).Google Scholar
[12]Karatsuba, E. A. and Vuorinen, M., “On hypergeometric functions and generalizations of Legendre's relation”, J. Math. Anal. Appl. 260 (2001) 623–640.CrossRefGoogle Scholar
[13]Lochs, G., “Abschätzung spezieller Werte der unvollständigen Betafunktion”, Anz. Österreich. Akad. Wiss. Math.-Natur Kl. 123 (1986) 59–63.Google Scholar
[14]Maligranda, L., “Indices and interpolation”, Dissertationes Math. (Rozprawy Mat.) 234 (1985) 1–54.Google Scholar
[15]Mitrinović, D. S., Analytic inequalities (Springer, New York, 1970).CrossRefGoogle Scholar
[16]Raab, W., “Die Ungleichungen von Vietoris”, Monatsh. Math. 98 (1984) 311–322.CrossRefGoogle Scholar
[17]Roberts, A. W. and Varberg, D. E., Convex functions (Academic Press, New York, 1973).Google Scholar
[18]Rosenbaum, R. A., “Sub-additive functions”, Duke Math. J. 17 (1950) 227–247.CrossRefGoogle Scholar
[19]Vietoris, L., “Über gewisse die unvollständige Betafunktion betreffende Ungleichungen”, Sirzungsber. Österreich. Akad. Wiss. Math.-Natur Kl. 191 (1982) 85–92.Google Scholar