Hostname: page-component-7bb8b95d7b-dtkg6 Total loading time: 0 Render date: 2024-09-12T09:22:40.702Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Completely and Absolutely Monotone Functions

Published online by Cambridge University Press:  20 November 2018

Arvind Mahajan  and
Dieter K. Ross
Arvind Mahajan
Affiliation:
Department of Applied Mathematics La Trobe University, Bundoora, 3083, Australia
Dieter K. Ross
Affiliation:
Department of Applied Mathematics La Trobe University, Bundoora, 3083, Australia
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.

The solutions of a certain class of first order linear differential equations are shown to be either completely or absolutely monotone depending on the nature of its coefficients. This is a simple theorem which is used to deduce a number of new and interesting results dealing with the complete and absolute monotonicity of functions. In particular, a partial answer is supplied to a question posed by Askey and Pollard: “When is completely monotone?”

Keywords

26D1534A1033A1533A40Completely monotoneabsolutely monotoneInequalitiesBessel functionsgamma functions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 2 , 01 June 1982 , pp. 143 - 148
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Askey, R., Summability of Jacobi series, Trans. Amer. Math. Soc. 179 (1973), pp. 71-84.Google Scholar
2. Askey, R.. and Pollard, H., Some absolutely monotonie and completely monotonie functions, SIAM J. Math. Anal. 5 (1974), pp. 58-63.Google Scholar
3. Fields, J. L. and Ismail, M., On some conjectures of Askey concerning completely monotonie functions, Spline Functions and Approximation Theory, edited by Meir, A. and Sharma, A., ISNM Vol. 21, Birkhauser Verlag, Basel, 1973, pp. 101-111.Google Scholar
4. Fields, J. L. and Ismail, M. E.-H., On the positivity of some 1F2's, SIAM J. Math. Anal., 6 (1975), pp. 551-559.Google Scholar
5. Gasper, G., Nonnegative sums of cosine, ultraspherical and Jacobi polynomials, J. Math. Anal. Appl. 26 (1969), pp. 60-68.Google Scholar
6. Muldoon, M. E., Some monotonicity properties and characterisations of the gamma function, Aequationes Math. 18 (1978), pp. 54-63.Google Scholar
7. Watson, G. N., A treatise on the theory of Bessel functions, Cambridge Univ. Press, 1966.Google Scholar
8. Widder, D. V., Laplace Transforms, Princeton University Press, Princeton, N.J., 1946.Google Scholar