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INEQUALITIES ASSOCIATED WITH RATIOS OF GAMMA FUNCTIONS

Part of: Elementary classical functions Computational number theory Convergence and divergence of infinite limiting processes

Published online by Cambridge University Press:  23 April 2018

JENICA CRINGANU Open the ORCID record for JENICA CRINGANU [Opens in a new window]
JENICA CRINGANU*
Affiliation:
Department of Mathematics and Computer Science, ‘Dunarea de Jos’ University of Galati, Domneasca No. 111, Galati, Romania email jcringanu@ugal.ro
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Abstract

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We use properties of the gamma function to estimate the products $\prod _{k=1}^{n}(4k-3)/4k$ and $\prod _{k=1}^{n}(4k-1)/4k$, motivated by the work of Chen and Qi [‘Completely monotonic function associated with the gamma function and proof of Wallis’ inequality’, Tamkang J. Math.36(4) (2005), 303–307] and Mortici et al. [‘Completely monotonic functions and inequalities associated to some ratio of gamma function’, Appl. Math. Comput.240 (2014), 168–174].

Keywords

gamma functionWallis ratioinequalitiescomplete monotonicity

MSC classification

Primary: 33B15: Gamma, beta and polygamma functions
Secondary: 11Y60: Evaluation of constants 40A05: Convergence and divergence of series and sequences
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 453 - 458
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1972).Google Scholar
Chen, C. P. and Qi, F., ‘Completely monotonic function associated with the gamma function and proof of Wallis’ inequality’, Tamkang J. Math. 36(4) (2005), 303–307.Google Scholar
Mortici, C., Cristea, V. G. and Lu, D., ‘Completely monotonic functions and inequalities associated to some ratio of gamma function’, Appl. Math. Comput. 240 (2014), 168–174.Google Scholar
Widder, D., The Laplace Transform (Princeton University Press, Princeton, NJ, 1941).Google Scholar