Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Remarks on a paper by Chao-Ping Chen and Feng Qi


Author: Stamatis Koumandos
Journal: Proc. Amer. Math. Soc. 134 (2006), 1365-1367
MSC (2000): Primary 33B15; Secondary 26D20
Published electronically: October 6, 2005
MathSciNet review: 2199181
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence $ P_{n}:=\frac{1.3\ldots (2n-1)}{2.4\ldots 2n}$. We show that their result follows easily from a theorem of G. N Watson published in 1959. We also show that the main result of Chen and Qi's paper is a special case of a more general inequality which admits a very short proof.


References [Enhancements On Off] (What's this?)

  • 1. George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
  • 2. Chao-Ping Chen and Feng Qi, The best bounds in Wallis’ inequality, Proc. Amer. Math. Soc. 133 (2005), no. 2, 397–401. MR 2093060, 10.1090/S0002-9939-04-07499-4
  • 3. Chao-Ping Chen and Feng Qi, Best upper and lower bounds in Wallis' inequality, J. Indones.  Math.  Soc. 12 (2006), to appear.
  • 4. Chao-Ping Chen and Feng Qi, Completely monotonic function associated with the gamma function and proof of Wallis' inequality, Tamkang J.  Math. 36, (2005), no. 4, to appear.
  • 5. Chao-Ping Chen and Feng Qi, The best bounds to $ \frac{(2n)!}{2^{2n}\,(n!)^2}$, Math.  Gaz. 88 (2004), 54-55.
  • 6. Chao-Ping Chen and Feng Qi, Improvement of lower bound in Wallis' inequality, RGMIA Res.  Rep.  Coll. 5 (2002), suppl., Art.  23. Available online at http://rgmia.vu.edu.au/v5(E).html.
  • 7. Chao-Ping Chen and Feng Qi, A new proof of the best bounds in Wallis' inequality, RGMIA Res.  Rep.  Coll. 6 (2003), no. 2, Art. 2. Available online at http://rgmia.vu.edu.au/v6n2.html.
  • 8. Neven Elezović, Carla Giordano, and Josip Pečarić, The best bounds in Gautschi’s inequality, Math. Inequal. Appl. 3 (2000), no. 2, 239–252. MR 1749300, 10.7153/mia-03-26
  • 9. S.  Koumandos and S.  Ruscheweyh, Positive Gegenbauer polynomial sums and applications to starlike functions, Constr. Approx., to appear.
  • 10. S.  Koumandos, An extension of Vietoris's inequalities, Ramanujan J., to appear.
  • 11. S. -L.  Qiu and M.  Vuorinen, Some properties of the gamma and psi functions, with applications, Math. Comp. 74 (2005), 723-742.
  • 12. Zoltán Sasvári, Inequalities for binomial coefficients, J. Math. Anal. Appl. 236 (1999), no. 1, 223–226. MR 1702663, 10.1006/jmaa.1999.6420
  • 13. Dušan V. Slavić, On inequalities for Γ(𝑥+1)/Γ(𝑥+1/2), Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 498-541 (1975), 17–20. MR 0385182
  • 14. Pantelimon Stănică, Good lower and upper bounds on binomial coefficients, JIPAM. J. Inequal. Pure Appl. Math. 2 (2001), no. 3, Article 30, 5 pp. (electronic). MR 1876263
  • 15. G. N. Watson, A note on Gamma functions, Proc. Edinburgh Math. Soc. (2) 11 (1958/1959), no. Edinburgh Math. Notes 42 (misprinted 41) (1959), 7–9. MR 0117358

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33B15, 26D20

Retrieve articles in all journals with MSC (2000): 33B15, 26D20


Additional Information

Stamatis Koumandos
Affiliation: Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
Email: skoumand@ucy.ac.cy

DOI: https://doi.org/10.1090/S0002-9939-05-08104-9
Keywords: Wallis' inequality, Gamma function, monotonicity, best bounds
Received by editor(s): September 15, 2004
Received by editor(s) in revised form: November 30, 2004
Published electronically: October 6, 2005
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2005 American Mathematical Society