Advertisement

Results in Mathematics

, 74:44 | Cite as

Inequalities and Asymptotic Expansions Related to the Volume of the Unit Ball in \(\pmb {\mathbb {R}}^{{{\varvec{n}}}}\)

  • Chao-Ping ChenEmail author
  • Richard B. Paris
Article
  • 6 Downloads

Abstract

Let \(\Omega _{n}=\pi ^{n/2}/\Gamma (\frac{n}{2}+1) \, (n \in \mathbb {N})\) denote the volume of the unit ball in \(\mathbb {R}^{n}\). In this paper, we present asymptotic expansions and inequalities related to \(\Omega _{n}\) and the quantities:
$$\begin{aligned} \frac{\Omega _{n-1}}{\Omega _{n}}, \quad \frac{\Omega _{n}}{\Omega _{n-1}+\Omega _{n+1}} \quad \text {and}\quad \frac{\Omega _n^{1/n}}{\Omega _{n+1}^{1/(n+1)}}. \end{aligned}$$

Keywords

Volume of the unit n-dimensional ball gamma function asymptotic expansions inequalities 

Mathematics Subject Classification

Primary 33B15 Secondary 41A60 26D15 

Notes

References

  1. 1.
    Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alzer, H.: Inequalities for the volume of the unit ball in \(\mathbb{R}^{n}\). J. Math. Anal. Appl. 252, 353–363 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alzer, H.: Inequalities for the volume of the unit ball in \(\mathbb{R}^{n}\). II. Mediterr. J. Math. 5, 395–413 (2008)CrossRefGoogle Scholar
  4. 4.
    Anderson, G.D., Qiu, S.-L.: A monotoneity property of the gamma function. Proc. Am. Math. Soc. 125, 3355–3362 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Special functions of quasiconformal theory. Expo. Math. 7, 97–136 (1989)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ban, T., Chen, C.-P.: New inequalities for the volume of the unit ball in \({\mathbb{R}}^{n}\). J. Math. Inequalities 11, 527–542 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Böhm, J., Hertel, E.: Polyedergeometrie in n-dimensionalen Räumen konstanter Krmmung. Birkhäuser, Basel (1981)zbMATHGoogle Scholar
  8. 8.
    Borgwardt, K.H.: The Simplex Method. Springer, Berlin (1987)CrossRefGoogle Scholar
  9. 9.
    Chen, C.-P.: Inequalities and asymptotic expansions associated with the Ramanujan and Nemes formulas for the gamma function. Appl. Math. Comput. 261, 337–350 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, C.-P., Elezović, N., Vukšć, L.: Asymptotic formulae associated with the Wallis power function and digamma function. J. Class. Anal. 2, 151–166 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, C.-P., Lin, L.: Inequalities for the volume of the unit ball in \(\mathbb{R}^{n}\). Mediterr. J. Math. 11, 299–314 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, C.-P., Paris, R.B.: Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function. Appl. Math. Comput. 250, 514–529 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dubourdieu, J.: Sur un théorème de M. S. Bernstein relatif \(\grave{a}\) la transformation de Laplace-Stieltjes. Compos. Math. 7, 96–111 (1939). (in French) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Guo, B.-N., Qi, F.: Monotonicity and logarithmic convexity relating to the volume of the unit ball. Optim. Lett. 7, 1139–1153 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    van Haeringen, H.: Completely monotonic and related functions. J. Math. Anal. Appl. 204, 389–408 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Klain, D.A., Rota, G.-C.: A continuous analogue of Sperner’s theorem. Commun. Pure Appl. Math. 50, 205–223 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Koumandos, S.: Remarks on some completely monotonic functions. J. Math. Anal. Appl. 324, 1458–1461 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Koumandos, S., Pedersen, H.L.: Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function. J. Math. Anal. Appl. 355, 33–40 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lu, D., Zhang, P.: A new general asymptotic formula and inequalities involving the volume of the unit ball. J. Number Theory 170, 302–314 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Luke, Y.L.: The Special Functions and their Approximations, vol. I. Academic Press, New York (1969)zbMATHGoogle Scholar
  21. 21.
    Merkle, M.: Gurland’s ratio for the gamma function. Comput. Math. Appl. 49, 389–406 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mortici, C.: Monotonicity properties of the volume of the unit ball in \(\mathbb{R}^{n}\). Optim. Lett. 4, 457–464 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mortici, C.: Estimates of the function and quotient by minc-Sathre. Appl. Math. Comput. 253, 52–60 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mortici, C.: Series associated to some expressions involving the volume of the unit ball and applications. Appl. Math. Comput. 294, 121–138 (2017)MathSciNetGoogle Scholar
  25. 25.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  26. 26.
    Paris, R.B., Kaminski, D.: Asymptotics and Mellin–Barnes Integrals. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  27. 27.
    Temme, N.M.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York (1996)CrossRefGoogle Scholar
  28. 28.
    Yin, L., Huang, L.-G.: Some inequalities for the volume of the unit ball. J. Class. Anal. 6, 39–46 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoChina
  2. 2.Division of Computing and MathematicsAbertay UniversityDundeeUK

Personalised recommendations