## Banach Journal of Mathematical Analysis

### Complete monotonicity of a function involving the ratio of gamma functions and applications

#### Abstract

In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result of Guo and Qi [Taiwanese J. Math. 7 (2003), no. 2, 239--247] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.

#### Article information

Source
Banach J. Math. Anal., Volume 6, Number 1 (2012), 35-44.

Dates
First available in Project Euclid: 14 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1337014663

Digital Object Identifier
doi:10.15352/bjma/1337014663

Mathematical Reviews number (MathSciNet)
MR2862541

Zentralblatt MATH identifier
1245.33003

#### Citation

Qi, Feng; Wei, Chun-Fu; Guo, Bai-Ni. Complete monotonicity of a function involving the ratio of gamma functions and applications. Banach J. Math. Anal. 6 (2012), no. 1, 35--44. doi:10.15352/bjma/1337014663. https://projecteuclid.org/euclid.bjma/1337014663

#### References

• S. Abramovich, J. Barić, M. Matić and J. Pečarić, On van de Lune-Alzer's inequality, J. Math. Inequal. 1 (2007), no. 4, 563–587.
• H. Alzer, Inequalities for the volume of the unit ball in $\mathbb{R}^n$, J. Math. Anal. Appl. 252 (2000), 353–363.
• H. Alzer, Inequalities for the volume of the unit ball in $\mathbb{R}^n$, I\!I, Mediterr. J. Math. 5 (2008), 395–413.
• R.D. Atanassov and U.V. Tsoukrovski, Some properties of a class of logarithmically completely monotonic functions, C. R. Acad. Bulgare Sci. 41 (1988), no. 2, 21–23.
• G. Bennett, Meaningful inequalities, J. Math. Inequal. 1 (2007), no. 4, 449–471.
• C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433–439.
• C. Berg and H.L. Pedersen, A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in $n$-space, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2121–2132.
• C. Berg and H.L. Pedersen, A Pick function related to the sequence of volumes of the unit ball in $n$-space, Available online at http://arxiv.org/abs/0912.2185.
• C.-P. Chen, F. Qi, P. Cerone and S.S. Dragomir, Monotonicity of sequences involving convex and concave functions, Math. Inequal. Appl. 6 (2003), no. 2, 229–239.
• A.Z. Grinshpan and M.E.H. Ismail, Completely monotonic functions involving the gamma and $q$-gamma functions, Proc. Amer. Math. Soc. 134 (2006), 1153–1160.
• B.-N. Guo, R.-J. Chen and F. Qi, A class of completely monotonic functions involving the polygamma functions, J. Math. Anal. Approx. Theory 1 (2006), no. 2, 124–134.
• B.-N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc. 48 (2011), no. 3, 655–667.
• B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 2, 21–30.
• B.-N. Guo and F. Qi, An extension of an inequality for ratios of gamma functions, J. Approx. Theory 163 (2011), no. 9, 1208–1216.
• B.-N. Guo and F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2003), no. 2, 239–247.
• B.-N. Guo and F. Qi, Monotonicity of sequences involving geometric means of positive sequences with monotonicity and logarithmical convexity, Math. Inequal. Appl. 9 (2006), no. 1, 1–9.
• B.-N. Guo and F. Qi, Refinements of lower bounds for polygamma functions, Proc. Amer. Math. Soc. (2012), in press.
• B.-N. Guo and F. Qi, Some properties of the psi and polygamma functions, Hacet. J. Math. Stat. 39 (2010), 219–231.
• B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), no. 1, 103–111.
• B.-N. Guo, F. Qi and H.M. Srivastava, Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct. 21 (2010), no. 11, 103–111.
• R.A. Horn, On infinitely divisible matrices, kernels and functions, Z. Wahrscheinlichkeitstheorie und Verw. Geb 8 (1967), 219–230.
• D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.
• F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages.
• }.tex-soochow} F. Qi, Inequalities and monotonicity of sequences involving $\sqrt[n]{(n+k)!/k!}\,$, Soochow J. Math. 29 (2003), no. 4, 353–361.
• F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603–607.
• F. Qi and B.-N. Guo, A logarithmically completely monotonic function involving the gamma function, Taiwanese J. Math. 14 (2010), no. 4, 1623–1628.
• F. Qi and B.-N. Guo, An inequality involving the gamma and digamma functions, Available online at http://arxiv.org/abs/1101.4698.
• F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, 63–72.
• F. Qi and B.-N. Guo, Monotonicity of sequences involving convex function and sequence, Math. Inequal. Appl. 9 (2006), no. 2, 247–254.
• F. Qi and B.-N. Guo, Necessary and sufficient conditions for a function involving a ratio of gamma functions to be logarithmically completely monotonic, Available online at http://arxiv.org/abs/0904.1101.
• F. Qi and B.-N. Guo, Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic, Adv. Appl. Math. 44 (2010), no. 1, 71–83.
• F. Qi and B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (2010), no. 6, 1283–1297.
• F. Qi and B.-N. Guo, Monotonicity and logarithmic convexity relating to the volume of the unit ball, Available online at http://arxiv.org/abs/0902.2509.
• F. Qi, B.-N. Guo and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), 81–88.
• F. Qi, B.-N. Guo and C.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, 31–36.
• F. Qi and S. Guo, On a new generalization of Martins' inequality, J. Math. Inequal. 1 (2007), no. 4, 503–514.
• F. Qi, S. Guo and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149–2160.
• F. Qi, W. Li and B.-N. Guo, Generalizations of a theorem of I. Schur, RGMIA Res. Rep. Coll. 9 (2006), no. 3, Art. 15.
• F. Qi and Q.-M. Luo, Generalization of H. Minc and L. Sathre's inequality, Tamkang J. Math. 31 (2000), no. 2, 145–148.
• F. Qi and J.-S. Sun, A monotonicity result of a function involving the gamma function, Anal. Math. 32 (2006), no. 4, 279–282.
• R.L. Schilling, R. Song and Z. Vondraček, Bernstein Functions, de Gruyter Studies in Mathematics 37, De Gruyter, Berlin, Germany, 2010.
• D.V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
• Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. 352 (2009), no. 2, 967–970.
• T.-H. Zhao, Y.-M. Chu and Y.-P. Jiang, Monotonic and logarithmically convex properties of a function involving gamma functions, J. Inequal. Appl. 2009 (2009), Article ID 728612, 13 pages.