A class of completely mixed monotonic functions involving the gamma function with applications

Yang, Zhen-Hang; Tian, Jing-Feng

doi:10.1090/proc/14199

A class of completely mixed monotonic functions involving the gamma function with applications
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by Zhen-Hang Yang and Jing-Feng Tian PDF

Proc. Amer. Math. Soc. 146 (2018), 4707-4721 Request permission

Abstract:

In this paper, we introduce the notion of completely mixed monotonicity of a function of several variables, very few of which have appeared. We give a necessary and sufficient condition for a function constructed by ratios of gamma functions to be completely mixed monotonic. From this, some new inequalities for gamma, psi, and polygamma functions are derived.

References

Serge Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), no. 1, 1–66 (French). MR 1555269, DOI 10.1007/BF02547400
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
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. MR 939205
F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (1) (2004), Art. 8, 63–72, http://rgmia.org/v7n1.php.
Salomon Bochner, Harmonic analysis and the theory of probability, University of California Press, Berkeley-Los Angeles, Calif., 1955. MR 0072370
Mourad E. H. Ismail, Lee Lorch, and Martin E. Muldoon, Completely monotonic functions associated with the gamma function and its $q$-analogues, J. Math. Anal. Appl. 116 (1986), no. 1, 1–9. MR 837337, DOI 10.1016/0022-247X(86)90042-9
Joaquin Bustoz and Mourad E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), no. 176, 659–667. MR 856710, DOI 10.1090/S0025-5718-1986-0856710-6
M. Merkle, Gurland’s ratio for the gamma function, Comput. Math. Appl. 49 (2005), no. 2-3, 389–406. MR 2123415, DOI 10.1016/j.camwa.2004.01.016
Feng Qi, Qiao Yang, and Wei Li, Two logarithmically completely monotonic functions connected with gamma function, Integral Transforms Spec. Funct. 17 (2006), no. 7, 539–542. MR 2242416, DOI 10.1080/10652460500422379
Feng Qi, Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct. 18 (2007), no. 7-8, 503–509. MR 2348595, DOI 10.1080/10652460701358976
F. Qi and B.-N. Guo, Wendel-Gautschi-Kershaw’s inequalities and sufficient and necessary conditions that a class of functions involving ratio of gamma functions are logarithmically completely monotonic, RGMIA Res. Rep. Coll. 10 (1) (2007), Art. 2; Available online at http://www.staff.vu.edu.au/rgmia/v10n1.asp.
Feng Qi, A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw’s double inequality, J. Comput. Appl. Math. 206 (2007), no. 2, 1007–1014. MR 2333728, DOI 10.1016/j.cam.2006.09.005
Feng Qi and Bai-Ni Guo, Wendel’s and Gautschi’s inequalities: refinements, extensions, and a class of logarithmically completely monotonic functions, Appl. Math. Comput. 205 (2008), no. 1, 281–290. MR 2466631, DOI 10.1016/j.amc.2008.07.005
Feng Qi and Bai-Ni Guo, A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality, J. Comput. Appl. Math. 212 (2008), no. 2, 444–456. MR 2383033, DOI 10.1016/j.cam.2006.12.022
Feng Qi, A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi-Kershaw’s inequality, J. Comput. Appl. Math. 224 (2009), no. 2, 538–543. MR 2492886, DOI 10.1016/j.cam.2008.05.030
Feng Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. , posted on (2010), Art. ID 493058, 84. MR 2611044, DOI 10.1155/2010/493058
Feng Qi and Qiu-Ming Luo, Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem, J. Inequal. Appl. , posted on (2013), 2013:542, 20. MR 3339476, DOI 10.1186/1029-242x-2013-542
F. Qi, Bounds for the ratio of two gamma functions: from Gautschi’s and Kershaw’s inequalities to complete monotonicity, Turkish J. Anal. Number Theory 2 (5) (2014), 152–164, DOI:10.12691/tjant-2-5-1.
Zhen-Hang Yang and Yu-Ming Chu, Monotonicity and absolute monotonicity for the two-parameter hyperbolic and trigonometric functions with applications, J. Inequal. Appl. , posted on (2016), Paper No. 200, 10. MR 3538004, DOI 10.1186/s13660-016-1143-8
Zhen-Hang Yang and Shen-Zhou Zheng, Monotonicity of a mean related to polygamma functions with an application, J. Inequal. Appl. , posted on (2016), Paper No. 216, 10. MR 3545603, DOI 10.1186/s13660-016-1155-4
Zhen-Hang Yang and Shen-Zhou Zheng, Complete monotonicity involving some ratios of gamma functions, J. Inequal. Appl. , posted on (2017), Paper No. 255, 17. MR 3711629, DOI 10.1186/s13660-017-1527-4
Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948), 563–564. MR 29448, DOI 10.2307/2304460
Walter Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. and Phys. 38 (1959/60), 77–81. MR 103289, DOI 10.1002/sapm195938177
Jovan D. Kečkić and Petar M. Vasić, Some inequalities for the gamma function, Publ. Inst. Math. (Beograd) (N.S.) 11(25) (1971), 107–114. MR 308446
D. Kershaw, Some extensions of W. Gautschi’s inequalities for the gamma function, Math. Comp. 41 (1983), no. 164, 607–611. MR 717706, DOI 10.1090/S0025-5718-1983-0717706-5
Lee Lorch, Inequalities for ultraspherical polynomials and the gamma function, J. Approx. Theory 40 (1984), no. 2, 115–120. MR 732692, DOI 10.1016/0021-9045(84)90020-0
Andrea Laforgia, Further inequalities for the gamma function, Math. Comp. 42 (1984), no. 166, 597–600. MR 736455, DOI 10.1090/S0025-5718-1984-0736455-1
Horst Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337–346. MR 1149288, DOI 10.1090/S0025-5718-1993-1149288-7
Neven Elezović, Carla Giordano, and Josip Pe arić, The best bounds in Gautschi’s inequality, Math. Inequal. Appl. 3 (2000), no. 2, 239–252. MR 1749300, DOI 10.7153/mia-03-26
Necdet Batir, Some new inequalities for gamma and polygamma functions, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 103, 9. MR 2178284
Feng Qi, Bai-Ni Guo, and Chao-Ping Chen, The best bounds in Gautschi-Kershaw inequalities, Math. Inequal. Appl. 9 (2006), no. 3, 427–436. MR 2242773, DOI 10.7153/mia-09-41
Feng Qi, A new lower bound in the second Kershaw’s double inequality, J. Comput. Appl. Math. 214 (2008), no. 2, 610–616. MR 2398355, DOI 10.1016/j.cam.2007.03.016
Feng Qi, Xiao-Ai Li, and Shou-Xin Chen, Refinements, extensions and generalizations of the second Kershaw’s double inequality, Math. Inequal. Appl. 11 (2008), no. 3, 457–465. MR 2431210, DOI 10.7153/mia-11-35
M. Merkle, Inequalities for the gamma function via convexity, in Advances in Inequalities for Special Functions, P. Cerone and S. S. Dragomir, Eds., pp. 81–100, Nova Science, New York, NY, USA, 2008.
G. J. O. Jameson, Inequalities for gamma function ratios, Amer. Math. Monthly 120 (2013), no. 10, 936–940. MR 3139572, DOI 10.4169/amer.math.monthly.120.10.936
Zhen-Hang Yang and Jing-Feng Tian, Monotonicity and inequalities for the gamma function, J. Inequal. Appl. , posted on (2017), Paper No. 317, 15. MR 3740576, DOI 10.1186/s13660-017-1591-9
Zhen-Hang Yang and Jingfeng Tian, Monotonicity and sharp inequalities related to gamma function, J. Math. Inequal. 12 (2018), no. 1, 1–22. MR 3771780, DOI 10.7153/jmi-2018-12-01
D. S. Mitrinović and I. B. Lacković, Hermite and convexity, Aequationes Math. 28 (1985), no. 3, 229–232. MR 791622, DOI 10.1007/BF02189414