참고문헌
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Third printing, with corrections. National Bureau of Standards Applied Mathematics Series, 55 Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1965.
- H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373-389. https://doi.org/10.1090/S0025-5718-97-00807-7
- H. Alzer and C. Berg, Some classes of completely monotonic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 445-460.
- H. Alzer and C. Berg, Some classes of completely monotonic functions. II, Ramanujan J. 11 (2006), no. 2, 225-248. https://doi.org/10.1007/s11139-006-6510-5
- 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.
- C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433-439. https://doi.org/10.1007/s00009-004-0022-6
- C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer-Verlag, New York-Heidelberg, 1975.
- C. Berg and H. L. Pedersen, Pick functions related to the gamma function, Rocky Mountain J. Math. 32 (2002), no. 2, 507-525. https://doi.org/10.1216/rmjm/1030539684
- S. Bochner, Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley and Los Angeles, 1955.
- P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics, 97. Longman, Harlow, 1998.
- W. E. Clark and M. E. H. Ismail, Inequalities involving gamma and psi functions, Anal. Appl. (Singap.) 1 (2003), no. 1, 129-140. https://doi.org/10.1142/S0219530503000041
- M. J. Dubourdieu, Sur un theoreme de M. S. Bernstein relatif a la transformation de Laplace-Stieltjes, Compositio Math. 7 (1939), 96-111.
- A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.
- I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series and Products, 5th Edition, Academic Press, 1994.
- A. Z. Grinshpan and M. E. H. Ismail, Completely monotonic functions involving the gamma and q-gamma functions, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1153-1160. https://doi.org/10.1090/S0002-9939-05-08050-0
- R. A. Horn, On infinitely divisible matrices, kernels, and functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 219-230. https://doi.org/10.1007/BF00531524
- D. Kershaw and A. Laforgia, Monotonicity results for the gamma function, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 119 (1985), no. 3-4, 127-133.
- J.-Ch. Kuang, Applied Inequalities, Second edition. With a preface by Shan Zhen Lu. Hunan Jiaoyu Chubanshe, Changsha, 1993.
- J.-Ch. Kuang, Applied Inequalities, Third edition. Shandong Science and Technology Press, Ji’nan City, Shandong Province, China, 2004.
- W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, Berlin, 1966.
- K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transform. Spec. Funct. 12 (2001), no. 4, 389-402. https://doi.org/10.1080/10652460108819360
-
H. Minc and L. Sathre, Some inequalities involving
$(r!)^{1/r}$ , Proc. Edinburgh Math. Soc. (2) 14 (1964/1965), 41-46. https://doi.org/10.1017/S0013091500011214 - D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers Group, Dordrecht, 1993.
- J. Pecaric, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.
- F. Qi, Monotonicity results and inequalities for the gamma and incomplete gamma functions, Math. Inequal. Appl. 5 (2002), no. 1, 61-67.
- F. Qi, J. Cao, D.-W. Niu, and N. Ujevic, An upper bound of a function with two independent variables, Appl. Math. E-Notes 6 (2006), 148-152.
- F. Qi and Ch.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), no. 2, 603-607. https://doi.org/10.1016/j.jmaa.2004.04.026
- F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8, 63-72; Available online at http://rgmia.org/v7n1.php.
- F. Qi and B.-N. Guo, Monotonicity and convexity of ratio between gamma functions to different powers, J. Indones. Math. Soc. 11 (2005), no. 1, 39-49.
- F. Qi and B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, Available online at http://arxiv.org/abs/0903.5123.
- F. Qi, B.-N. Guo, and Ch.-P. Chen, The best bounds in Gautschi-Kershaw inequalities, Math. Inequal. Appl. 9 (2006), no. 3, 427-436.
- F. Qi, B.-N. Guo, and Ch.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 5, 31-36; Available online at http://rgmia.org/v7n1.php.
- F. Qi, B.-N. Guo, and Ch.-P. Chen, Some completely monotonic functions involving the gamma and polygamma functions, J. Aust. Math. Soc. 80 (2006), no. 1, 81-88. https://doi.org/10.1017/S1446788700011393
- F. Qi and S. Guo, On a new generalization of Martins’ inequality, J. Math. Inequal. 1 (2007), no. 4, 503-514.
- F. Qi, D.-W. Niu, and J. Cao, An infimum and an upper bound of a function with two independent variables, Octogon Math. Mag. 14 (2006), no. 1, 248-250.
- J. Sandor, Sur la fonction gamma, Publ. C. R. M. P. Neuchatel, Serie I 21 (1989), 4-7.
- H. van Haeringen, Completely Monotonic and Related Functions, Report 93-108, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands, 1993.
-
H. Vogt and J. Voigt, A monotonicity property of the
${\Gamma}-function$ , J. Inequal. Pure Appl. Math. 3 (2002), no. 5, Article 73, 3 pp.; Available online at http://jipam.vu.edu.au/article.php?sid=225. - Zh.-X. Wang and D.-R. Guo, Special Functions, Translated from the Chinese by Guo and X. J. Xia. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
- Zh.-X. Wang and D.-R. Guo, A Panorama of Special Functions, The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000.
- D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.
피인용 문헌
- Complete monotonicity of a function involving the divided difference of digamma functions vol.56, pp.11, 2013, https://doi.org/10.1007/s11425-012-4562-0
- Completely monotonic functions and inequalities associated to some ratios of the Pochhammer k-symbol vol.432, pp.1, 2015, https://doi.org/10.1016/j.jmaa.2015.06.055
- Complete Monotonicity of Functions Connected with the Exponential Function and Derivatives vol.2014, 2014, https://doi.org/10.1155/2014/851213
- Monotonicity and logarithmic convexity relating to the volume of the unit ball vol.7, pp.6, 2013, https://doi.org/10.1007/s11590-012-0488-2
- A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS vol.48, pp.3, 2011, https://doi.org/10.4134/JKMS.2011.48.3.655
- Some exact constants for the approximation of the quantity in the Wallis’ formula vol.2013, pp.1, 2013, https://doi.org/10.1186/1029-242X-2013-67
- Complete monotonicity of some functions involving polygamma functions vol.233, pp.9, 2010, https://doi.org/10.1016/j.cam.2009.09.044
- Monotonicity, concavity, and inequalities related to the generalized digamma function vol.2018, pp.1, 2018, https://doi.org/10.1186/s13662-018-1695-7