Mathematics of Computation

Author(s): Horst Alzer.
Journal: Math. Comp. 66 (1997), 373-389.
MSC (1991): Primary 33B15; Secondary 26D07
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Abstract: We present new inequalities for the gamma and psi functions, and we provide new classes of completely monotonic, star-shaped, and super-additive functions which are related to $\Gamma$ and $\psi$ .

1.: M. Abramowitz and I. A. Stegun, eds., Handbook of mathematical functions with formulas, graphs and mathematical tables, Dover, New York, 1965. MR 31:1400
2.: H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), 337-346. MR 93f:33001
3.: G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc. 347 (1995), 1713-1723. MR 95m:33002
4.: T. M. Apostol, Introduction to analytic number theory, Springer, New York, 1976. MR 55:7892
5.: S. Bochner, Harmonic analysis and the theory of probability, Univ. of California Press, Berkeley/Los Angeles, 1955. MR 17:273d
6.: A. V. Boyd, Gurland's inequality for the gamma function, Skand. Aktuarietidskr. 1960 (1961), 134-135. MR 24:A2058
7.: -, Note on a paper by Uppuluri, Pacific J. Math. 22 (1967), 9-10. MR 35:6872
8.: A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1962), 1203-1215. MR 26:6326
9.: J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), 659-667. MR 87m:33002
10.: P. J. Davis, Leonhard Euler's integral: A historical profile of the gamma function, Amer. Math. Monthly 66 (1959), 849-869. MR 21:5540
11.: W. A. Day, On monotonicity of the relaxation functions of viscoelastic materials, Proc. Cambridge Philos. Soc. 67 (1970), 503-508. MR 40:3779
12.: C. J. Eliezer and D. E. Daykin, Generalizations and applications of Cauchy-Schwarz inequalities, Quart. J. Math. Oxford Ser. (2) 18 (1967), 357-360. MR 37:1541
13.: T. Erber, The gamma function inequalities of Gurland and Gautschi, Skand. Aktuarietidskr. 1960 (1961), 27-28. MR 24:A2682
14.: A. Erdélyi, ed., Higher transcendental functions, vol. 1, McGraw-Hill, New York, 1953. MR 15:419i
15.: W. Feller, An introduction to probability theory and its applications, Vol. 2, Wiley, New York, 1966. MR 35:1048
16.: G. M. Fichtenholz, Differential- und Integralrechnung II, Dt. Verlag Wiss., Berlin, 1978. MR 80f:26001
17.: A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), 251-258. MR 84e:26017
18.: W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959), 77-81. MR 21:2067
19.: -, A harmonic mean inequality for the gamma function, SIAM J. Math. Anal. 5 (1974), 278-281. MR 50:2570
20.: -, Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282-292. MR 50:2571
21.: D. V. Gokhale, On an inequality for gamma functions, Skand. Aktuarietidskr. 1962 (1963), 213-215. MR 28:4151
22.: L. Gordon, A stochastic approach to the gamma function, Amer. Math. Monthly 101 (1994), 858-865. MR 95k:33003
23.: J. Gurland, An inequality satisfied by the gamma function, Skand. Aktuarietidskr. 39 (1956), 171-172. MR 20:1797
24.: M. E. H. Ismail, L. Lorch, and M. E. Muldoon, Completely monotonic functions associated with the gamma function and its q-analogues, J. Math. Anal. Appl. 116 (1986), 1-9. MR 88b:33002
25.: J. D. Ke\v{c}ki\'{c} and P. M. Vasi\'{c}, Some inequalities for the gamma function, Publ. Inst. Math. (Beograd) (N.S.) 11 (1971), 107-114. MR 46:7560
26.: D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), 607-611. MR 84m:33003
27.: D. Kershaw and A. Laforgia, Monotonicity results for the gamma function, Atti Accad. Sci. Torino 119 (1985), 127-133. MR 87i:33006
28.: C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Math. 10 (1974), 152-164. MR 50:5899
29.: A. Laforgia, Further inequalities for the gamma function, Math. Comp. 42 (1984), 597-600. MR 85i:33001
30.: I. B. Lazarevi\'{c} and A. Lupas, Functional equations for Wallis and gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Electr. Telec. Autom. 461-497 (1979), 245-251. MR 50:13631
31.: L. Lorch, Inequalities for ultraspherical polynomials and the gamma function, J. Approx. Theory 40 (1984), 115-120. MR 85d:33024
32.: L. G. Lucht, Mittelwertungleichungen für Lösungen gewisser Differenzengleichungen, Aequationes Math. 39 (1990), 204-209. MR 91h:39004
33.: Y. L. Luke, Inequalities for the gamma function and its logarithmic derivative, Math. Balkanica 2 (1972), 118-123. MR 50:10338
34.: W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Springer, Berlin, 1966. MR 38:1291
35.: L. Maligranda, J. E. Pe\v{c}ari\'{c}, and L. E. Persson, Stolarsky's inequality with general weights, Proc. Amer. Math. Soc. 123 (1995), 2113-2118. MR 95i:26026
36.: A. W. Marshall and I. Olkin, Inequalities: Theory of majorization and its applications, Academic Press, New York, 1979. MR 81b:00002
37.: H. Minc and L. Sathre, Some inequalities involving $(r!)^{1/r}$ , Edinburgh Math. Soc. 14 (1964/65), 41-46. MR 29:55
38.: D. S. Mitrinovi\'{c}, Analytic inequalities, Springer, New York, 1970. MR 43:448
39.: M. E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978), 54-63. MR 58:11536
40.: I. Olkin, An inequality satisfied by the gamma function, Skand. Aktuarietidskr. 1958 (1959), 37-39. MR 21:4257
41.: H. Ruben, Variance bounds and orthogonal expansions in Hilbert space with an application to inequalities for gamma functions and $\pi$ , J. Reine Angew. Math. 225 (1967), 147-153. MR 35:421
42.: J. Sándor, Sur la fonction gamma, Publ. C.R.M.P. Neuchâtel, Sér. I, 21 (1989), 4-7.
43.: E. Schmidt, Über die Ungleichung, welche die Integrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet, Math. Ann. 117 (1940), 301-326. MR 2:218e
44.: J. B. Selliah, An inequality satisfied by the gamma function, Canad. Math. Bull. 19 (1976), 85-87. MR 54:3050
45.: D. V. Slavi\'{c}, On inequalities for $\Gamma (x+1)/\Gamma (x+1/2)$ , Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 498-541 (1975), 17-20. MR 52:6047
46.: K. B. Stolarsky, From Wythoff's Nim to Chebyshev's inequality, Amer. Math. Monthly 98 (1991), 889-900. MR 93b:90132
47.: S. Y. Trimble, J. Wells, and F. T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal. 20 (1989), 1255-1259. MR 91a:26019
48.: D. V. Widder, The Laplace transform, Princeton Univ. Press, Princeton, 1941. MR 3:232d
49.: J. Wimp, Sequence transformations and their applications, Academic Press, New York, 1981. MR 84e:65005
50.: S. Zimering, On a Mercerian theorem and its application to the equiconvergence of Cesàro and Riesz transforms, Publ. Inst. Math. (Beograd) (N.S.) 1 (1962), 83-91. MR 31:543

Horst Alzer
Affiliation: Morsbacher Str. 10, 51545 Waldbröl, Germany

DOI: 10.1090/S0025-5718-97-00807-7
PII: S 0025-5718(97)00807-7
Keywords: Gamma function, psi function, complete monotonicity, inequalities, star-shaped functions, super-additive functions, infinite divisibility, Laplace transform.
Received by editor(s): October 13, 1995
Received by editor(s) in revised form: March 4, 1996
Copyright of article: Copyright 1997, American Mathematical Society

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