On some inequalities for the gamma and psi functions

Alzer, Horst

doi:10.1090/S0025-5718-97-00807-7

On some inequalities for the gamma and psi functions
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Math. Comp. 66 (1997), 373-389 Request permission

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 $\psi$.

References

Milton Abramowitz (ed.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1965. Superintendent of Documents. MR 0177136
Horst Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), no. 201, 337–346. MR 1149288, DOI 10.1090/S0025-5718-1993-1149288-7
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), no. 5, 1713–1723. MR 1264800, DOI 10.1090/S0002-9947-1995-1264800-3
Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
A. V. Boyd, Gurland’s inequality for the gamma function, Skand. Aktuarietidskr. 1960 (1960), 134–135 (1961). MR 132212, DOI 10.1080/03461238.1960.10410585
A. V. Boyd, Note on a paper by Uppuluri, Pacific J. Math. 22 (1967), 9–10. MR 216037
A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1962), 1203–1215. MR 148822
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
Philip J. Davis, Leonhard Euler’s integral: A historical profile of the gamma function, Amer. Math. Monthly 66 (1959), 849–869. MR 106810, DOI 10.2307/2309786
W. A. Day, On monotonicity of the relaxation functions of viscoelastic materials, Proc. Cambridge Philos. Soc. 67 (1970), 503–508. MR 250545, DOI 10.1017/s0305004100045771
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 225951, DOI 10.1093/qmath/18.1.357
Thomas Erber, The gamma function inequalities of Gurland and Gautschi, Skand. Aktuarietidskr. 1960 (1960), 27–28 (1961). MR 132846, DOI 10.1080/03461238.1960.10410596
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
G. M. Fichtenholz, Differential- und Integralrechnung. II, 7th ed., Hochschulbücher für Mathematik [University Books for Mathematics], vol. 62, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978 (German). Translated from the Russian by Brigitte Mai and Walter Mai. MR 524565
A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), no. 1, 251–258. MR 680877, DOI 10.1016/0022-247X(82)90057-9
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
Walter Gautschi, A harmonic mean inequality for the gamma function, SIAM J. Math. Anal. 5 (1974), 278–281. MR 350077, DOI 10.1137/0505030
Walter Gautschi, Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282–292. MR 350078, DOI 10.1137/0505031
D. V. Gokhale, On an inequality for gamma functions, Skand. Aktuarietidskr. 1962 (1963), 213–215 (1963). MR 160942, DOI 10.1080/03461238.1962.10405948
Louis Gordon, A stochastic approach to the gamma function, Amer. Math. Monthly 101 (1994), no. 9, 858–865. MR 1300491, DOI 10.2307/2975134
John Gurland, An inequality satisfied by the Gamma function, Skand. Aktuarietidskr. 39 (1956), 171–172. MR 95293, DOI 10.1080/03461238.1956.10414947
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
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
Donald Kershaw and Andrea Laforgia, Monotonicity results for the gamma function, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 119 (1985), no. 3-4, 127–133 (1986) (English, with Italian summary). MR 845134
Clark H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Math. 10 (1974), 152–164. MR 353416, DOI 10.1007/BF01832852
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
Ilija B. Lazarević and Alexandru Lupaş, Functional equations for Wallis and gamma functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 461-497 (1974), 245–251. MR 0361185
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
Lutz G. Lucht, Mittelwertungleichungen für Lösungen gewisser Differenzengleichungen, Aequationes Math. 39 (1990), no. 2-3, 204–209 (German, with English summary). MR 1045668, DOI 10.1007/BF01833151
Yudell L. Luke, Inequalities for the gamma function and its logarithmic derivative, Math. Balkanica 2 (1972), 118–123. MR 357872
Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968
Lech Maligranda, Josip E. Pečarić, and Lars Erik Persson, Stolarsky’s inequality with general weights, Proc. Amer. Math. Soc. 123 (1995), no. 7, 2113–2118. MR 1243171, DOI 10.1090/S0002-9939-1995-1243171-8
Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 552278
Henryk Minc and Leroy Sathre, Some inequalities involving $(r!)^{1/r}$, Proc. Edinburgh Math. Soc. (2) 14 (1964/65), 41–46. MR 162751, DOI 10.1017/S0013091500011214
D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686
Martin E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aequationes Math. 18 (1978), no. 1-2, 54–63. MR 492418, DOI 10.1007/BF01844067
Ingram Olkin, An inequality satisfied by the gamma function, Skand. Aktuarietidskr. 1958 (1958), 37–39 (1959). MR 105517, DOI 10.1080/03461238.1958.10405978
Harold 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 209524, DOI 10.1515/crll.1967.225.147
J. Sándor, Sur la fonction gamma, Publ. C.R.M.P. Neuchâtel, Sér. I, 21 (1989), 4–7.
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
J. B. Selliah, An inequality satisfied by the gamma function, Canad. Math. Bull. 19 (1976), no. 1, 85–87. MR 414958, DOI 10.4153/CMB-1976-011-8
Dušan V. Slavić, On inequalities for $\Gamma (x+1)/\Gamma (x+1/2)$, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 498-541 (1975), 17–20. MR 0385182
Kenneth B. Stolarsky, From Wythoff’s Nim to Chebyshev’s inequality, Amer. Math. Monthly 98 (1991), no. 10, 889–900. MR 1137536, DOI 10.2307/2324146
S. Y. Trimble, Jim Wells, and F. T. Wright, Superadditive functions and a statistical application, SIAM J. Math. Anal. 20 (1989), no. 5, 1255–1259. MR 1009357, DOI 10.1137/0520082
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Jet Wimp, Sequence transformations and their applications, Mathematics in Science and Engineering, vol. 154, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 615250
Shimshon Zimering, On a Mercerian theorem and its application to the equiconvergence of Cesàro and Riesz transforms, Publ. Inst. Math. (Beograd) (N.S.) 1(15) (1962), 83–91 (1962). MR 176268