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Volumn 66, Issue 217, 1997, Pages 373-389

On some inequalities for the gamma and psi functions

(1)  Alzer, Horst a  

a NONE   (Germany)

Author keywords

Complete nonotonicity; Gamma function; Inequalities; Infinite divisibility; Laplace transform; Psi function; Star shaped functions; Super additive functions

Indexed keywords


EID: 0042376839     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-97-00807-7     Document Type: Article
Times cited : (384)

References (50)
  • 2
    • 84968470274 scopus 로고
    • Some gamma function inequalities
    • MR 93f:33001
    • H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), 337-346. MR 93f:33001
    • (1993) Math. Comp. , vol.60 , pp. 337-346
    • Alzer, H.1
  • 6
    • 84945602615 scopus 로고
    • Gurland's inequality for the gamma function
    • MR 24:A2058
    • A. V. Boyd, Gurland's inequality for the gamma function, Skand. Aktuarietidskr. 1960 (1961), 134-135. MR 24:A2058
    • (1961) Skand. Aktuarietidskr. , vol.1960 , pp. 134-135
    • Boyd, A.V.1
  • 7
    • 0011676174 scopus 로고
    • Note on a paper by Uppuluri
    • MR 35:6872
    • _, Note on a paper by Uppuluri. Pacific J. Math. 22 (1967), 9-10. MR 35:6872
    • (1967) Pacific J. Math. , vol.22 , pp. 9-10
  • 8
    • 84972535737 scopus 로고
    • Some function classes related to the class of convex functions
    • MR 26:6326
    • A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math. 12 (1902), 1203-1215. MR 26:6326
    • (1902) Pacific J. Math. , vol.12 , pp. 1203-1215
    • Bruckner, A.M.1    Ostrow, E.2
  • 9
    • 84966216618 scopus 로고
    • On gamma function inequalities
    • MR 87m:33002
    • J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), 659-667. MR 87m:33002
    • (1986) Math. Comp. , vol.47 , pp. 659-667
    • Bustoz, J.1    Ismail, M.E.H.2
  • 10
    • 0011035926 scopus 로고
    • Leonhard Euler's integral: A historical profile of the gamma function
    • MR 21:5540
    • P. J. Davis, Leonhard Euler's integral: A historical profile of the gamma function, Amer. Math. Monthly 66 (1959), 849-869. MR 21:5540
    • (1959) Amer. Math. Monthly , vol.66 , pp. 849-869
    • Davis, P.J.1
  • 11
    • 0014926481 scopus 로고
    • On monotonicity of the relaxation functions of viscoelastic materials
    • MR 40:3779
    • W. A. Day, On monotonicity of the relaxation functions of viscoelastic materials, Proc. Cambridge Philos. Soc. 67 (1970), 503-508. MR 40:3779
    • (1970) Proc. Cambridge Philos. Soc. , vol.67 , pp. 503-508
    • Day, W.A.1
  • 12
    • 0042766961 scopus 로고
    • Generalizations and applications of Cauchy-Schwarz inequalities
    • MR 37:1541
    • 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
    • (1967) Quart. J. Math. Oxford Ser. (2) , vol.18 , pp. 357-360
    • Eliezer, C.J.1    Daykin, D.E.2
  • 13
    • 84912062631 scopus 로고
    • The gamma function inequalities of Gurland and Gautschi
    • MR 24:A2682
    • T. Erber, The gamma function inequalities of Gurland and Gautschi, Skand. Aktuarietidskr. 1960 (1961), 27-28. MR 24:A2682
    • (1961) Skand. Aktuarietidskr. , vol.1960 , pp. 27-28
    • Erber, T.1
  • 17
    • 0001308973 scopus 로고
    • Kolmogorov-Landau inequalities for monotone functions
    • MR 84e:26017
    • A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), 251-258. MR 84e:26017
    • (1982) J. Math. Anal. Appl. , vol.90 , pp. 251-258
    • Fink, A.M.1
  • 18
    • 85171841264 scopus 로고
    • Some elementary inequalities relating to the gamma and incomplete gamma function
    • MR 21:2067
    • W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38 (1959), 77-81. MR 21:2067
    • (1959) J. Math. Phys. , vol.38 , pp. 77-81
    • Gautschi, W.1
  • 19
    • 0001639155 scopus 로고
    • A harmonic mean inequality for the gamma function
    • MR 50:2570
    • _, A harmonic mean inequality for the gamma function, SIAM J. Math. Anal. 5 (1974), 278-281. MR 50:2570
    • (1974) SIAM J. Math. Anal. , vol.5 , pp. 278-281
  • 20
    • 0001639154 scopus 로고
    • Some mean value inequalities for the gamma function
    • MR 50:2571
    • _, Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (1974), 282-292. MR 50:2571
    • (1974) SIAM J. Math. Anal. , vol.5 , pp. 282-292
  • 21
    • 84945612133 scopus 로고
    • On an inequality for gamma functions
    • MR 28:4151
    • D. V. Gokhale, On an inequality for gamma functions, Skand. Aktuarietidskr. 1962 (1963), 213-215. MR 28:4151
    • (1963) Skand. Aktuarietidskr. , vol.1962 , pp. 213-215
    • Gokhale, D.V.1
  • 22
    • 21844488155 scopus 로고
    • A stochastic approach to the gamma function
    • MR 95k:33003
    • L. Gordon, A stochastic approach to the gamma function, Amer. Math. Monthly 101 (1994), 858-865. MR 95k:33003
    • (1994) Amer. Math. Monthly , vol.101 , pp. 858-865
    • Gordon, L.1
  • 23
    • 84890486698 scopus 로고
    • An inequality satisfied by the gamma function
    • MR 20:1797
    • J. Gurland, An inequality satisfied by the gamma function, Skand. Aktuarietidskr. 39 (1956), 171-172. MR 20:1797
    • (1956) Skand. Aktuarietidskr. , vol.39 , pp. 171-172
    • Gurland, J.1
  • 24
    • 0022721836 scopus 로고
    • Completely monotonic functions associated with the gamma function and its q-analogues
    • MR 88b:33002
    • M. E. H. Ismail, L. Lurch, 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
    • (1986) J. Math. Anal. Appl. , vol.116 , pp. 1-9
    • Ismail, M.E.H.1    Lurch, L.2    Muldoon, M.E.3
  • 25
    • 0013247342 scopus 로고
    • Some inequalities for the gamma function
    • Beograd (N.S.) MR 46:7560
    • J. D. Kečkić and P. M. Vasić, Some inequalities for the gamma function, Publ. Inst. Math. (Beograd) (N.S.) 11 (1971), 107-114. MR 46:7560
    • (1971) Publ. Inst. Math. , vol.11 , pp. 107-114
    • Kečkić, J.D.1    Vasić, P.M.2
  • 26
    • 84966228652 scopus 로고
    • Some extensions of W. Gautschi's inequalities for the gamma function
    • MR 84m:33003
    • D. Kershaw, Some extensions of W. Gautschi's inequalities for the gamma function, Math. Comp. 41 (1983), 607-611. MR 84m:33003
    • (1983) Math. Comp. , vol.41 , pp. 607-611
    • Kershaw, D.1
  • 27
    • 0013246187 scopus 로고
    • Monotonicity results for the gamma function
    • MR 871:33006
    • D. Kershaw and A. Laforgia, Monotonicity results for the gamma function, Atti Accad. Sci. Torino 119 (1985), 127-133. MR 871:33006
    • (1985) Atti Accad. Sci. Torino , vol.119 , pp. 127-133
    • Kershaw, D.1    Laforgia, A.2
  • 28
    • 0001502378 scopus 로고
    • A probabilistic interpretation of complete rnonotonicity
    • MR 50:5899
    • C. H. Kimberling, A probabilistic interpretation of complete rnonotonicity, Aequationes Math. 10 (1974), 152-164. MR 50:5899
    • (1974) Aequationes Math. , vol.10 , pp. 152-164
    • Kimberling, C.H.1
  • 29
    • 84966209797 scopus 로고
    • Further inequalities for the gamma function
    • MR 851:33001
    • A. Lafbrgia, Further inequalities for the gamma function, Math. Comp. 42 (1981), 597-600. MR 851:33001
    • (1981) Math. Comp. , vol.42 , pp. 597-600
    • Lafbrgia, A.1
  • 31
    • 0002422392 scopus 로고
    • Inequalities for ultraspherical polynomials and the gamma function
    • MR 85d:33024
    • L. Lorch, Inequalities for ultraspherical polynomials and the gamma function, J. Approx. Theory 40 (1984), 115-120. MR 85d:33024
    • (1984) J. Approx. Theory , vol.40 , pp. 115-120
    • Lorch, L.1
  • 32
    • 0011153058 scopus 로고
    • Mittelwertungleichungen für Lösungen gewisser Differezengleichungen
    • MR 91h:39004
    • L. G. Lucht, Mittelwertungleichungen für Lösungen gewisser Differezengleichungen, Aequationes Math. 39 (1990), 204-209. MR 91h:39004
    • (1990) Aequationes Math. , vol.39 , pp. 204-209
    • Lucht, L.G.1
  • 33
    • 0002192812 scopus 로고
    • Inequalities for the gamma function and its logarithmic derivative
    • MR 50:10338
    • V. L. Luke, Inequalities for the gamma function and its logarithmic derivative, Math. Balkanica 2 (1972), 118-123. MR 50:10338
    • (1972) Math. Balkanica , vol.2 , pp. 118-123
    • Luke, V.L.1
  • 35
  • 39
    • 0002146064 scopus 로고
    • Some monotonicity properties and characterizations of the gamma function
    • MR 58:11536
    • M. E. Muldoon, Some monotonicity properties and characterizations of the gamma function, Aeqnationes Math. 18 (1978), 54-63. MR 58:11536
    • (1978) Aeqnationes Math. , vol.18 , pp. 54-63
    • Muldoon, M.E.1
  • 40
    • 17644366554 scopus 로고
    • An inequality satisfied by the gamma function
    • MR 21:4257
    • I. Olkin, An inequality satisfied by the gamma function, Skand. Aktuarietidskr. 1958 (1959), 37-39. MR 21:4257
    • (1959) Skand. Aktuarietidskr. , vol.1958 , pp. 37-39
    • Olkin, I.1
  • 41
    • 0042265720 scopus 로고
    • Variance bounds and orthogonal expansions in Hilbert space with an application to inequalities for gamma functions and π
    • MR 35:421
    • H. Ruben, Variance bounds and orthogonal expansions in Hilbert space with an application to inequalities for gamma functions and π, J. Reine Angew. Math 225 (1967), 147-153. MR 35:421
    • (1967) J. Reine Angew. Math , vol.225 , pp. 147-153
    • Ruben, H.1
  • 43
    • 0011604435 scopus 로고
    • Über die Ungleichung, welche die Integrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet
    • MR 2:218e
    • 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
    • (1940) Math. Ann. , vol.117 , pp. 301-326
    • Schmidt, E.1
  • 44
    • 0042766962 scopus 로고
    • An inequality satisfied by the gamma function
    • MR 54:3050
    • J. B. Selliah, An inequality satisfied by the gamma function, Canad. Math. Bull. 19 (1976), 85-87. MR 54:3050
    • (1976) Canad. Math. Bull. , vol.19 , pp. 85-87
    • Selliah, J.B.1
  • 46
    • 0011369884 scopus 로고
    • From Wythoff's Nim to Chebyshey's inequality
    • MR 93b:90132
    • K B. Stolarsky, From Wythoff's Nim to Chebyshey's inequality. Amer. Math. Monthly 98 (1991), 889-900. MR 93b:90132
    • (1991) Amer. Math. Monthly , vol.98 , pp. 889-900
    • Stolarsky, K.B.1
  • 47
    • 0012100570 scopus 로고
    • Superadditive functions and a statistical application
    • MR 91a:26019
    • 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
    • (1989) SIAM J. Math. Anal. , vol.20 , pp. 1255-1259
    • Trimble, S.Y.1    Wells, J.2    Wright, F.T.3
  • 48
    • 0004201980 scopus 로고
    • Princeton Univ. Press, Princeton, MR 3:232d
    • D. V. Widder, The Laplace transform, Princeton Univ. Press, Princeton, 1941. MR 3:232d
    • (1941) The Laplace Transform
    • Widder, D.V.1
  • 50
    • 0042265724 scopus 로고
    • On a Mercerian theorem and its application to the equiconvergence of Cesàro and Riesz transforms
    • Beograd (N.S.) MR 31:543
    • 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
    • (1962) Publ. Inst. Math. , vol.1 , pp. 83-91
    • Zimering, S.1


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