﻿ A Bibliography on Gamma-Function-Inequalities

## A bibliography on inequalities for the gamma function. ~ Compiled by Alexandru Lupas ~

```[ 1] M. Abramowitz and I. A. Stegun (eds.),
Handbook of Mathematical Functions with Formulas,
Graphs and Mathematical Tables, Dover, New York, 1965.

[ 2] P. Ahern, W. Rudin,
Geometric properties of the Gamma function,
Amer. Math. Monthly 103 (1996) 678-681.

[ 3] N. I. Akhiezer,
The Classical Moment Problem and Some Related Questions in Analysis.
(English translation), Oliver and Boyd, Edinburgh, 1965.

[ 4] H. Alzer,
Some gamma function inequalities,
Math. Comp. 60 (1993), 337--346.

[ 5] H. Alzer,
On some inequalities for the gamma and psi functions,
Math. Comp. 66 (1997)373--389.

[ 6] Alzer, H.,
Some inequalities for the incomplete gamma function,
Math. Computation , 66(1997), 771-778.

[ 7] H. Alzer,
Inequalities for the gamma and polygamma functions,
Abhandl. Math. Sem.Univ. Hamburg 68 (1998) 363--372.

[ 8] Alzer, H.,
A harmonic mean inequality for the gamma function,
J.Comp. Appl. Math.,97(1997), 195--198.

[ 9] H. Alzer,
Mean-value inequalities for the polygamma functions,
Aequationes Math. 61 (2001) 151--161.

[10] H. Alzer and C. Berg,
Some classes of completely monotonic functions,
Ann. Acad. Scient. Fennicae 27 (2002),445--460.

[11] H. Alzer and C. Berg,
Some classes of completely monotonic functions (II),
(to appear)

[12] H. Alzer and O. G. Ruehr,
A submultiplicative property of the psi function,
J. Comp. Appl. Math. 101 (1999), 53--60.

[13] H. Alzer and J. Wells,
Inequalities for the polygamma functions,
SIAM J. Math. Anal.29 (1998), 1459--1466.

[14] G. D. Anderson, R. W. Barnard, K. C. Richards, M. K. Vamanamurthy and M. Vuorinen,
Inequalities for the zero-balanced hypergeometric functions,
Trans. Amer. Math.Soc. 347 (1995), 1713--1723.

[15] G. D. Anderson and S. -L. Qiu,
A monotonicity property of the gamma function,
Proc. Amer. Math. Soc. 125 (1997), 3355--3362.

[16] W. J. Anderson, H. J. Haubold  and A. M. Mathai,
Astrophysical  thermonuclear functions,
Astrophys. Space Sci., 49--70, 214, 1994.

[17] Andrews, L.C.,
Special functions for Engineers and Applied Mathematics,
Macmillan Publishing Company, New York, 1985.

[18] Andrews, G., Askey, R. and Roy, R.,
Special Functions,
Cambridge University Press, New York, 1999.

[19] E. Artin,
The Gamma Function,
Holt, Rinehart and Winston, New York, 1964.

The q-gamma and q-beta functions,
Appl. Anal.(1978) 125--141.

Ramanujan's extension of the gamma and beta functions,
Amer. Math. Monthly, 87(1980) 346-359.

A g-extension of Cauchy's form of the beta integral,
Quart.J.Math. (Oxford),328 (1981) 255-266.

[23] E. W. Barnes,
The theory of the gamma function,
Messenger Math. 29 (2),(1900), 64-128.

[24] C. Berg and H. L. Pedersen,
Pick functions related to the gamma function,
Rocky Mount. J. Math. 32 (2002), 507--525.

[25] H. Bohr and J. Mollerup,
Laerebog i Matematisk Analyse,
Jul. Gjellerups Forlag, Copenhagen,1922 (vol.3).

[26] A. V. Boyd,
Gurland's inequality for the gamma function,
Skand. Aktuarietidskr. 1960 (1961),134--135.

[27] A. V. Boyd, Note on a paper by Uppuluri,
Pacific J. Math. 22 (1967), 9--10.

[28] P. S. Bullen, D. S. Mitrinovic and P. M. Vasic,
Means and Their Inequalities,
Reidel,Dordrecht, 1988.

[29] J. Bustoz and M. E. H. Ismail,
On gamma function inequalities,
Math. Comp. 47 (1986),659--667.

[30] M.A. Chaudhry and Syed M.Zubair,
On a Class of Incomplete Gamma functions  with Applications,
Chapman Hall,2002.

[31] W. E. Clark and M.Ismail,
Binomial and q-binomial coefficient inequalities related to
the Hamiltonicity  of the Knesser graphs and their q-analogues,

[32] W. E. Clark and M.Ismail,
Inequalities involving Gamma and Psi functions,
Analysis and Applications, Vol. 1, No. 1 (2003) 129--140.

[33] P. Czinder and Z. Pales,
A general Minkowski-type inequality for two  variable Gini means,
Publ. Math.Debrecen 57 (2000), 203-216.

[34] H. Dang and G. Weerakkody,
Bounds for the maximum likelihood estimates in two-parameter gamma distribution,
J. Math. Anal. Appl. 245 (2000), 1-6.

[35] P. J. Davis,  Leonhard Euler's integral:
A historical profile of  the gamma function,
Amer.Math. Monthly 66 (1959), 849--869.

[36] J. Dutka,
On some gamma function inequalities,
SIAM J. Math. Anal.,  16(1985),180--185.

[37] A. Elbert, A. Laforgia,
On some properties of the gamma function,
Proc. Amer. Math. Soc. 128 (2000) 2667--2673.

[38] T. Erber,
The gamma function inequalities of Gurland and Gautschi,
Skand. Aktuarietidskr.1960 (1961), 27--28.

[39] A. M. Fink,
Kolmogorov-Landau inequalities for monotone functions,
J. Math. Anal. Appl.90 (1982), 251--258.

[40] W. Gautschi,
Some elementary inequalities relating to the gamma and incomplete gamma function,
J. Math. Phys. 38 (1959), 77--81.

[41] W. Gautschi,
A harmonic mean inequality for the  gamma function,
SIAM J. Math. Anal. 5 (1974),278--281.

[42] W. Gautschi,
Some mean value inequalities for the gamma function,
SIAM J. Math. Anal. 5 (1974),282--292.

[43] W. Gautschi,
The incomplete gamma function since Tricomi,
in: Tricomi's Ideas and Contemporary Applied Mathematics,
Atti Convegni Lincei, 147, Accad. Naz. Lincei, Rome, 1998, 207-237.

[44] M. Godefroy,
La fonction Gamma ; Theorie, Histoire, Bibliographie,
Gauthier-Villars, Paris, (1901).

[45] D. V. Gokhale,
On an inequality for gamma functions,
Skand. Aktuarietidskr. 1962 (1963),213--215.

[46] J. Gurland,
An inequality satisfied by the gamma function,
Skand. Aktuarietidskr. 39 (1956),171--172.

[47] G. H. Hardy, J.E. Littlewood, and G. Polya, Inequalities,
Cambridge Univ. Press, Cambridge,1952.

[48] 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.

[49] M. E. H. Ismail and M.E.Mouldon ,
Inequalities for gamma and q-gamma functions, Approximation and Computation,
A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar ed.,
ISNM 119, Birkhauser,Boston,Basel,Berlin 1994,309--323.

[50] F. John,
Special solutions of certain difference equations,
Acta Math.71(1939)175--189.

[51] D. G. Kabe,
On some inequalities satisfied by beta and gamma functions,
South African Statist. J.,12(1978)25--31.

[52] H. H. Kairies, Convexity in the theory of the gamma function,
General Inequalities ,Proc.First Internat. Conf.1976
Oberwolfach, Birkhauser, Basel,1978,49--62.

[53] H. H. Kairies and M. E. Muldoon ,
Some characterizations of q-factorial  functions,
Aequationes Math. 25(1982) 67--76.

[54] J. D. Keckic and P. M. Vasic,
Some inequalities for the gamma function,
Publ. Inst. Math.(Beograd) (N.S.) 11 (1971), 107--114.

[55] D. Kershaw,
Some extensions of W. Gautschi's inequalities for the gamma function,
Math.Comp. 41 (1983), 607--611.

[56] D. Kershaw and A. Laforgia,
Monotonicity results for the gamma function,
Atti Accad. Sci.Torino 119 (1985), 127--133.

[57] A. Laforgia,
Further inequalities for the gamma function,
Math. Comp. 42 (1984), 597--600.

[58] A. Laforgia and S. Sismondi,
Monotonicity results and inequalities for the gamma and error functions,
J. Comput. Appl. Math., 23( 1988)25--33.

[59] A. Laforgia and S. Sismondi,
A geometric mean inequality for the  gamma function,
Boll. Un. Mat. Ital.,  A7(3),( 1989),339--342.

[60] I. B. Lazarevic and A. Lupas,
Functional equations for Wallis  and gamma functions,
Univ.Beograd. Publ. Elektrotehn. Fak. Ser.A. 461-497 (1979), 245--251.

[61] L. Lorch,
Inequalities for ultraspherical polynomials and the gamma function,
J. Approx.Theory 40 (1984), 115--120.

[62] L. G. Lucht,
Mittelwertungleichungen fuer Loesungen gewisser Differenzengleichungen,
Aequationes Math. 39 (1990), 204--209.

[63] Y. L. Luke,
Inequalities for the gamma function and its logarithmic derivative,
Math. Balkanica 2 (1972), 118--123.

[64] A. W. Marshall and I. Olkin,
Inequalities: Theory of majorization and its applications,

[65] M. Merkle,
Logarithmic convexity and inequalities for the gamma function,
J. Math. Anal.Appl. 203 (1996), 369--380.

[66] M. Merkle,
On log-convexity of a ratio of gamma functions,
Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.8 (1997), 114-119.

[67] M. Merkle,
Convexity, Schur-convexity and bounds for the gamma function
involving the digamma function,
Rocky Mountain J. Math. 28 (1998), 1053--1066.

[68] M. Merkle,
Conditions for convexity of a derivative and some applications to the Gamma function,
Aequationes Math. 55 (1998) 273--280.

[69] H. Minc and L. Sathre,
Some inequalities involving (r!)^1/r,
Edinburgh Math. Soc. 14(1964/65), 41--46.

[70] D. S. Mitrinovic,
Analytic inequalities,
Springer, New York, 1970.

[71] M. E. Muldoon,
Some monotonicity properties and characterizations of the gamma function,
Aequationes Math. 18 (1978), 54--63.

[72] M. E. Muldoon,
Convexity properties of special functions and their zeros,
in Recent Progress in Inequalities, A Volume Dedicated to Prof. D. S. Mitrinovic ,

[73] N. Nielsen,
Handbuch der Theorie der Gammafunktion,
B.G. Teubner, Leipzig, 1906.

[74] I. Olkin,
An inequality satisfied by the gamma function,
Skand. Aktuarietidskr. 1958 (1959), 37--39.

[75] B. Palumbo,
A generalization of some inequalities for the gamma function,
J. Comput. Appl. Math., 88(1998), 255-268.

[76] B. Palumbo,
Determinantal inequalities for the psi function,
Math. Inequal. Appl. 2 (1999), 223-231.

[77] T. Popoviciu,
Les fonctions convexes,
Actualites Sci. Indust. 992, Paris, 1944.

[78] A.W. Roberts and D.E. Varberg,
Convex functions,

[79] J. Sandor,
Sur la fonction gamma,
Publ. C.R.M.P. Neuchatel, Ser. I, 21 (1989), 4--7.

[80] E. Schmidt,
Ueber die Ungleichung, welche die Integrale ueber eine Potenz einer
Funktion und ueber eine andere Potenz ihrer Ableitung verbindet,
Math. Ann. 117 (1940), 301--326.

[81] P. Sebah and X. Gourdon,
Introduction to the Gamma Function,
(preprint-2002)

[82] J. B. Selliah,
An inequality satisfied by the gamma function,
Canad. Math. Bull. 19 (1976), 85--87.

[83] W. Sibagaki,
Theory and applications of the gamma function,
Iwanami Syoten, Tokyo, Japan, (1952).

[84] D. V. Slavic,
On inequalities for G(x+1)/G(x+1/2),
Univ. Beograd. Publ. Elektrotehn, Fak.Ser. Mat. Fiz. 498-541 (1975), 17--20.

[85] N. Sonine,
Note sur une formule de Gauss,
Bulletin de la S.M.F., 9(1881), 162--166.

[86] Z. Starc,
Power product inequalities for the Gamma function,
Kragujevac J. Math. 24 (2002), 81-84.

[87] N. M. Temme and Olde Daalhius, A.B.,
Uniform asymptotic approximation of Fermi-Dirac integrals,
Journal of Computational and Applied Mathematics,31(1990), 383--387.

[88] N. M. Temme,
Traces to Tricomi in recent work on special functions and asymptotics of integrals,
in: Mathematical Analysis (J.M. Rassias, ed.),
Teubner-Texte Math., Teubner, Leipzig, 236-249, 79,1985.

[89] N. M. Temme,
Special Functions: An Introduction to the Classical Functions of Mathematical Physics,
Wiley, New York, 1996.

[90] G. N. Watson,
A note on gamma functions,
Proc.Edinburgh Math.Soc.(2)11(1958/59)Edinburgh Math.Notes 42(1959),7--9 .
[ The author shows: "If 1/G(x) := Gamma(x+1/2)/Gamma(x+1) = (x+H(x))^{-1/2}
then 1/4 <= H(x) <= 1/2  for x >= -1/2 and  1/4 <= H(x) <= 1/Pi when x >= 0." ]

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