[ 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. [20] R. Askey, The q-gamma and q-beta functions, Appl. Anal.(1978) 125--141. [21] R. Askey, Ramanujan's extension of the gamma and beta functions, Amer. Math. Monthly, 87(1980) 346-359. [22] R. Askey, 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, J. Combinatorial Theory A 76(1996), 83--88, (see also correction). [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, Academic Press, New York, 1979. [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 , Kluwer Academic Publishers,1997,(pp.1-15). [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, Academic Press, New York-London, 1973. [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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