SOME COMPLETELY MONOTONIC FUNCTIONS ASSOCIATED WITH THE q-GAMMA AND THE q-POLYGAMMA FUNCTIONS

doi:10.1016/S0252-9602(15)30050-3

Abstract

Abstract:

In this paper, the q-analogue of the Stirling formula for the q-gamma function (Moak formula) is exploited to prove the complete monotonicity properties of some functions involving the q-gamma and the q-polygamma functions for all real number q > 0. The monotonicity of these functions is used to establish sharp inequalities for the q-gamma and the q-polygamma functions and the q-Harmonic number. Our results are shown to be a generalization of results which were obtained by Selvi and Batir [23].

Key words: completely monotonic functions, inequalities, q-gamma function, q-polygamma function

CLC Number:

33D05

Ahmed SALEM, Eid S. KAMEL. SOME COMPLETELY MONOTONIC FUNCTIONS ASSOCIATED WITH THE q-GAMMA AND THE q-POLYGAMMA FUNCTIONS[J].Acta mathematica scientia,Series B, 2015, 35(5): 1214-1224.

Trendmd

References

[1] Abramowitz M, Stegun C A. Handbook of Mathematical Functions with Formulas, Graphs, Mathematical Tables. Applied Mathematics Series, Vol 55. Washington, DC:National Bureau of Standards, 1964
[2] Alzer H, Felder G. A Turán-type inequality for the gamma function. J Math Anal Appl, 2009, 350:276-282
[3] Alzer H, Grinshpan A Z. Inequalities for the gamma and q-gamma functions. J Appr Theory, 2007, 144:67-83
[4] Alzer H, Batir N. Monotonicity properties of the gamma function. Appl Math Lett, 2007, 20:778-781
[5] Askey R. The q-gamma and q-beta functions. Appl Anal, 1978, 8:125-141
[6] Elezovic N, Giordano C, Pecaric J. Convexity and q-gamma function. Rendiconti del Circolo Matematico di Palermo, Series II, 1999, 48:285-298
[7] Gao P. Some monotonicity properties of gamma and q-gamma functions. ISRN Math Anal, 2011, 2011:1-15
[8] Grinshpan A Z, Ismail M E H. Completely monotonic functions involving the gamma and q-gamma functions. Proc Amer Math Soc, 2006, 134:1153-1160
[9] Ismail M E H, Muldoon M E. Inequalities and monotonicity properties for gamma and q-gamma functions//Zahar R V M, ed. Approximation and Computation, International Series of Numerical Mathematics. Boston, MA:Birkhauser, 1994, 119:309-323
[10] Ismail M E H, Lorch L, Muldoon M E. Completely monotonic functions associated with the gamma function and its q-analogues. J Math Anal Appl, 1986, 116:1-9
[11] Krattenthaler C, Srivastava H M. Summations for basic hypergeometric series involving a q-analogue of the digamma function. Comput Math Appl, 1996, 32(2):73-91
[12] Moak D S. The q-analogue of Stirling's formula. Rocky Mountain J Math, 1984, 14:403-413
[13] Moak D S. The q-gamma function for q > 1. Aequationes Math, 1980, 20:278-285
[14] Mortici C. Improved asymptotic formulas for the gamma function. Comput Math Appl, 2011, 61:3364-3369
[15] Mortici C. Estimating gamma function by digamma function. Comput Math Appl, 2010, 52:942-946
[16] Olde Daalhuis A B. asymptotic expansions of q-gamma, q-exponential and q-Bessel functions. J Math Anal Appl, 1994, 186:896-913
[17] Salem A. A completely monotonic function involving q-gamma and q-digamma functions. J Appr Theory, 2012, 164:971-980
[18] Salem A. Some properties and expansions associated with q-digamma function. Quaest Math, 2013, 36(1):67-77
[19] Salem A. A q-analogue of the exponential integral. Afrika Matematika, 2013, 24(2):117-125
[20] Salem A. An infinite class of completely monotonic functions involving the q-gamma function. J Math Anal Appl, 2013, 406(2):392-399
[21] Salem A. Two classes of bounds for the q-gamma and the q-digamma functions in terms of the q-zeta functions. Banach J Math Anal, 2014, 8(1):109-117
[22] Salem A. Complete monotonicity properties of functions involving q-gamma and q-digamma functions. Math Inequalities Appl, 2014, 17(3):801-811
[23] Sevli H, Batir N. Complete monotonicity results for some functions involving the gamma and polygamma functions. Math Comput Modelling, 2011, 53:1771-1775
[24] Wei C, Gu Q. q-generalizations of a family of harmonic number identities. Adv Appl Math, 2010, 45:24-27