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A Completely Monotonic Function Involving Q-Gamma and Q-Digamma Functions

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A Completely Monotonic Function Involving Q-Gamma and Q-Digamma Functions View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Journal of Approximation Theory 164 (2012) 971–980 www.elsevier.com/locate/jat Full length article A completely monotonic function involving q-gamma and q-digamma functions Ahmed Salem Department of Mathematics, Faculty of Science, Al-Jouf University, Sakaka, Al-Jouf, Saudi Arabia Received 4 September 2011; received in revised form 25 January 2012; accepted 19 March 2012 Available online 21 April 2012 Communicated by Roderick Wong Abstract In this paper, the monotonicity property for a function involving q-gamma and q-digamma functions is investigated for q > 0. An application of this result provides a sharp inequality for the q-gamma function. Our results are shown to be a generalization of results which were obtained by Alzer and Batir (2007) [2]. ⃝c 2012 Elsevier Inc. All rights reserved. Keywords: Completely monotonic functions; Inequalities; q-gamma function; q-digamma function 1. Introduction The q-gamma function is defined for positive real numbers x and q 6D 1 as 1 nC1 1−x Y 1 − q 0q .x/ D .1 − q/ ; 0 < q < 1 (1.1) 1 − qnCx kD0 and 1 −.nC1/ 1−x x.x−1/ Y 1 − q 0 .x/ D .q − 1/ q 2 ; q > 1: (1.2) q − q−.nCx/ kD0 1 E-mail address: [email protected]. 0021-9045/$ - see front matter ⃝c 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jat.2012.03.014 972 A. Salem / Journal of Approximation Theory 164 (2012) 971–980 The close connection between the ordinary gamma function 0 and the q-gamma function 0q is given by the limit relations Z 1 lim 0 .x/ D lim 0 .x/ D 0.x/ D t x−1e−t dt: (1.3) − q C q q!1 q!1 0 Many of the classical facts concerning the ordinary gamma function have been extended to the q-gamma function (see [5,12,13,16] and the references therein). An important fact regarding the gamma function in applied mathematics as well as in probability is the Stirling formula, which gives a fairly accurate idea of the size of the gamma function. With the Euler–Maclaurin formula, Moak [13] obtained the following q-analogue of the Stirling formula for 0 < q < 1: x 1 Li2.1 − q / log 0q .x/ ∼ x − logTxUq C C Cq 2 log q 1 2k−1 X B2k log q x x C q P k− .q /; x ! 1 (1.4) .2k/W qx − 1 2 3 kD1 x where the Bk; k D 1; 2;:::, are the Bernoulli numbers, TxUq D .1 − q /=.1 − q/; Li2.z/ is the dilogarithm function defined for complex argument z as [1] Z z log.1 − t/ Li2.z/ D − dt; z 62 .1; 1/; (1.5) 0 t Pk.z/ is a polynomial of degree k satisfying 2 0 Pk.z/ D .z − z /Pk−1.z/ C .kz C 1/Pk−1.z/; P0.z/ D 1; k D 1; 2;::: (1.6) and 1 1 q − 1 1 Cq D log.2π/ C log − log q 2 2 log q 24 1 ! X C log r m.6mC1/ − r .2mC1/.3mC1/ (1.7) m=−∞ where r D exp.4π 2= log q/. It is easy to see that x 1 Li2.1 − q / lim Cq D C1 D log.2π/; lim D −x and Pk.1/ D .k C 1/W (1.8) q!1 2 q!1 log q and so (1.4) on letting q ! 1 tends to the ordinary Stirling formula [1] 1 1 1 X B2k 1 log 0.x/ ∼ x − log x − x C log.2π/ C ; x ! 1: (1.9) 2 2 2k.2k − 1/ x2k−1 kD1 It is worth noting that the Moak formula, (1.4), and the constant Cq , (1.7), are given in a slightly different form to the Moak formula, (2.14), and the constant Cq , (2.16), in [13]. To compare the two forms, we observe that, on changing the variable u D − log.1 − t/, we obtain the integral x 1 Z −x log q udu −1 Z 1−q log.1 − t/dt Li .1 − qx / D D D 2 I .x/ u : log q 0 e − 1 log q 0 t log q A. Salem / Journal of Approximation Theory 164 (2012) 971–980 973 Moak considered the integral I .x/ − I .1/ in his formula (2.14) mentioned in [13] and added the integral I .1/ to the constant. But here we add the integral I .x/ to (1.4) and thus the constant in (1.7) is not needed. In order to derive a formula for q > 1, we need the following identities: .x−1/.x−2/ 0q .x/ D q 2 0q−1 .x/; (1.10) x2 − 3x C 2 log 0q .x/ D log q C log 0 −1 .x/; (1.11) 2 q and z − 1 1 Li D −Li .1 − z/ − log2 z: (1.12) 2 z 2 2 Substituting into (1.4) after replacing q by q−1 yields x 1 Li2.1 − q / 1 log 0q .x/ ∼ x − logTxUq C C log q C C −1 2 log q 2 q 1 −1 2k−1 X B2k log q −x −x C q P k− .q /; x ! 1: (1.13) .2k/W q−x − 1 2 3 kD1 In view of (1.4), (1.9) and (1.13), the Moak formula for q > 0 can be read as x 1 Li2.1 − q / 1 log 0q .x/ ∼ x − logTxUq C C H.q − 1/ log q C C O 2 log q 2 q 1 2k−1 X B2k log qO x x C qO P k− .qO /; x ! 1 (1.14) .2k/W qO x − 1 2 3 kD1 where H.·/ denotes the Heaviside step function and q if 0 < q ≤ 1 qO D q−1 if q ≥ 1: A real-valued function f , defined on an interval I , is called completely monotonic if f has derivatives of all orders and satisfies .−1/n f .n/.x/ ≥ 0; n D 0; 1; 2;::: I x 2 I: (1.15) These functions have numerous applications in various branches, for instance, numerical analysis and probability theory. Numerous papers have appeared providing inequalities for the gamma and various related functions. Many properties of the gamma and q-gamma functions and some developments in this area are given in [8–10,6,4,2,3,14,15,7] and the references therein. The q-digamma function (the q-Psi function q ) is an important function related to the q-gamma function and is defined as the logarithmic derivative of the q-gamma function 0 d 0q .z/ q .z/ D .ln 0q .z// D : (1.16) dz 0q .z/ 974 A. Salem / Journal of Approximation Theory 164 (2012) 971–980 From (1.1), we get for 0 < q < 1 and for all real variables x > 0, 1 C X qk x q .x/ D − log.1 − q/ C log q (1.17) 1 − qkCx kD0 1 X qkx D − log.1 − q/ C log q (1.18) 1 − qk kD1 and from (1.2) we obtain for q > 1 and x > 0, " 1 − C # 1 X q .k x/ q .x/ D − log.q − 1/ C log q x − − (1.19) 2 − q−.kCx/ kD0 1 " 1 − # 1 X q kx D − log.q − 1/ C log q x − − : (1.20) 2 1 − q−k kD1 Krattenthaler and Srivastava [11] proved that q .x/ tends to .x/ on letting q ! 1 where .x/ is the ordinary Psi (digamma) function (the logarithmic derivative of the gamma function). From 0 (1.18) and (1.20), Alzer and Grinshpan [4] concluded that q .x/ is strictly completely monotonic on .0; 1/ for any q > 0, that is, n 0 .n/ .−1/ . q .x// > 0; for x > 0; q > 0 and n D 0; 1; 2;:::: (1.21) The main purpose of this paper is to present monotonicity properties of the function x Li2.1 − q / Ga.xI q/ D log 0q .x/ − x logTxUq − − C O log q q 1 1 − λH.q − 1/ log q C q .x C a/; (1.22) 2 2 ≥ D D C 1 where q > 0; x > 0; a 0 and λ 1 or λ a 2 . It is worth mentioning that Alzer and Batir [2] considered the function 1 1 Ga.x/ D log 0.x/ − x log x C x − log.2π/ C .x C a/; x > 0I a ≥ 0 (1.23) 2 2 which is a special case of the function Ga.xI q/ on letting q ! 1 and proved that Ga.x/ is 1 ≥ 1 − completely monotonic on .0; / if and only if a 3 and Ga.x/ is completely monotonic on .0; 1/ if and only if a D 0. Therefore, in our study, we concentrate on just two cases: when 0 < q < 1 and q > 1. An application of their result leads to sharp upper and lower bounds for D 1 0.x/ in terms of the -function, i.e., they determined the smallest number α 3 and the largest number β D 0 such that for x > 0, the following two-sided inequality is valid: p 1 p 1 2πx x exp −x − .x C α/ < 0.x/ < 2πx x exp −x − .x C β/ : (1.24) 2 2 Also, as an application of our results, the above inequality will be extended to the q-gamma and the q-digamma functions.
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