Some inequalities for q-polygamma function and ζ q -Riemann zeta functions

Transcription

1 Aales Mahemaicae e Iformaicae 37 (1) h://ami.ekf.hu Some iequaliies for q-olygamma fucio ad ζ q -Riema zea fucios Valmir Krasiqi a, Toufik Masour b Armed Sh. Shabai a a Dearme of Mahemaics, Uiversiy of Prishia, Prishië 1, Reublic of Kosova b Dearme of Mahemaics, Uiversiy of Haifa, 3195 Haifa, Israel Submied 6 February 1; Acceed 1 Jue 1 Absrac I his aer, we rese some iequaliies for q-olygamma fucios ad ζ q-riema Zea fucios, usig a q-aalogue of Holder ye iequaliy. Keywords: q-olygamma fucios, q-zea fucio. MSC: 33D5, 11S4, 6D Iroducio ad relimiaries I his secio, we rovide a summary of oaios ad defiiios used i his aer. For deails, oe may refer o [3, 5]. For = 1,,... we deoe by ψ (x) = ψ () (x) he olygamma fucios as he -h derivaive of he si fucio ψ(x) = Γ (x) Γ(x), x >, where Γ(x) deoes he usual gamma fucio. Throughou his aer we will fix q (, 1). Le a be a comlex umber. The q-shifed facorials are defied by 1 (a; q) = (1 aq k ), = 1,,..., k= (a; q) = lim (a; q) = (1 aq k ). k 95

2 96 V. Krasiqi, T. Masour, A. Sh. Shabai Jackso [4] defied he q-gamma fucio as I saisfies he fucioal equaio Γ q (x) = (q; q) (q x ; q) (1 q) 1 x, x, 1,... (1.1) Γ q (x + 1) = [x] q Γ q (x), Γ q (1) = 1, (1.) where for x comlex [x] q = 1 qx 1 q. The q-gamma fucio has he followig iegral rereseaio (see []) Γ q (x) = 1 1 q x 1 Eq q 1 q d q = x 1 Eq q d q, x >., which is he q-aalogue of he classical exoeial fucio. The q-aalogue of he ψ fucio is defied as he logarihmic derivaive of he q-gamma fucio where Eq x = j= q j(j 1) x j [j] q! = (1 + (1 q)x) q ψ q (x) = Γ q(x), x >. (1.3) Γ q (x) The q-jackso iegral from o a is defied by (see [4, 5]) a f(x)d q x = (1 q)a f(aq )q. (1.4) = For a = he q-jackso iegral is defied by (see [4, 5]) f(x)d q x = (1 q) = f(q )q (1.5) rovided ha sums i (1.4) ad (1.5) coverge absoluely. I [] he q-riemma zea fucio is defied as follows (see Secio.3 for he defiiios) 1 q (+α([]q))s ζ q (s) = {} s = q [] s. (1.6) q I relaio o (1.3) ad (1.6), K. Brahim [1], usig a q-aalogue of he geeralized Schwarz iequaliy, roved he followig Theorems. Theorem 1.1. For = 1,..., where ψ q, = ψ () q ψ q, (x)ψ q,m (x) ψ (x), q, m+ is -h derivaive of ψ q ad m+ is a ieger.

3 Some iequaliies for q-olygamma fucio ad ζ q-riema zea fucios 97 Theorem 1.. For all s > 1, ζ q (s) [s + 1] q ζ q (s + 1) q[s] ζ q (s + 1) q ζ q (s + ). The aim of his aer is o rese some iequaliies for q-olygamma fucios ad q-zea fucios by usig a q-aalogue of Holder ye iequaliy, similar o hose i [1].. Mai resuls.1. A lemma I order o rove our mai resuls, we eed he followig lemma. Lemma.1. Le a R + { }, le f ad g be wo oegaive fucios ad le, > 1 such ha = 1. The followig iequaliy holds a ( a f(x)g(x)d q x Proof. Le a >. By (1.4) we have ha a f(x)g(x)d q x = (1 q)a ) 1 f (x)d q x ( a g (x)d q x ) 1 f(aq )g(aq )q. (.1) = By he use of he Holder s iequaliy for ifiie sums, we obai ( f(aq )g(aq )q ) ( f (aq )q ) 1 ( = Hece ( (1 q)a f(aq )g(aq )q ) = = = =. g (aq )q ) 1. (.) ( ((1 q)a) 1 f (aq )q ) 1 ( ((1 q)a) 1 g (aq )q )1. (.3) The resul he follows from (.1), (.) ad (.3)... The q-olygamma fucio From (1.1) oe ca derive he followig series rereseaio for he fucio ψ q (x) = Γ q (x) Γ q(x) : ψ q (x) = log(1 q) + log q 1 = q x, x >, (.4) 1 q

4 98 V. Krasiqi, T. Masour, A. Sh. Shabai which imlies ha ψ q (x) = log(1 q) + log q q x 1 1 q 1 d q. (.5) Theorem.. For =, 4, 6... se ψ q, (x) = ψ q () (x) he -h derivaive of he fucio ψ q. The for, > 1 such ha = 1 he followig iequaliy holds ψ q, + y ) ψ q, (x) 1 ψq, (y) 1. (.6) Proof. From (.5) we deduce ha hece ψ q, (x) = log q q 1 q ψ q, + y By Lemma.1 wih a = q we have ψ q, + y ) = log q 1 q ( log q 1 q where f(u) = q q ) = log q q 1 q [ (log u) ] 1 = (ψ q, (x)) 1 (ψq, (y)) 1 ( (log u) 1 u u x 1 (log u) u x 1 d q u, (.7) 1 u (log u) u x + y 1 d q u. 1 u [ (log u) 1 u 1 u (log u) u x 1 ) 1 ( log q d q u 1 u 1 q ) u x 1 ad g(u) = ( (log u) 1 u For = = i (.6) oe has he followig resul. Corollary.3. We have + y ) ψ q, ] 1 u y 1 q dq u q ψ q, (x) ψ q, (y)..3. q-zea fucio ( ) For x > we se α(x) = log x log q E log x log q he ieger ar of log x log q. I [] he q-zea fucio is defied as follows ad {x} q = (log u) u y 1 1 u )1 d q u ) u y 1. ( ) [x]q, where E log x q x+α([x]q) log q is 1 ζ q (s) = {} s q = q (+α([]q))s [] s q.

5 Some iequaliies for q-olygamma fucio ad ζ q-riema zea fucios 99 There ([]) i is roved ha ζ q is a q-aalogue of he classical Riemma Zea fucio, ad for all s C such ha R(s) > 1, ad for all u > oe has where Z q () = K q () = e {}q q ζ q (s) = 1 Γ q (s), Γ q () = Γq() K, ad q() u s 1 Z q (u)d q u, (1 q) s 1 + (1 q) 1 ( (1 q); q) ( (1 q) 1 ; q) ( (1 q)q s ; q) ( (1 q) 1 q 1 s. ; q) Theorem.4. For = 1 ad x + y > 1, ) (x Γ q + y Γ q 1 (x) Γ q 1 (y) Proof. From Lemma.1 we have ha u x + y 1 Z q (u)d q u = ( u x 1 ζ 1 q (x) ζ 1 ζ q (x + y q (y) ). (Z q (u)) 1 u y 1 (Z q (u)) 1 dq u. ) 1 u x 1 (Z q (u))d q u ( For f(u) = u x 1 (Z q (u)) 1 y 1 ad g(u) = u (Z q (u)) 1 we obai ha Γ q + y ) ζ q + y ) Γ 1 q (x) Γ 1 q (y) ζ 1 q (x) ζ 1 q (y), )1 u y 1 (Z q (u))d q u. which comlees he roof. Ackowledgemes. The auhors would like o hak he aoymous referees for heir commes ad suggesios. Refereces [1] Brahim, K., Turá-Tye Iequaliies for some q-secial fucios, J. Ieq. Pure Al. Mah., 1() (9) Ar. 5. [] Fiouhi, A., Beaibi, N., Brahim, K., The Melli rasform i quaum calculus, Cosrucive Aroximaio, 3(3) (6) [3] Gaser, G., Rahma, M., Basic Hyergeomeric Series, d Ediio, (4), Ecycloedia of Mahemaics ad Alicaios, 96, Cambridge Uiversiy Press, Cambridge. [4] Jackso, F.H., O a q-defiie iegrals, Quar. J. Pure ad Al. Mah., 41 (191) [5] Kac, V.G., Cheug, P., Quaum Calculus, Uiversiex, Sriger-Verlag, New York, ().

6 1 V. Krasiqi, T. Masour, A. Sh. Shabai Valmir Krasiqi Dearme of Mahemaics Uiversiy of Prishia Prishië 1, Reublic of Kosova Toufik Masour Dearme of Mahemaics Uiversiy of Haifa 3195 Haifa, Israel Armed Sh. Shabai Dearme of Mahemaics Uiversiy of Prishia Prishië 1, Reublic of Kosova