Aales Mahemaicae e Iformaicae 37 (1). 95 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, 6D15. 1. 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
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.
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 + 1 = 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
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 + 1 = 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.
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 + 1 = 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) 35 33. [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) 193 3. [5] Kac, V.G., Cheug, P., Quaum Calculus, Uiversiex, Sriger-Verlag, New York, ().
1 V. Krasiqi, T. Masour, A. Sh. Shabai Valmir Krasiqi Dearme of Mahemaics Uiversiy of Prishia Prishië 1, Reublic of Kosova e-mail: vali.99@homail.com Toufik Masour Dearme of Mahemaics Uiversiy of Haifa 3195 Haifa, Israel e-mail: oufik@mah.haifa.ac.il Armed Sh. Shabai Dearme of Mahemaics Uiversiy of Prishia Prishië 1, Reublic of Kosova e-mail: armed_shabai@homail.com