Annales Mathematicae et Informaticae
37 (2010) pp. 95–100
http://ami.ektf.hu
Some inequalities for q-polygamma function
and ζq -Riemann zeta functions
Valmir Krasniqia , Toufik Mansourb
Armend Sh. Shabania
a
Department of Mathematics, University of Prishtina, Prishtinë 10000, Republic of
Kosova
b
Department of Mathematics, University of Haifa, 31905 Haifa, Israel
Submitted 26 February 2010; Accepted 10 June 2010
Abstract
In this paper, we present some inequalities for q-polygamma functions and
ζq -Riemann Zeta functions, using a q-analogue of Holder type inequality.
Keywords: q-polygamma functions, q-zeta function.
MSC: 33D05, 11S40, 26D15.
1. Introduction and preliminaries
In this section, we provide a summary of notations and definitions used in this
paper. For details, one may refer to [3, 5].
For n = 1, 2, . . . we denote by ψn (x) = ψ (n) (x) the polygamma functions as the
′
(x)
, x > 0, where Γ(x) denotes the
n-th derivative of the psi function ψ(x) = ΓΓ(x)
usual gamma function.
Throughout this paper we will fix q ∈ (0, 1). Let a be a complex number. The
q-shifted factorials are defined by
(a; q)n =
n−1
Y
(1 − aq k ),
k=0
(a; q)∞ = lim (a; q)n =
n→∞
n = 1, 2, . . . ,
Y
k>0
95
(1 − aq k ).
96
V. Krasniqi, T. Mansour, A. Sh. Shabani
Jackson [4] defined the q-gamma function as
Γq (x) =
(q; q)∞
(1 − q)1−x ,
(q x ; q)∞
(1.1)
x 6= 0, −1, . . .
It satisfies the functional equation
Γq (x + 1) = [x]q Γq (x),
(1.2)
Γq (1) = 1,
x
where for x complex [x]q = 1−q
1−q .
The q-gamma function has the following integral representation (see [2])
Γq (x) =
Z
1
1−q
tx−1 Eq−qt dq t =
0
Z
∞
1−q
tx−1 Eq−qt dq t,
x > 0.
0
P
j(j−1)
xj
∞
2
where Eqx = ∞
j=0 q
[j]q ! = (1 + (1 − q)x)q , which is the q-analogue of the
classical exponential function.
The q-analogue of the ψ function is defined as the logarithmic derivative of the
q-gamma function
Γ′q (x)
ψq (x) =
, x > 0.
(1.3)
Γq (x)
The q-Jackson integral from 0 to a is defined by (see [4, 5])
Z
a
f (x)dq x = (1 − q)a
0
∞
X
f (aq n )q n .
(1.4)
n=0
For a = ∞ the q-Jackson integral is defined by (see [4, 5])
Z
∞
f (x)dq x = (1 − q)
0
∞
X
f (q n )q n
(1.5)
n=−∞
provided that sums in (1.4) and (1.5) converge absolutely.
In [2] the q-Riemman zeta function is defined as follows (see Section 2.3 for the
definitions)
∞
∞
X
X
1
q (n+α([n]q ))s
ζq (s) =
=
.
(1.6)
{n}sq
[n]sq
n=1
n=1
In relation to (1.3) and (1.6), K. Brahim [1], using a q-analogue of the generalized
Schwarz inequality, proved the following Theorems.
Theorem 1.1. For n = 1, 2 . . .,
2
ψq,n (x)ψq,m (x) > ψq,
m+n (x),
2
(n)
where ψq,n = ψq
is n-th derivative of ψq and
m+n
2
is an integer.
97
Some inequalities for q-polygamma function and ζq -Riemann zeta functions
Theorem 1.2. For all s > 1,
[s + 1]q
ζq (s)
ζq (s + 1)
> q[s]q
.
ζq (s + 1)
ζq (s + 2)
The aim of this paper is to present some inequalities for q-polygamma functions
and q-zeta functions by using a q-analogue of Holder type inequality, similar to
those in [1].
2. Main results
2.1. A lemma
In order to prove our main results, we need the following lemma.
Lemma 2.1. Let a ∈ R+ ∪ {∞}, let f and g be two nonnegative functions and let
p, t > 1 such that p−1 + t−1 = 1. The following inequality holds
Z a
1t
Z a
p1 Z a
g t (x)dq x .
f (x)g(x)dq x 6
f p (x)dq x
0
0
0
Proof. Let a > 0. By (1.4) we have that
Z a
∞
X
f (x)g(x)dq x = (1 − q)a
f (aq n )g(aq n )q n .
0
(2.1)
n=0
By the use of the Holder’s inequality for infinite sums, we obtain
∞
X
n=0
∞
∞
X
p1 X
1t
f (aq n )g(aq n )q n 6
f p (aq n )q n
·
g t (aq n )q n .
n=0
(2.2)
n=0
Hence
(1 − q)a
∞
X
f (aq n )g(aq n )q n
n=0
1
6 ((1 − q)a) p
∞
X
n=0
f p (aq n )q n
p1
1
· ((1 − q)a) t
∞
X
g t (aq n )q n
n=0
The result then follows from (2.1), (2.2) and (2.3).
1t
.
(2.3)
2.2. The q-polygamma function
From (1.1) one can derive the following series representation for the function
ψq (x) =
Γ′q (x)
Γq (x) :
ψq (x) = − log(1 − q) + log q
X q nx
,
1 − qn
n>1
x > 0,
(2.4)
98
V. Krasniqi, T. Mansour, A. Sh. Shabani
which implies that
log q
ψq (x) = − log(1 − q) +
1−q
Z
q
tx−1
dq t.
1−t
0
(2.5)
(n)
Theorem 2.2. For n = 2, 4, 6 . . . set ψq,n (x) = ψq (x) the n-th derivative of the
function ψq . Then for p, t > 1 such that p1 + 1t = 1 the following inequality holds
ψq,n
x
p
+
1
1
y
6 ψq,n (x) p · ψq,n (y) t .
t
(2.6)
Proof. From (2.5) we deduce that
ψq,n (x) =
hence
ψq,n
log q
1−q
Z
q
0
Z
x
y
log q
+
=
p
t
1−q
(log u)n ux−1
dq u,
1−u
q
0
x
(2.7)
y
(log u)n u p + t −1
dq u.
1−u
By Lemma 2.1 with a = q we have
Z
x y h (log u)n i 1t y−1
log q q h (log u)n i p1 x−1
ψq,n
+
=
u p
u q dq u
p
t
1−q 0
1−u
1−u
log q Z q (log u)n ux−1
p1 log q Z q (log u)n uy−1
1t
6
dq u
dq u
1−q 0
1−u
1−q 0
1−u
1
1
= (ψq,n (x)) p (ψq,n (y)) t
where f (u) =
(log u)n
1−u
p
u
x−1
p
and g(u) =
(log u)n
1−u
t
u
y−1
t
.
For p = t = 2 in (2.6) one has the following result.
Corollary 2.3. We have
ψq,n
x + y
2
6
q
ψq,n (x) · ψq,n (y).
2.3. q-zeta function
For x > 0 we set α(x) =
log x
log q
−E
log x
log q .
log x
log q
and {x}q =
[x]q
,
qx+α([x]q )
the integer part of
In [2] the q-zeta function is defined as follows
ζq (s) =
∞
X
∞
X
1
q (n+α([n]q ))s
=
.
s
{n}q
[n]sq
n=1
n=1
where E
log x
log q
is
Some inequalities for q-polygamma function and ζq -Riemann zeta functions
99
There ([2]) it is proved that ζq is a q-analogue of the classical Riemman Zeta
function, and for all s ∈ C such that R(s) > 1, and for all u > 0 one has
Z ∞
1
ζq (s) =
us−1 Zq (u)dq u,
e q (s) 0
Γ
P∞ −{n}q t e
Γ (t)
where Zq (t) = n=1 eq
, Γq (t) = Kqq (t) , and
Kq (t) =
(−(1 − q); q)∞ (−(1 − q)−1 ; q)∞
(1 − q)−s
·
.
1 + (1 − q)−1 (−(1 − q)q s ; q)∞ (−(1 − q)−1 q 1−s ; q)∞
Theorem 2.4. For
1
p
+
1
t
= 1 and xp +
eq x + y
Γ
p
t
y
t
> 1,
1
1
ζqp (x) · ζqt (y)
.
6
1
1
fq p (x) · Γ
fq t (y)
ζq xp + yt
Γ
Proof. From Lemma 2.1 we have that
Z ∞
Z ∞
y
x−1
y−1
1
x
1
u p + t −1 Zq (u)dq u =
u p · (Zq (u)) p u t · (Zq (u)) t dq u.
0
0
1t
Z ∞
p1 Z ∞
6
ux−1 · (Zq (u))dq u ·
uy−1 · (Zq (u))dq u .
0
For f (u) = u
x−1
p
1
p
0
y−1
t
1
t
· (Zq (u)) we obtain that
· (Zq (u)) and g(u) = u
x y x y
1
1
1
1
eq
e qp (x) · Γ
e qt (y) · ζqp (x) · ζqt (y),
Γ
+
· ζq
+
6Γ
p
t
p
t
which completes the proof.
Acknowledgements. The authors would like to thank the anonymous referees
for their comments and suggestions.
References
[1] Brahim, K., Turán-Type Inequalities for some q-special functions, J. Ineq. Pure Appl.
Math., 10(2) (2009) Art. 50.
[2] Fitouhi, A., Bettaibi, N., Brahim, K., The Mellin transform in quantum calculus,
Constructive Approximation, 23(3) (2006) 305–323.
[3] Gasper, G., Rahman, M., Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Applications, 96, Cambridge University Press, Cambridge.
[4] Jackson, F.H., On a q-definite integrals, Quart. J. Pure and Appl. Math., 41 (1910)
193–203.
[5] Kac, V.G., Cheung, P., Quantum Calculus, Universitext, Springer-Verlag, New
York, (2002).
100
Valmir Krasniqi
Department of Mathematics
University of Prishtina
Prishtinë 10000, Republic of Kosova
e-mail: vali.99@hotmail.com
Toufik Mansour
Department of Mathematics
University of Haifa
31905 Haifa, Israel
e-mail: toufik@math.haifa.ac.il
Armend Sh. Shabani
Department of Mathematics
University of Prishtina
Prishtinë 10000, Republic of Kosova
e-mail: armend_shabani@hotmail.com
V. Krasniqi, T. Mansour, A. Sh. Shabani