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

(2010) pp. 95–100

http://ami.ektf.hu

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

Valmir Krasniqi

, Toufik Mansour

Armend Sh. Shabani

aDepartment of Mathematics, University of Prishtina, Prishtinë 10000, Republic of Kosova

bDepartment of Mathematics, University of Haifa, 31905 Haifa, Israel Submitted 26 February 2010; Accepted 10 June 2010

Abstract

In this paper, we present some inequalities forq-polygamma functions and ζq-Riemann Zeta functions, using aq-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].

Forn= 1,2, . . .we denote byψn(x) =ψ⁽ⁿ⁾(x)the polygamma functions as the n-th derivative of the psi function ψ(x) = ^Γ

′(x)

Γ(x), x > 0, where Γ(x) denotes the usual gamma function.

Throughout this paper we will fixq∈(0,1). Letabe a complex number. The q-shifted factorials are defined by

(a;q)n =

n−1Y

k=0

(1−aq^k), n= 1,2, . . . ,

(a;q)∞= lim

n→∞(a;q)n =Y

k>0

(1−aq^k).

95

Jackson [4] defined theq-gamma function as Γq(x) = (q;q)∞

(q^x;q)∞(1−q)^1−x, x6= 0,−1, . . . (1.1) It satisfies the functional equation

Γq(x+ 1) = [x]qΓq(x), Γq(1) = 1, (1.2) where forxcomplex[x]q = ^1−q_1−q^x.

Theq-gamma function has the following integral representation (see [2]) Γq(x) =

Z ₁₋¹_q

0

t^x−1Eq^−qtdqt= Z ₁₋^∞_q

0

t^x−1Eq^−qtdqt, x >0.

where E^x_q = P∞

j=0q^j(j2−1)_[j]^xj

q! = (1 + (1−q)x)^∞_q , which is the q-analogue of the classical exponential function.

Theq-analogue of theψfunction is defined as the logarithmic derivative of the q-gamma function

ψq(x) =Γ^′q(x)

Γq(x), x >0. (1.3)

Theq-Jackson integral from0to ais defined by (see [4, 5]) Z a

0

f(x)dqx= (1−q)a X∞

n=0

f(aqⁿ)qⁿ. (1.4)

Fora=∞theq-Jackson integral is defined by (see [4, 5]) Z ∞

0

f(x)dqx= (1−q) X∞

n=−∞

f(qⁿ)qⁿ (1.5)

provided that sums in (1.4) and (1.5) converge absolutely.

In [2] theq-Riemman zeta function is defined as follows (see Section 2.3 for the definitions)

ζq(s) = X∞

n=1

1 {n}^s_q =

X∞

n=1

q^{(n+α([n]q))s}

[n]^s_q . (1.6)

In relation to (1.3) and (1.6), K. Brahim [1], using aq-analogue of the generalized Schwarz inequality, proved the following Theorems.

Theorem 1.1. Forn= 1,2. . .,

ψq,n(x)ψq,m(x)>ψ_q,²m+n 2 (x),

whereψq,n=ψq⁽ⁿ⁾ isn-th derivative ofψq and ^m+n₂ is an integer.

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 forq-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₊∪ {∞}, letf andg be two nonnegative functions and let p, t >1 such thatp⁻¹+t⁻¹= 1. The following inequality holds

Z a 0

f(x)g(x)dqx6 Z ^a

0

f^p(x)dqx¹_p Z ^a

0

g^t(x)dqx¹_t . Proof. Leta >0. By (1.4) we have that

Z a 0

f(x)g(x)dqx= (1−q)a X∞

n=0

f(aqⁿ)g(aqⁿ)qⁿ. (2.1) By the use of the Holder’s inequality for infinite sums, we obtain

X^∞

n=0

f(aqⁿ)g(aqⁿ)qⁿ 6

X^∞

n=0

f^p(aqⁿ)qⁿ_p¹

·X^∞

n=0

g^t(aqⁿ)qⁿ¹_t

. (2.2)

Hence

(1−q)aX^∞

n=0

f(aqⁿ)g(aqⁿ)qⁿ

6((1−q)a)^1pX^∞

n=0

f^p(aqⁿ)qⁿ¹_p

·((1−q)a)^1tX^∞

n=0

g^t(aqⁿ)qⁿ¹_t

. (2.3) The result then follows from (2.1), (2.2) and (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) + logqX

n>1

q^nx

1−qⁿ, x >0, (2.4)

which implies that

ψq(x) =−log(1−q) + logq 1−q

Z q 0

t^x−1

1−tdqt. (2.5)

Theorem 2.2. For n= 2,4,6. . . setψq,n(x) =ψq⁽ⁿ⁾(x)the n-th derivative of the function ψq. Then for p, t >1 such that ¹_p +¹_t = 1 the following inequality holds

ψq,n

x p+y

t

6ψq,n(x)^1p·ψq,n(y)^1t. (2.6) Proof. From (2.5) we deduce that

ψq,n(x) = logq 1−q

Z q 0

(logu)ⁿu^x−1

1−u dqu, (2.7)

hence

ψq,n

x p+y

t

= logq 1−q

Z q 0

(logu)ⁿu^xp+yt−1 1−u dqu.

By Lemma 2.1 witha=qwe have ψq,n

x p+y

t

= logq 1−q

Z q 0

h(logu)ⁿ 1−u

i¹_p u^x

−1 p

h(logu)ⁿ 1−u

i¹_t u^y

−1 q dqu

6logq 1−q

Z q 0

(logu)ⁿu^x−1

1−u dqu¹_plogq 1−q

Z q 0

(logu)ⁿu^y−1 1−u dqu¹_t

= (ψq,n(x))^1p(ψq,n(y))^1t wheref(u) =

(logu)ⁿ 1−u

p

u^x

−1

p andg(u) =

(logu)ⁿ 1−u

t

u^y−t1.

Forp=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

Forx >0 we setα(x) = ^log_log^x_q −E

logx logq

and{x}q = _qx+α([x]^[x]q q), whereE

logx logq

is the integer part of ^log_log^x_q.

In [2] theq-zeta function is defined as follows ζq(s) =

X∞

n=1

1 {n}^s_q =

X∞

n=1

q^{(n+α([n]q))s} [n]^s_q .

There ([2]) it is proved that ζq is a q-analogue of the classical Riemman Zeta function, and for all s∈Csuch thatR(s)>1, and for allu >0one has

ζq(s) = 1 Γeq(s)

Z ∞ 0

u^s−1Zq(u)dqu, whereZq(t) =P∞

n=1e^−{n}q ^qt,Γeq(t) = _K^Γq(t)

q(t), and Kq(t) = (1−q)^−s

1 + (1−q)⁻¹ · (−(1−q);q)∞(−(1−q)⁻¹;q)∞

(−(1−q)q^s;q)∞(−(1−q)⁻¹q^1−s;q)∞

.

Theorem 2.4. For ¹_p+¹_t = 1 and ^x_p+^y_t >1, eΓq

x p+^y_t Γfq

1 p(x)·fΓq

1 t(y)

6ζ

1

qp(x)·ζq^1t(y) ζq

x

p+^y_t . Proof. From Lemma 2.1 we have that

Z ∞ 0

u^xp+yt−1Zq(u)dqu= Z ∞

0

u^x−1p ·(Zq(u))^1pu^y−1t ·(Zq(u))^1tdqu.

6 Z ^∞

0

u^x−1·(Zq(u))dqu_p¹

· Z ^∞

0

u^y−1·(Zq(u))dqu¹_t .

Forf(u) =u^x−1p ·(Zq(u))^1p andg(u) =u^y−1t ·(Zq(u))^1t we obtain that Γeq

x p+y

t ·ζq

x p+y

t

6eΓ

1

qp(x)·eΓq^1t(y)·ζ

1

qp(x)·ζ

1

qt(y),

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 someq-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), En- cyclopedia of Mathematics and Applications, 96, Cambridge University Press, Cam- bridge.

[4] Jackson, F.H., On aq-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).

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