ON SOME q-INTEGRAL INEQUALITIES

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q-Integral Inequalities Kamel Brahim vol. 9, iss. 4, art. 106, 2008

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ON SOME q-INTEGRAL INEQUALITIES

KAMEL BRAHIM

Institut Préparatoire aux Études d’Ingénieur Tunisia

EMail:Kamel.Brahim@ipeit.rnu.tn

Received: 24 June, 2008

Accepted: 10 November, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26D15.

Key words: q-integral, Inequalities.

Abstract: In this paper, we provide aq-analogue of an open problem posed by Q. A. Ngô et al. in the paper, Note on an integral inequality, J. Inequal. Pure and Appl. Math., 7(4)(2006), Art. 120, by using analytic and elementary methods in Quantum Calculus.

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1. Introduction

In [9], Q.A. Ngô et al. studied an interesting integral inequality and proved the following result:

Theorem 1.1. Letf(x)≥0be a continuous function on[0,1]satisfying

(1.1)

Z 1

x

f(t)dt ≥ Z 1

x

tdt, ∀x∈[0,1].

Then the inequalities

(1.2)

Z 1

0

f^α+1(x)dx≥ Z 1

0

x^αf(x)dx

and

(1.3)

Z 1

0

f^α+1(x)dx≥ Z 1

0

xf^α(x)dx

hold for every positive real numberα >0.

Then, they proposed the following open problem: Under what condition does the inequality

(1.4)

Z 1

0

f^α+β(x)dx≥ Z 1

0

x^βf^α(x)dx

hold forαandβ?

In view of the interest in this type of inequalities, much attention has been paid to the problem and many authors have extended the inequality to more general cases (see [1,3,7,8]). In this paper, we shall discuss aq-analogue of Ngô’s problem.

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This paper is organized as follows: In Section2, we present definitions and facts from theq-calculus necessary for understanding this paper. In Section3, we discuss aq-analogue of the inequalities given in [9] and [3].

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2. Notations and Preliminaries

Throughout this paper, we will fixq ∈(0,1). For the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper (see [4] and [6]). We write fora∈C,

[a]_q = 1−q^a 1−q .

Theq-derivativeDqf of a functionf is given by (2.1) (Dqf)(x) = f(x)−f(qx)

(1−q)x , if x6= 0, (D_qf)(0) =f⁰(0),providedf⁰(0)exists.

Theq-Jackson integral from0toais defined by (see [5]) (2.2)

Z a

0

f(x)d_qx= (1−q)a

∞

X

n=0

f(aqⁿ)qⁿ, provided the sum converges absolutely.

Theq-Jackson integral in a generic interval[a, b]is given by (see [5]) (2.3)

Z b

a

f(x)dqx= Z b

0

f(x)dqx− Z a

0

f(x)dqx.

We recall that for any functionf, we have (see [6])

(2.4) D_q

Z x

a

f(t)d_qt

=f(x).

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If F is any anti q-derivative of the function f, namely D_qF = f, continuous at x= 0, then

(2.5)

Z a

0

f(x)d_qx=F(a)−F(0).

Aq-analogue of the integration by parts formula is given by (2.6)

Z b

a

f(x)(D_qg(x))d_qx=f(a)g(a)−f(b)g(b)− Z b

a

(D_qf(x))g(qx)d_qx.

Finally, we denote

[0,1]_q ={q^k: k = 0,1,2, . . . ,∞}.

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3. Main Results

Let us begin with the following useful result:

Lemma 3.1 ([9] General Cauchy inequality). Letαandβbe positive real numbers satisfyingα+β = 1.Then for all positive real numbersxandy, we always have

(3.1) αx+βy ≥x^αy^β.

Theorem 3.2. Letf be a nonnegative function defined on[0,1]_qsatisfying

(3.2)

Z 1

x

f^β(t)d_qt≥ Z 1

x

t^βd_qt, ∀x∈[0,1]_q. Then the inequality

(3.3)

Z 1

0

f^α+β(x)d_qx≥ Z 1

0

x^αf^β(x)d_qx,

holds for all positive real numbersα >0andβ >0.

To prove Theorem3.2, we need the following lemma.

Lemma 3.3. Under the conditions of Theorem3.2, we have (3.4)

Z 1

0

x^αf^β(x)d_qx≥ 1 [α+β+ 1]q

.

Proof. By using aq-integration by parts, we obtain Z 1

0

x^α−1 Z 1

x

f^β(t)d_qt

d_qx= 1 [α]_q

x^α

Z 1

x

f^β(t)d_qt ^x=1

x=0

+ q^α [α]_q

Z 1

0

x^αf^β(x)d_qx

= q^α [α]q

Z 1

0

x^αf^β(x)d_qx,

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which yields (3.5)

Z 1

0

x^αf^β(x)d_qx= [α]_q q^α

Z 1

0

x^α−1 Z 1

x

f^β(t)d_qt

d_qx.

On the other hand, from condition (3.2), we get Z 1

0

x^α−1 Z 1

x

f^β(t)d_qt

d_qx≥ Z 1

0

x^α−1 Z 1

x

t^βd_qt

d_qx

= 1

[β+ 1]q

Z 1

0

(x^α−1−x^α+β)d_qx

= q^α

[α]q[α+β+ 1]q

.

Therefore, from (3.5), we obtain (3.6)

Z 1

0

x^αf^β(x)d_qx≥ 1 [α+β+ 1]_q.

We now give the proof of Theorem3.2.

Proof of Theorem3.2. Using Lemma3.1, we obtain

(3.7) β

α+βf^α+β(x) + α

α+βx^α+β ≥x^αf^β(x), which gives

(3.8) β Z 1

0

f^α+β(x)d_qx+α Z 1

0

x^α+βd_qx≥(α+β) Z 1

0

x^αf^β(x)d_qx.

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Moreover, by using Lemma3.3, we get (α+β)

Z 1

0

x^αf^β(x)d_qx=α Z 1

0

x^αf^β(x)d_qx+β Z 1

0

x^αf^β(x)d_qx (3.9)

≥ α

[α+β+ 1]_q +β Z 1

0

x^αf^β(x)d_qx.

Then, from relation (3.8), we obtain (3.10) β

Z 1

0

f^α+β(x)d_qx+ α

[α+β+ 1]_q ≥ α

[α+β+ 1]_q +β Z 1

0

x^αf^β(x)d_qx, which completes the proof.

Takingβ = 1in Theorem3.2, we obtain

Corollary 3.4. Letf be a nonnegative function defined on[0,1]qsatisfying

(3.11)

Z 1

x

f(t)dqt≥ Z 1

x

tdqt, ∀x∈[0,1]q.

Then the inequality

(3.12)

Z 1

0

f^α+1(x)d_qx≥ Z 1

0

x^αf(x)d_qx

holds for every positive real numberα >0.

Theorem 3.5. Letf be a nonnegative function defined on[0,1]_qsatisfying

(3.13)

Z 1

x

f(t)d_qt≥ Z 1

x

td_qt, ∀x∈[0,1]_q.

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Then the inequality

(3.14)

Z 1

0

f^α+1(x)d_qx≥ Z 1

0

xf^α(x)d_qx

holds for every positive real numberα >0.

Proof. We have

(3.15) ∀x∈[0,1]q, (f^α(x)−x^α)(f(x)−x)≥0, so

(3.16) f^α+1(x) +x^α+1 ≥x^αf(x) +xf^α(x).

By integrating with some simple calculations we deduce that (3.17)

Z 1

0

f^α+1(x)d_qx+ 1 [α+ 2]_q ≥

Z 1

0

x^αf(x)d_qx+ Z 1

0

xf^α(x)d_qx.

Then, from Lemma3.3forβ= 1, the result follows.

Theorem 3.6. Letf be a nonnegative function defined on[0,1]qsatisfying

(3.18)

Z 1

x

f(t)d_qt≥ Z 1

x

td_qt, ∀x∈[0,1]_q. Then the inequality

(3.19)

Z 1

0

f^α+β(x)dqx≥ Z 1

0

x^αf^β(x)dqx holds for all real numbersα >0andβ ≥1.

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Lemma 3.7. Under the conditions of Theorem3.6, we have (3.20)

Z 1

0

x^αf^β(x)d_qx≥ 1 [α+β+ 1]_q for all real numbersα >0andβ ≥1.

Proof. Using Lemma3.1, we obtain

(3.21) 1

βf^β(x) + β−1

β x^β ≥x^β−1f(x), which implies

(3.22)

Z 1

0

x^αf^β(x)d_qx+ (β−1) Z 1

0

x^α+βd_qx≥β Z 1

0

x^α+β−1f(x)d_qx.

Therefore, from Lemma3.3, we get (3.23)

Z 1

0

x^αf^β(x)d_qx+ β−1

[α+β+ 1]_q ≥ β [α+β+ 1]_q. Thus (3.20) is proved.

We now give the proof of Theorem3.6.

Proof of Theorem3.6. By using Lemma3.1, we obtain

(3.24) β

α+βf^α+β(x) + α

α+βx^α+β ≥x^αf^β(x), which implies

(3.25) β

Z 1

0

f^α+β(x)d_qx+ α

[α+β+ 1]_q ≥(α+β) Z 1

0

x^αf^β(x)d_qx.

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Then, from Lemma3.7, we obtain (3.26) β

Z 1

0

f^α+β(x)d_qx+ α

[α+β+ 1]_q ≥ α

[α+β+ 1]_q +β Z 1

0

x^αf^β(x)d_qx, which completes the proof.

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References

[1] L. BOUGOFFA, Note on an open problem, J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 58. [ONLINE: http://jipam.vu.edu.au/article.

php?sid=871].

[2] L. BOUGOFFA, Corrigendum of the paper entitled: Note on an open problem, J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 121. [ONLINE:http://

jipam.vu.edu.au/article.php?sid=910].

[3] K. BOUKERRIOUA AND A. GUEZANE-LAKOUD, On an open question re- garding an integral inequality, J. Inequal. Pure and Appl. Math., 8(3) (2007), Art.

77. [ONLINE:http://jipam.vu.edu.au/article.php?sid=885].

[4] G. GASPER AND M. RAHMAN, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge Uni- versity Press, Cambridge.

[5] F.H. JACKSON, Onq-definite integrals, Quarterly Journal of Pure and Applied Mathematics, 41 (1910), 193–203.

[6] V.G. KACANDP. CHEUNG, Quantum Calculus, Universitext, Springer-Verlag, New York, (2002).

[7] W.J. LIU, C.C LIANDJ.W. DONG, On an open problem concerning an integral inequality, J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 74. [ONLINE:

http://jipam.vu.edu.au/article.php?sid=882].

[8] W.J. LIU, G.S. CHENGANDC.C LI, Further development of an open problem, J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 14. [ONLINE: http://

jipam.vu.edu.au/article.php?sid=952].

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[9] Q.A. NGÔ, D.D. THANG, T.T. DAT AND D.A. TUAN, Note on an integral inequality, J. Inequal. Pure and Appl. Math., 7(4) (2006), Art. 120. [ONLINE:

http://jipam.vu.edu.au/article.php?sid=737].