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volume 4, issue 4, article 77, 2003.

Received 01 July, 2003;

accepted 08 October, 2003.

Communicated by:F. Qi

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Journal of Inequalities in Pure and Applied Mathematics

NOTES ON QI TYPE INTEGRAL INEQUALITIES

LAZHAR BOUGOFFA

King Khalid University Faculty of Science Department of Mathematics P. O. Box 9004

Abha Saudi Arabia.

EMail:abogafah@kku.edu.sa

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Notes on Qi Type Integral Inequalities Lazhar Bougoffa

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J. Ineq. Pure and Appl. Math. 4(4) Art. 77, 2003

http://jipam.vu.edu.au

Abstract

By utilizing a reversed Hölder inequality and a reversed convolution inequality, some new integral inequalities which originate from an open problem posed in [F. Qi, Several integral inequalities, J. Inequal. Pure Appl. Math. 1(2) (2000), Art. 19. Available online athttp://jipam.vu.edu.au/v1n2.html/001_

00.html. RGMIA Res. Rep. Coll. 2 (1999), no. 7, Art. 9, 1039–1042. Available online athttp://rgmia.vu.edu.au/v2n7.html] are established.

2000 Mathematics Subject Classification: Primary 26D15; Secondary 39B62, 60E15.

Key words: Qi type integral inequality, Reversed Hölder inequality, Reversed convo- lution inequality.

The author would like to thank the anonymous referees for their valuable help to re- new the first section, add and correct several references, and improve the typesetting of this paper.

1. Introduction

In [6] F. Qi posed the following open problem:

Under what conditions does the inequality

(1.1)

Z b

a

f(x)t

dx≥ Z b

a

f(x)dx ^t−1

hold for t >1?

In response, affirmative answers, generalizations, reversed forms, and inter- pretations of inequality (1.1) have been obtained by several mathematicians.

The paper [9] first gave an affirmative answer to this open problem using the integral version of Jensen’s inequality and a lemma of convexity. The second affirmative answer to (1.1) was given by Towghi in [8]. Motivated by (1.1), Pogány [5] proved the following

(1.2)

Z b

a

f(x)α

dx≥ Z b

a

f(x)dx ^β

and its reversed form under assumptions of the bounds, depending onb−a,α andβ, and convexity off, which contains an answer to the above open problem and some reversed forms of (1.1).

In [3, 4], by employing a functional inequality which is an abstract general- ization of the classical Jessen’s inequality, the following functional inequality (1.4) was established, from which inequality (1.1), some integral inequalities,

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LetLbe a linear vector space of real-valued functions, pandqbe two real numbers such that p ≥q ≥ 1. Assume that f andg are two positive functions in L and Gis a positive linear form on L such that G(g) > 0, f g ∈ L, and gf^p ∈ L. If

(1.3) [G(gf)]^p−q ≥[G(g)]^p−1, then

(1.4) G(gf^p)≥[G(gf)]^q.

Very recently, Csiszár and Móri [1] interpreted inequality (1.2) in terms of moments as

(1.5) E(X^α)≥C(EX)^β,

where C = (b−a)^β−1 and X = f is a random variable. To demonstrate the power of the convexity method in probability theory, among other things, they found sharp conditions on the range of X, under which (1.5) or the converse inequality holds for fixed0 < β < α. Hence, the results by Pogány [5] were improved slightly by a factor of at least ³₂_α−β^α

.

In this short note, by utilizing the reversed Hölder inequality in [2] and a reversed convolution inequality in [7], we establish some new Qi type integral inequalities which extend related results in references.

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2. Two Lemmas

In order to prove our main results, the following two lemmas are necessary.

Lemma 2.1 ([2]). For two positive functionsf andg satisfying0< m≤ ^f_g ≤ M <∞on the setX, and forp > 1andq >1with ¹_p +¹_q = 1, we have

(2.1)

Z

X

f dµ ¹_pZ

X

gdµ ¹_q

≤A_p,qm M

Z

X

f¹^pg¹^qdµ,

where

(2.2) A_p,q(t) = 1

p¹^pq¹^q

· 1−t t^pq¹

1−t¹^p¹_p

1−t¹^q¹_q .

Inequality (2.1) is called the reverse Hölder inequality.

Lemma 2.2 ([7]). For two positive functionsf andg satisfying0< m≤ ^f_g^pq ≤ M <∞on the setX, and forp > 1andq >1with ¹_p +¹_q = 1, we have

(2.3)

Z

X

f^pdµ ¹_pZ

X

g^qdµ ¹_q

≤ M

m _pq¹ Z

X

f gdµ.

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3. Main Results and Proofs

In this section, we will state our main results and give their proofs as follows.

Using Lemma2.1and an estimation due to Lars-Erik Persson, we obtain:

Theorem 3.1. If0< m≤f ≤M < ∞on[a, b], then

(3.1)

Z

[a,b]

f¹^pdµ≥B Z

[a,b]

f dµ ¹_p−1

,

whereB =m[µ(b)−µ(a)]¹⁺¹^q _M^m_pq¹

andp >1,q >1with ¹_p +¹_q = 1.

Proof. In Lemma2.1, puttingg ≡1yields (3.2) [µ(b)−µ(a)]¹^q

Z

[a,b]

f dµ ¹_p

≤A_p,q m

M Z

[a,b]

f¹^pdµ,

and so (3.3)

Z

[a,b]

f¹^pdµ≥ [µ(b)−µ(a)]¹^q A_p,q _M^m

Z

[a,b]

f dµ

¹_p⁻¹Z

[a,b]

f dµ

.

Since0< m≤f, we have (3.4)

Z

[a,b]

f¹^pdµ≥ m[µ(b)−µ(a)]¹⁺¹^q A_p,q _M^m

Z

[a,b]

f dµ _p¹−1

.

Now, using the estimationA_p,q(t)< t⁻^pq¹ due to Lars-Erik Persson (see [7]), we obtain inequality (3.1).

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Corollary 3.2. Letp >1andq >1with ¹_p +¹_q = 1. If

(3.5) mm

M _pq¹

= 1

[µ(b)−µ(a)]¹⁺¹^q and0< m≤f ≤M <∞on[a, b], then

(3.6)

Z

[a,b]

f¹^pdµ≥ Z

[a,b]

f dµ ¹_p−1

.

Using Lemma2.1, Lemma2.2and the estimation due to Lars-Erik Persson, we have the following:

Theorem 3.3. Letp > 1andq >1with ¹_p+¹_q = 1. If0< m¹^p ≤f ≤M¹^p <∞ on[a, b], then

(3.7)

Z

[a,b]

f¹^pdµ p

≥[µ(b)−µ(a)]^p+1^q m M

_pq² Z

[a,b]

f^pdµ ¹_p

.

Proof. Puttingg ≡1into Lemma2.2, we obtain (3.8)

Z

[a,b]

f^pdµ ¹_p

[µ(b)−µ(a)]¹^q ≤m M

−_pq¹ Z

[a,b]

f dµ.

Therefore

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Again, substituting g ≡ 1 in the reverse Hölder inequality in Lemma 2.1 leads to

Z

[a,b]

f dµ ¹_p

≤[µ(b)−µ(a)]⁻¹^q A_p,q m¹^p M^p¹

!Z

[a,b]

f¹^pdµ,

and so, (3.10)

Z

[a,b]

f dµ ≤[µ(b)−µ(a)]⁻^p^q A^p_p,q m¹^p M¹^p

!Z

[a,b]

f¹^pdµ p

.

Combining (3.9) with (3.10), we obtain (3.11)

Z

[a,b]

f¹^pdµ p

≥[µ(b)−µ(a)]^p+1^q m M

_pq¹

A^−p_p,q m¹^p M¹^p

!Z

[a,b]

f^pdµ ¹_p

.

Then, by using the estimationAp,q(t)< t⁻^pq¹ due to Lars-Erik Persson (see [7]), we obtain inequality (3.7).

Corollary 3.4. If 0< m^p¹ ≤f ≤M¹^p <∞on[a, b]and_M^m = [µ(b)−µ(a)]

−p(p+1) 2

forp >1, then

(3.12)

Z

[a,b]

f¹^pdµ p

≥ Z

[a,b]

f^pdµ ¹_p

.

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References

[1] V. CSISZÁR AND T.F. MÓRI, The convexity method of proving moment- type inequalities, Statist. Probab. Lett., (2003), accepted.

[2] X.-H. LIU, On the inverses of Hölder inequalities, Math. Practice Theory, 20(1) (1990), 84–88. (Chinese)

[3] S. MAZOUZI AND F. QI, On an open problem by Feng Qi regarding an integral inequality, RGMIA Res. Rep. Coll., 6(1) (2003), Art. 6. Available online athttp://rgmia.vu.edu.au/v6n1.html.

[4] S. MAZOUZI AND F. QI, On an open problem regarding an integral in- equality, J. Inequal. Pure Appl. Math., 4(2) (2003), Art. 31. Available online athttp://jipam.vu.edu.au/v4n2/029_03.html.

[5] T.K. POGÁNY, On an open problem of F. Qi, J. Inequal. Pure Appl. Math., 3(4) (2002), Art. 54. Available online athttp://jipam.vu.edu.au/

v3n4/016_01.html.

[6] F. QI, Several integral inequalities, J. Inequal. Pure Appl. Math., 1(2) (2000), Art. 19. Available online at http://jipam.vu.edu.au/

v1n2.html/001_00.html. RGMIA Res. Rep. Coll., 2(7) (1999), Art. 9, 1039–1042. Available online at http://rgmia.vu.edu.au/

v2n7.html.

[7] S. SAITOH, V.K. TUAN ANDM. YAMAMOTO, Reverse convolution in- equalities and applications to inverse heat source problems, J. Inequal. Pure

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[8] N. TOWGHI, Notes on integral inequalities, RGMIA Res. Rep. Coll., 4(2) (2001), Art. 12, 277–278. Available online athttp://rgmia.vu.edu.

au/v4n2.html.

[9] K.-W. YU AND F. QI, A short note on an integral inequality, RGMIA Res. Rep. Coll., 4(1) (2001), Art. 4, 23–25. Available online at http:

//rgmia.vu.edu.au/v4n1.html.