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volume 7, issue 5, article 178, 2006.

Received 25 January, 2006;

accepted 22 March, 2006.

Communicated by:L. Pick

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Journal of Inequalities in Pure and Applied Mathematics

Aq-ANALOGUE OF AN INEQUALITY DUE TO KONRAD KNOPP

MEHMET ZEKI SARIKAYA AND HÜSEYIN YILDIRIM

Department of Mathematics Faculty of Science and Arts Kocatepe University Afyon-TURKEY

EMail:sarikaya@aku.edu.tr EMail:hyildir@aku.edu.tr

c

2000Victoria University ISSN (electronic): 1443-5756 085-06

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Some Hardy Type Integral Inequalities

Mehmet Zeki Sarikaya and Hüseyin Yildirim

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J. Ineq. Pure and Appl. Math. 7(5) Art. 178, 2006

http://jipam.vu.edu.au

Abstract

In this note, we obtain some new generalizations of the Hardy’s integral inequality by using a fairly elementary analysis. These inequalities generalize some known results and simplify the proofs of some existing results.

2000 Mathematics Subject Classification:26D10 and 26D15.

Key words: Hardy integral inequalities and Hölder’s inequality.

1. Introduction

In [4], Hardy proved the following inequality. Ifp >1, f ≥0and F(x) =

Z x

0

f(t)dt,

then (1.1)

Z ∞

0

F x

p

dx < q^p Z ∞

0

f^p(x)dx

unlessf ≡0.The constantq=p(p−1)⁻¹ is the best possible. This inequality plays an important role in analysis and its applications. It is obvious that, for parametersaandb such that0 < a < b < ∞,the following inequality is also valid

(1.2)

Z b

a

F x

p

dx < q^p Z b

a

f^p(x)dx,

where 0 < R∞

0 f^p(x)dx < ∞. The classical Hardy inequality asserts that if p > 1andf is a nonnegative measurable function on (a, b), then (1.2) is true unless f ≡ 0 a.e. in (a, b), where the constant here is best possible. This inequality remains true provided that0< a < b <∞.

In particular, Hardy [3] in 1928 gave a generalized form of the inequality (1.1) when he showed that for anym 6= 1, p > 1and any integrable function f(x)≥0on(0,∞)for which

F(x) = ( Rx

0 f(t)dt for m >1, R∞

x f(t)dt for m <1,

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then (1.3)

Z ∞

0

x^−mF^p(x)dx <

p

|m−1|

pZ ∞

0

x^−m[xf(x)]^pdx

unlessf ≡0,where the constant is also best possible.

Because of their fundamental importance in the discipline over the years much effort and time has been devoted to the improvement and generalization of Hardy’s inequalities (1.1), (1.2) and (1.3). These include, among others, the works in [1] – [9].

The objective of this paper is to obtain further generalizations of the classical Hardy integral inequality which will be useful in applications by using some elementary methods of analysis. Throughout this paper, the left-hand sides of the inequalities exist if the right-hand sides exist.

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2. Main Results

The following theorems are the main results of the present paper.

Theorem 2.1. Let p > 1, m > 1 be constants. Let f(x) be a nonnegative and integrable function on (0,∞)and letz(x)be differentiable in(0,∞)with z⁰(x)>0andz(0₊)>0.Letw(x)andr(x)be positive and absolutely contin- uous functions on(0,∞).Let

1− 1

m−1 z(x) z⁰(x)

w⁰(x)

w(x) + p m−1

z(x) z⁰(x)

r⁰(x) r(x) ≥ 1

λ >0.

Leta∈(0,∞)be fixed and set

F(x) := 1 r(x)

Z x

a

r(t)z⁰(t)f(t)

z(t) dt, x∈(0,∞).

Then the inequality (2.1)

Z b

a

w(x) z⁰(x)

z^m(x)F^p(x)dx≤

λp m−1

pZ b

a

w(x)z⁰(x)

z^m(x)f^p(x)dx holds for allb≥a.

Proof. Integrating the left-hand side of inequality (2.1) by parts gives Z b

a

w(x) z⁰(x)

z^m(x)F^p(x)dx

=w(b)[z(b)]^−m+1

−m+ 1 F^p(b) + 1 m−1

Z b

a

z^−m+1w⁰F^pdx

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− p m−1

Z b

a

z^−m+1wr⁰(x)

r(x)F^pdx+ p m−1

Z b

a

wz⁰(x)

z^m(x)F^p−1f dx.

Sincem >1andF(b)≥0,we have Z b

a

w(x)z⁰(x)

z^m(x)F^p(x)

1− 1

m−1 z(x) z⁰(x)

w⁰(x)

w(x) + p m−1

z(x) z⁰(x)

r⁰(x) r(x)

dx

=w(b)[z(b)]^−m+1

−m+ 1 F^p(b) + p m−1

Z b

a

w z⁰(x)

z^m(x)F^p−1f dx

≤ p m−1

Z b

a

w z⁰(x)

z^m(x)F^p−1f dx.

Here, using the assumption onλ,we have Z b

a

w(x)z⁰(x)

z^m(x)F^p(x)1 λdx

≤ Z b

a

w(x)z⁰(x)

z^m(x)F^p(x)

1− 1

m−1 z(x) z⁰(x)

w⁰(x)

w(x) + p m−1

z(x) z⁰(x)

r⁰(x) r(x)

dx

≤ p m−1

Z b

a

wz⁰(x)

z^m(x)F^p−1f dx.

By Hölder’s inequality, Z b

a

w(x) z⁰(x)

z^m(x)F^p(x)dx≤ λp m−1

Z b

a

w z⁰(x) z^m(x)F^pdx

1

q Z b

a

w z⁰(x) z^m(x)f^pdx

1 p

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and thus on simplification, we have Z b

a

w(x) z⁰(x)

z^m(x)F^p(x)dx≤

λp m−1

pZ b

a

w z⁰(x) z^m(x)f^pdx.

This proves the theorem.

Theorem 2.2. Letp, m, f, z, z⁰, w andrbe as in Theorem2.1. Let

1− 1

m−1 z(x) z⁰(x)

w⁰(x)

w(x) − p m−1

z(x) z⁰(x)

r⁰(x) r(x) ≥ 1

λ >0.

F(x) :=r(x) Z x

a

z⁰(t)f(t)

z(t)r(t)dt, x∈(0,∞).

Z b

a

w(x) z⁰(x)

z^m(x)F^p(x)dx≤

λp m−1

pZ b

a

w(x)z⁰(x)

z^m(x)f^p(x)dx

holds for allb≥a.

Proof. This is similar to the proof of the Theorem2.1.

1− 1

m−1 z(x) z⁰(x)

w⁰(x)

w(x) + p m−1

z(x) z⁰(x)

r⁰(x) r(x) ≥ 1

λ >0.

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F(x) := 1 r(x)

Z x

x 2

r(t)z⁰(t)f(t)

z(t) dt, x∈(0,∞).

Z b

0

w(x)z⁰(x)

z^m(x)F^p(x)dx≤

λp m−1

pZ b

0

w(x)[z(x)]^p−m

[z⁰(x)]^p^q |g(x)|^pdx

holds where

g(x) = 1 r(x)

r(x)z⁰(x)f(x) z(x) − 1

2

r(^x₂)z⁰(^x₂)f(^x₂) z(^x₂)

.

Proof. Upon integrating by parts we have

Z b

0

w(x) z⁰(x)

z^m(x)F^p(x)dx

=w(b)[z(b)]^−m+1

−m+ 1 F^p(b) + 1 m−1

Z b

0

z^−m+1w⁰F^pdx

− p m−1

Z b

0

z^−m+1wr⁰(x)

r(x)F^pdx+ p m−1

Z b

0

z^−m+1wF^p−1|g(x)|dx,

where

g(x) = 1 r(x)

r(x)z⁰(x)f(x) z(x) − 1

2

r(^x₂)z⁰(^x₂)f(^x₂) z(^x₂)

.

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Sincem >1andF(b)≥0we have Z b

0

w(x)z⁰(x)

z^m(x)F^p(x)

1− 1

m−1 z(x) z⁰(x)

w⁰(x)

w(x) + p m−1

z(x) z⁰(x)

r⁰(x) r(x)

dx

=w(b)[z(b)]^−m+1

−m+ 1 F^p(b) + p m−1

Z b

0

w z⁰(x)

z^m(x)F^p−1f dx

≤ p m−1

Z b

0

z^−m+1wF^p−1|g(x)|dx.

By the assumption onλ,we have Z b

0

w(x)z⁰(x)

z^m(x)F^p(x)1 λdx

≤ Z b

0

w(x)z⁰(x)

z^m(x)F^p(x)

1− 1

m−1 z(x) z⁰(x)

w⁰(x)

w(x) + p m−1

z(x) z⁰(x)

r⁰(x) r(x)

dx

≤ p m−1

Z b

0

z(x)

z⁰(x)w z⁰(x)

z^m(x)F^p−1|g(x)|dx.

By Hölder’s inequality Z b

0

w(x) z⁰(x)

z^m(x)F^p(x)dx

≤ λp m−1

Z b

0

w z⁰(x) z^m(x)F^pdx

1

q Z b

0

wz^p−m(x) [z⁰(x)]^p^q

|g(x)|^pdx

!¹_p

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and thus on simplification, we have Z b

0

w(x)z⁰(x)

z^m(x)F^p(x)dx≤

λp m−1

pZ b

0

wz^p−m(x)

[z⁰(x)]^p^q |g(x)|^pdx.

This proves the theorem.

1− 1

m−1 z(x) z⁰(x)

w⁰(x)

w(x) + p m−1

z(x) z⁰(x)

r⁰(x) r(x) ≥ 1

λ >0.

F(x) :=r(x) Z x

x 2

z⁰(t)f(t)

z(t)r(t)dt, x∈(0,∞).

Z b

0

w(x)z⁰(x)

z^m(x)F^p(x)dx≤

λp m−1

pZ b

0

w(x)[z(x)]^p−m

[z⁰(x)]^p^q |g(x)|^pdx

holds where

g(x) = r(x)

r(x)z⁰(x)f(x) z(x) − 1

2

r(^x₂)z⁰(^x₂)f(^x₂) z(^x₂)

. Proof. This is similar to the proof of the Theorem2.3.

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Theorem 2.5. Let p > q > 0, ¹_p + ¹_q = 1, and m < 1 be real numbers.

Let z(x) be differentiable in (0,∞) with z⁰(x) > 0and z(0₊) > 0, let w(x) and r(x)be positive and absolutely continuous functions on (0,∞), n and let f : [0,∞)→[0,∞)be integrable so that

1 + 1

m−1 z(x) z⁰(x)

w⁰(x) w(x) − p

q 1 m−1

z(x) z⁰(x)

r⁰(x) r(x) ≥ 1

λ >0

a.e. for someλ >0.Letb∈(0,∞)be fixed and set

F(x) := 1 r(x)

Z b

x

r(t)z⁰(t)f(t)

z(t) dt, x∈(0,∞).

Then the following inequality (2.5)

Z b

a

w(x) z⁰(x)

z^m(x)F^p^q (x)dx≤

λp m−1

pZ b

a

w(x)z⁰(x)

z^m(x)f^p^q(x)dx

holds for all0≤a≤b.

Proof. This is similar to the proof of the Theorem2.1.

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References

[1] W.S. CHEUNG, Z. HANJS ANDJ. PE ˇCARI ´C, Some Hardy-type inequali- ties, J. Math. Anal. Appl., 250(2) (2000), 621–634.

[2] G.H. HARDY, J.E. LITTLEWOOD AND G. PÓLYA, Inequalities, Cam- bridge University Press, (1952).

[3] G.H. HARDY, Notes on some points in the integral calculus, Messenger Math., 57 (1928), 12–16.

[4] G.H. HARDY, Notes on a theorem of Hilbert, Math. Z., 6 (1920), 314–317.

[5] B.G. PACHPATTE, On some variants of Hardy’s inequality, J. Math. Anal.

Appl., 124 (1987), 495–501.

[6] B.G. PACHPATTE, On some generalizations of Hardy’s integral inequality, J. Math. Anal. Appl., 234(1) (1999), 15–30.

[7] B.G. PACHPATTE, On a new class of Hardy type inequalities, Proc. R. Soc.

Edin., 105A (1987), 265–274.

[8] B. YANG, Z. ZENG AND L. DEBNATH, On New Generalizations of Hardy’s Integral Inequality, J. Math. Anal. Appl., 217(1) (1998), 321–327.

[9] J.E. PE ˇCARI ´CAND E.R. LOVE, Still more generalizations of Hardy’s in- equality, J. Austral. Math. Soc. Ser. A, 58 (1995), 1–11.