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On some Opial-type inequalities

Journal of Inequalities and Applications20112011:7

https://doi.org/10.1186/1029-242X-2011-7

Received: 28 December 2010

Accepted: 17 June 2011

Published: 17 June 2011

Abstract

In the present paper we establish some new Opial-type inequalities involving higher-order partial derivatives. Our results in special cases yield some of the recent results on Opial's inequality and also provide new estimates on inequalities of this type.

MR (2000) Subject Classification 26D15

Keywords

Opial's inequalityOpial-type integral inequalitiesHölder's inequality

1 Introduction

In the year 1960, Opial [1] established the following integral inequality:

Theorem 1.1. Suppose f C1[0, h] satisfies f(0) = f(h) = 0 and f(x) > 0 for all x (0, h). Then the integral inequality holds
(1.1)

where this constant is best possible.

Opial's inequality and its generalizations, extensions and discretizations play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [26]. The inequality (1.1) has received considerable attention, and a large number of papers dealing with new proofs, extensions, generalizations, variants and discrete analogues of Opial's inequality have appeared in the literature [722]. For an extensive survey on these inequalities, see [2, 6]. For Opial-type integral inequalities involving high-order partial derivatives see [2327]. The main purpose of the present paper is to establish some new Opial-type inequalities involving higher-order partial derivatives by an extension of Das's idea [28]. Our results in special cases yield some of the recent results on Opial's-type inequalities and provide some new estimates on such types of inequalities.

2 Main results

Let n ≥ 1, k ≥ 1. Our main results are given in the following theorems.

Theorem 2.1 Let x(s, t) C(n - 1)[0, a] × C(k - 1)[0, b] be such that , , σ [0, s], τ [0, t], 0 ≤ in - 1, 0 ≤ jk - 1. Further, let , be absolutely continuous, and . Then
(2.1)
where
and
Proof. For σ integration by parts (n - 1)-times and in view of , , 0 ≤ in - 1, 0 ≤ jk - 1 we have
(2.2)
Multiplying both sides of (2.2) by x(n,k)(s, t) and using the Cauchy-Schwarz inequality, we have
(2.3)
Thus, integrating both sides of (2.3) over t from 0 to b first and then integrating the resulting inequality over s from 0 to a and applying the Cauchy-Schwarz inequality again, we obtain

This completes the proof.

Remark 2.1. Let x(s, t) reduce to s(t) and with suitable modifications, Then (2.1) becomes the following inequality:
(2.4)
This is just an inequality established by Das [28]. Obviously, for n ≥ 2, (2.4) is sharper than the following inequality established by Willett [29].
(2.5)
Remark 2.2. Taking for n = k = 1 in (2.1), (2.1) reduces to
(2.6)
Let x(s, t) reduce to s(t) and with suitable modifications. Then (2.6) becomes the following inequality: If x(t) is absolutely continuous in [0, a] and x(0) = 0, then

This is just an inequality established by Beesack [30].

Remark 2.3. Let 0 ≤ α, β < n, but fixed, and let g(s, t) C(n-α- 1)[0, a] × C(k-β-1)[0, b] be such that , 0 ≤ in - α - 1, 0 ≤ ik - β -1 and suppose that , are absolutely continuous, and .

Then from (2.1) it follows that
Thus, for g(s, t) = x(α, β)(s, t), where x(s, t) C(n- 1)[0, a] × C(k- 1)[0, b], , , αin - 1, βjk - 1, and x(n- 1, k-1)(s, t) are absolutely continuous, and , then
(2.7)
Obviously, a special case of (2.7) is the following inequality:
(2.8)
Let x(s, t) reduce to s(t) and with suitable modifications. Then (2.8) becomes the following inequality:

This is just an inequality established by Agarwal and Thandapani [31].

Theorem 2.2. Let l and m be positive numbers satisfying l + m > 1. Further, let x(s, t) C(n- 1)[0, a] × C(k- 1)[0, b] be such that , , σ [0, s], τ [0, t], 0 ≤ in - 1, 0 ≤ jk - 1 and assume that , are absolutely continuous, and . Then
(2.9)
where
Proof. From (2.2), we have
by Hölder's inequality with indices l + m and , it follows that
where
Multiplying the both sides of above inequality by |x(n,k)(s, t)| m and integrating both sides over t from 0 to b first and then integrating the resulting inequality over s from 0 to a, we obtain
Now, applying Hölder's inequality with indices and to the integral on the right-side, we obtain

This completes the proof.

Remark 2.4. Let x(s, t) reduce to s(t) and with suitable modifications. Then (2.9) becomes the following inequality:
(2.10)
This is an inequality given by Das [28]. Taking for n = 1 in (2.10), we have
(2.11)
For m, l ≥ 1 Yang [32] established the following inequality:
(2.12)

Obviously, for m, l ≥ 1, (2.11) is sharper than (2.12).

Remark 2.5. For n = k = 1; (2.9) reduces to
Let x(s, t) reduce to s(t) and with suitable modifications. Then above inequality becomes the following inequality:

This is just an inequality established by Yang [32].

Remark 2.6. Following Remark 2.3, for x(s, t) C(n - 1)[0, a] × C(k - 1)[0, b], , , αin - 1, βjk - 1 and x(n - 1, k - 1)(s, t) are absolutely continuous, and , it is easy to obtain that
(2.13)
Obviously, a special case of (2.14) is the following inequality:
(2.14)
Let x(s, t) reduce to s(t) and with suitable modifications, then (2.14) becomes the following inequality:

This is just an inequality established by Agarwal and Thandapani [31].

Theorem 2.3. Let l and m be positive numbers satisfying l + m = 1. Further, let x(s, t) C(n - 1)[0, a] × C(k - 1)[0, b] be such that , , σ [0, s], τ [0, t], 0 ≤ in - 1, 0 ≤ jk - 1 and assume that , are absolutely continuous, and . Then
(2.15)
Proof. It is clear that
and hence
Now applying Hölder inequality with indices and , we obtain

This completes the proof.

Remark 2.7. Let x(s, t) reduce to s(t) and with suitable modifications. Then (2.16) becomes the following inequality:

This is an inequality given by Das [28].

Remark 2.8. Following Remark 2.3, for x(s, t) C(n - 1)[0, a] × C(k - 1)[0, b], , , αin - 1, βjk - 1, and x(n - 1, k - 1)(s, t) are absolutely continuous, and , from (2.16), it is easy to obtain that
(2.16)
Let x(s, t) reduce to s(t) and with suitable modifications, then (2.16) becomes the following inequality:

This is an inequality given by Das [28].

Declarations

Acknowledgements

The authors express their grateful thanks to the referee for his many very valuable suggestions and comments. Research of Chang-Jian Zhao was supported by National Natural Science Foundation of China (10971205). Research of Wing-Sum Cheung was partially supported by a HKU URC grant.

Authors’ Affiliations

(1)
Department of Mathematics, China Jiliang University, Hangzhou, China
(2)
Department of Mathematics, The University of Hong Kong, Hong Kong

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Copyright

© Zhao and Cheung; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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