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# On some Opial-type inequalities

- Chang-Jian Zhao
^{1}Email author and - Wing-Sum Cheung
^{2}

**2011**:7

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

© Zhao and Cheung; licensee Springer. 2011

**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 inequality
- Opial-type integral inequalities
- Hölder's inequality

## 1 Introduction

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

**Theorem 1.1**.

*Suppose f*∈

*C*

^{1}[0,

*h*]

*satisfies f*(0) =

*f*(

*h*) = 0

*and f*(

*x*) > 0

*for all x*∈ (0,

*h*).

*Then the integral inequality holds*

*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 [2–6]. 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 [7–22]. For an extensive survey on these inequalities, see [2, 6]. For Opial-type integral inequalities involving high-order partial derivatives see [23–27]. 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 ≤

*i*≤

*n*- 1, 0 ≤

*j*≤

*k*- 1.

*Further, let*,

*be absolutely continuous, and*.

*Then*

**Proof**. For

*σ*integration by parts (

*n*- 1)-times and in view of , , 0 ≤

*i*≤

*n*- 1, 0 ≤

*j*≤

*k*- 1 we have

*x*

^{(n,k)}(

*s*,

*t*) and using the Cauchy-Schwarz inequality, we have

*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:

*n*≥ 2, (2.4) is sharper than the following inequality established by Willett [29].

*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 ≤ *i* ≤ *n* - *α* - 1, 0 ≤ *i* ≤ *k* - *β* -1 and suppose that
,
are absolutely continuous, and
.

*g*(

*s*,

*t*) =

*x*

^{(α, β)}(

*s*,

*t*), where

*x*(

*s*,

*t*) ∈

*C*

^{(n- 1)}[0,

*a*] ×

*C*

^{(k}

^{- 1)}[0,

*b*], , ,

*α*≤

*i*≤

*n*- 1,

*β*≤

*j*≤

*k -*1, and

*x*

^{(n- 1, k-1)}(

*s*,

*t*) are absolutely continuous, and , then

*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 ≤

*i*≤

*n*- 1, 0 ≤

*j*≤

*k -*1

*and assume that*,

*are absolutely continuous, and*.

*Then*

*|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

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:

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

*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*], , ,

*α*≤

*i*≤

*n*- 1,

*β*≤

*j*≤

*k -*1 and

*x*

^{(n - 1, k - 1)}(

*s*,

*t*) are absolutely continuous, and , it is easy to obtain that

*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 ≤

*i*≤

*n*- 1, 0 ≤

*j*≤

*k*- 1

*and assume that*,

*are absolutely continuous, and*.

*Then*

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*], , ,

*α*≤

*i*≤

*n*- 1,

*β*≤

*j*≤

*k -*1, and

*x*

^{(n - 1, k - 1)}(

*s*,

*t*) are absolutely continuous, and , from (2.16), it is easy to obtain that

*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

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