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# On some Opial-type inequalities

*Journal of Inequalities and Applications*
**volume 2011**, Article number: 7 (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**

## 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*

*where*

*and*

**Proof**. For *σ* integration by parts (*n* - 1)-times and in view of , , 0 ≤ *i* ≤ *n* - 1, 0 ≤ *j* ≤ *k* - 1 we have

Multiplying both sides of (2.2) by *x*^{(n,k)}(*s*, *t*) and using the Cauchy-Schwarz inequality, we have

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:

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].

**Remark 2.2**. Taking for *n* = *k* = 1 in (2.1), (2.1) reduces to

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 ≤ *i* ≤ *n* - *α* - 1, 0 ≤ *i* ≤ *k* - *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*], , , *α* ≤ *i* ≤ *n* - 1, *β* ≤ *j* ≤ *k -* 1, and *x*^{(n- 1, k-1)}(*s*, *t*) are absolutely continuous, and , then

Obviously, a special case of (2.7) is the following inequality:

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 ≤ *i* ≤ *n* - 1, 0 ≤ *j* ≤ *k -* 1 *and assume that*, *are absolutely continuous, and* . *Then*

*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:

This is an inequality given by Das [28]. Taking for *n* = 1 in (2.10), we have

For *m*, *l* ≥ 1 Yang [32] established the following inequality:

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*], , , *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

Obviously, a special case of (2.14) is the following inequality:

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 ≤ *i* ≤ *n* - 1, 0 ≤ *j* ≤ *k* - 1 *and assume that*, *are absolutely continuous, and*. *Then*

**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*], , , *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

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].

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## 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.

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The authors declare that they have no competing interests.

### Authors' contributions

C-JZ and W-SC jointly contributed to the main results Theorems 2.1, 2.2, and 2.3. Both authors read and approved the final manuscript.

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Zhao, CJ., Cheung, WS. On some Opial-type inequalities.
*J Inequal Appl* **2011, **7 (2011). https://doi.org/10.1186/1029-242X-2011-7

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

### Keywords

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