# On some Opial-type inequalities

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## 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* Open image in new window *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, 3, 4, 5, 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, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. For an extensive survey on these inequalities, see [2, 6]. For Opial-type integral inequalities involving high-order partial derivatives see [23, 24, 25, 26, 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*Open image in new window , Open image in new window ,

*σ*∈ [0,

*s*],

*τ*∈ [0,

*t*], 0 ≤

*i*≤

*n*- 1, 0 ≤

*j*≤

*k*- 1.

*Further, let*Open image in new window , Open image in new window

*be absolutely continuous, and*Open image in new window .

*Then*

*where*

**Proof**. For

*σ*integration by parts (

*n*- 1)-times and in view of Open image in new window , Open image in new window , 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 Open image in new window , 0 ≤ *i* ≤ *n* - *α* - 1, 0 ≤ *i* ≤ *k* - *β* -1 and suppose that Open image in new window , Open image in new window are absolutely continuous, and Open image in new window .

*g*(

*s*,

*t*) =

*x*

^{(α, β)}(

*s*,

*t*), where

*x*(

*s*,

*t*) ∈

*C*

^{(n- 1)}[0,

*a*] ×

*C*

^{(k}

^{- 1)}[0,

*b*], Open image in new window , Open image in new window ,

*α*≤

*i*≤

*n*- 1,

*β*≤

*j*≤

*k -*1, and

*x*

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

*s*,

*t*) are absolutely continuous, and Open image in new window , 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*Open image in new window , Open image in new window ,

*σ*∈ [0,

*s*],

*τ*∈ [0,

*t*], 0 ≤

*i*≤

*n*- 1, 0 ≤

*j*≤

*k -*1

*and assume that*Open image in new window , Open image in new window

*are absolutely continuous, and*Open image in new window .

*Then*

*where*

**Proof**. From (2.2), we have

*|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*], Open image in new window , Open image in new window ,

*α*≤

*i*≤

*n*- 1,

*β*≤

*j*≤

*k -*1 and

*x*

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

*s*,

*t*) are absolutely continuous, and Open image in new window , 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*Open image in new window , Open image in new window ,

*σ*∈ [0,

*s*],

*τ*∈ [0,

*t*], 0 ≤

*i*≤

*n*- 1, 0 ≤

*j*≤

*k*- 1

*and assume that*Open image in new window , Open image in new window

*are absolutely continuous, and*Open image in new window .

*Then*

**Proof**. It is clear that

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*], Open image in new window , Open image in new window ,

*α*≤

*i*≤

*n*- 1,

*β*≤

*j*≤

*k -*1, and

*x*

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

*s*,

*t*) are absolutely continuous, and Open image in new window , 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].

## Notes

### 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.

### References

- 1.
- 2.Agarwal RP, Pang PYH:
*Opial Inequalities with Applications in Differential and Difference Equations.*Kluwer Academic Publishers, Dordrecht; 1995.CrossRefGoogle Scholar - 3.Agarwal RP, Lakshmikantham V:
**Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations.***World Scientific, Singapore*1993.Google Scholar - 4.Bainov D, Simeonov P:
*Integral Inequalities and Applications.*Kluwer Academic Publishers, Dordrecht; 1992.CrossRefGoogle Scholar - 5.Li JD:
**Opial-type integral inequalities involving several higher order derivatives.***J Math Anal Appl*1992,**167:**98–100. 10.1016/0022-247X(92)90238-9MATHMathSciNetCrossRefGoogle Scholar - 6.Mitrinovič DS, Pečarić JE, Fink AM:
*Inequalities involving Functions and Their Integrals and Derivatives.*Kluwer Academic Publishers, Dordrecht; 1991.CrossRefGoogle Scholar - 7.Cheung WS:
**On Opial-type inequalities in two variables.***Aequationes Math*1989,**38:**236–244. 10.1007/BF01840008MATHMathSciNetCrossRefGoogle Scholar - 8.Cheung WS:
**Some new Opial-type inequalities.***Mathematika*1990,**37:**136–142. 10.1112/S0025579300012869MATHMathSciNetCrossRefGoogle Scholar - 9.Cheung WS:
**Some generalized Opial-type inequalities.***J Math Anal Appl*1991,**162:**317–321. 10.1016/0022-247X(91)90152-PMATHMathSciNetCrossRefGoogle Scholar - 10.Cheung WS:
**Opial-type inequalities with****m****functions in****n****variables.***Mathematika*1992,**39:**319–326. 10.1112/S0025579300015047MATHMathSciNetCrossRefGoogle Scholar - 11.Cheung WS, Zhao DD, Pečarić JE:
**Opial-type inequalities for Differential Operators.***Nonlinear Anal*2011, in press.Google Scholar - 12.Godunova EK, Levin VI:
**On an inequality of Maroni.***Mat Zametki*1967,**2:**221–224.MathSciNetGoogle Scholar - 13.Mitrinovič DS:
*Analytic Inequalities.*Springer-Verlag, Berlin, New York; 1970.CrossRefGoogle Scholar - 14.Pachpatte BG:
**On integral inequalities similar to Opial's inequality.***Demonstratio Math*1989,**22:**21–27.MATHMathSciNetGoogle Scholar - 15.Pachpatte BG:
**On integral inequalities similar to Opial's inequality.***Demonstratio Math*1989,**22:**21–27.MATHMathSciNetGoogle Scholar - 16.Pachpatte BG:
**Some inequalities similar to Opial's inequality.***Demonstratio Math*1993,**26:**643–647.MATHMathSciNetGoogle Scholar - 17.Pachpatte BG:
**A note on generalized Opial type inequalities.***Tamkang J Math*1993,**24:**229–235.MATHMathSciNetGoogle Scholar - 18.Pečarić JE:
*An integral inequality, in Analysis, Geometry, and Groups: A Riemann Legacy Volume.*Edited by: Srivastava HM, Rassias ThM. Part II, Hadronic Press, Palm Harbor, Florida; 1993:472–478.Google Scholar - 19.Pečarić JE, Brnetić I:
**Note on generalization of Godunova-Levin-Opial inequality.***Demonstratio Math*1997,**30:**545–549.MATHMathSciNetGoogle Scholar - 20.Pečarić JE, Brnetić I:
**Note on the Generalization of Godunova-Levin-Opial inequality in Several independent Variables.***J Math Anal Appl*1997,**215:**274–282. 10.1006/jmaa.1997.5529MATHMathSciNetCrossRefGoogle Scholar - 21.Rozanova GI:
**Integral inequalities with derivatives and with arbitrary convex functions.***Moskov Gos Ped Inst Vcen Zap*1972,**460:**58–65.MathSciNetGoogle Scholar - 22.Yang GS:
**Inequality of Opial-type in two variables.***Tamkang J Math*1982,**13:**255–259.MATHMathSciNetGoogle Scholar - 23.Agarwal RP:
**Sharp Opial-type inequalities involving****r****-derivatives and their applications.***Tohoku Math J*1995,**47**(4):567–593. 10.2748/tmj/1178225462MATHMathSciNetCrossRefGoogle Scholar - 24.Agarwal RP, Pang PYH:
**Sharp opial-type inequalities in two variables.***Appl Anal*1996,**56**(3):227–242. 10.1080/00036819508840324MathSciNetGoogle Scholar - 25.Alzer H:
**An Opial-type inequality involving higher-order derivatives of two functions.***Appl Math Letters*1997,**10**(4):123–128. 10.1016/S0893-9659(97)00071-2MATHMathSciNetCrossRefGoogle Scholar - 26.Karpuz B, Kaymakcalan B, Özkan UM:
**Some multi-dimenstonal Opial-type inequalities on time scales.***J Math Ineq*2010,**4**(2):207–216.MATHCrossRefGoogle Scholar - 27.Zhao CJ, Cheung WS:
**Sharp integral inequalities involving high-order partial derivatives.***J Ineq Appl*2008,**2008:**10. Article ID 571417Google Scholar - 28.Das KM:
**An inequality similar to Opial's inequality.***Proc Amer Math Soc*1969,**22:**258–261.MATHMathSciNetGoogle Scholar - 29.Willett D:
**The existence-uniqueness theorem for an****n****-th order linear ordinary differential equation.***Amer Math Monthly*1968,**75:**174–178. 10.2307/2315901MATHMathSciNetCrossRefGoogle Scholar - 30.Beesack PR:
**On an integral inequality of Z.***Opial Trans Amer Math Soc*1962,**104:**470–475. 10.1090/S0002-9947-1962-0139706-1MATHMathSciNetCrossRefGoogle Scholar - 31.Agarwal RP, Thandapani E:
**On some new integrodifferential inequalities.***Anal sti Univ "Al. I. Cuza" din Iasi*1982,**28:**123–126.MATHMathSciNetGoogle Scholar - 32.Yang GS:
**On a certain result of Z.***Opial Proc Japan Acad*1966,**42:**78–83. 10.3792/pja/1195522120MATHCrossRefGoogle Scholar

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