Open Access

On Opial's type inequalities for an integral operator with homogeneous kernel

Journal of Inequalities and Applications20122012:123

https://doi.org/10.1186/1029-242X-2012-123

Received: 8 February 2012

Accepted: 30 May 2012

Published: 30 May 2012

Abstract

In this article, we establish some new Opial's type integral inequalities for an integral operator with homogeneous kernel. The results in special cases yield some of the interrelated results and provide new estimates on inequalities of this type.

MS (2000) Subject Classification: 26D15.

Keywords

Opial's inequalityHölder's inequalityJensen's inequalityintegral operatorhomogeneous kernel

1 Introduction

In 1960, Opial [1] established the following 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
0 h | f ( x ) f ( x ) | d x h 4 0 h f x 2 d x .
(1.1)

The Opial's type inequality was first established by Willett [2]:

Theorem 1.2 Let x(t) be absolutely continuous in [0, a], and x(0) = 0. Then
0 a | x ( t ) x ( t ) | d t a 2 0 a | x ( t ) | 2 d t .
(1.2)

A non-trivial generalization of Theorem 1.2 was established by Hua [3]:

Theorem 1.3 Let x(t) be absolutely continuous in [0, a], and x(0) = 0. Further, let l be a positive integer. Then
0 a | x ( t ) x ( t ) | d t a l l + 1 0 a | x ( t ) | l + 1 d t .
(1.3)

A sharper inequality was established by Godunova [4]:

Theorem 1.4 Let f(t) be convex and increasing functions on [0, ∞) with f(0) = 0. Further, let x(t) be absolutely continuous on [α,τ], and x(α) = 0. Then, following inequality holds
α τ f | x ( t ) | | x ( t ) | d t f α τ | x ( t ) | d t .
(1.4)

Opial's inequality and its generalizations, extensions, and discretizations play an important role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [513]. For Opial-type integral inequalities involving high-order partial derivatives, we refer the readers to see [14, 15]. For an extensive survey on these inequalities, see [16]. Mitrinović and Pečarić [17] proved some new extensions of Opial's type inequalities. The aim of this article is to establish some Opial-type inequalities, which are some extensions of Godunova and Levin's, Mitrnović and Pečarić inequalities.

2 Statement of results

Here, we shall extend some of the previous results for the singular integral operator T which have an integral representation. For this, we say that the integral operator T (f)y belongs to the class U(f, K) if it can be represented in the following form
T f y = 0 + K y , x f x d x , x [ 0 , + ) ,
(2.1)

where f(x) is a continuous function on [0, + ∞), and K(y, x) is a non-negative homogeneous kernel function defined on [0, + ∞) × [0, + ∞), such that T(f)y > 0 if f(x) > 0, where x [0, + ∞).

Theorem 2.1 For i = 1, 2, 3, let T(f i )y U(f i , K), where f2(x) > 0, x [0, + ∞). Further, let p(x) > 0, x [0, + ∞), and let g(y, x) be convex and increasing on [0, ∞) × [0, ∞). Then the following inequality holds
0 + p ( y ) g T ( f 1 ) y T ( f 2 ) y , T ( f 3 ) y T ( f 2 ) y d y 0 + ϕ ( y ) g f 1 ( y ) f 2 ( y ) , f 3 ( y ) f 2 ( y ) d y ,
(2.2)
where
ϕ ( y ) = f 2 ( y ) 0 + p ( x ) K ( x , y ) T ( f 2 ) x d x .
Remark 2.2 Let the singular integral operator T(f)y change to a function x(t), so we say that the function x(t) belongs to the class Ū ( y , K ) if it can be represented in the following form
x ( t ) = α τ K ( t , s ) y ( s ) d s , t [ α , τ ] ,
where y(t) is a continuous function on [α, τ], and K(t, s) is a non-negative kernel defined on [α, τ] × [α, τ]; such that x(t) > 0 if y(t) > 0, t [α, τ]. Taking these in Theorem 2.1 and with suitable modifications, then (2.2) becomes the following established by Mitrinović and Pečarić [17].
α τ p ( t ) f x 1 ( t ) x 2 ( t ) , x 3 ( t ) x 2 ( t ) d t α τ ϕ ( t ) f y 1 ( t ) y 2 ( t ) , y 3 ( t ) y 2 ( t ) d t ,
(2.3)
where
ϕ ( t ) = y 2 ( t ) α τ p ( s ) K ( s , t ) x 2 ( s ) d s .
Remark 2.3 In particular, if λ > 0 we have
K ( t , s ) = K λ ( t , s ) = ( t - s ) λ - 1 Γ ( λ ) , s t , 0 s > t .

then y(t) is the derivative of order λ of x(t) in the sense of Riemann-Liouville. Thus, if x(t) is differentiable, then for λ = 1, it follows that y(t) = x'(t).

Further, taking for f(x, y) = f(x) and K(t, s) = K λ (t, s) in (2.3), (2.3) reduces to a result of Godunova and Levin [18].

Now, let T(f)y U(f, K), where K(y, x) = 0 for x > y. Such the singular integral operator we shall say belong to the class U1(f, K). It is clear that in the case, we have
T ( f ) y = 0 y K ( y , x ) f ( x ) d x .
(2.4)
Theorem 2.4 Let the function f(x) be differentiable on [0, ∞), and that for v > 1 the function f(x1/v) be convex and f(0) = 0, and 1 μ + 1 v = 1 . Further, let T(f)y U1(f, K), where
0 y K ( y , x ) μ f ( x ) d x 1 / μ M .
Then
0 + | T ( f ) y | ( 1 - v ) f | T ( f ) y | | f ( y ) | v d y v M v f M 0 + | f ( y ) | v d y 1 / v .
(2.5)
Remark 2.5 Let the singular integral operator T(f)y change to a function x(t), so we say that the function x(t) belongs to the class Ū ( y , K ) . Taking these in Theorem 2.4 and with suitable modifications, then (2.5) becomes the following result.
α τ | x ( t ) | 1 - v f | x ( t ) | | y ( t ) | v d t v M v f M α τ | y ( t ) | v d t 1 / v .

This is just a new result established by Mitrinović and Pečarić [17].

3 Proofs of results

Proof of Theorem 2.1 From the hypotheses of Theorem 2.1, it turn out that
0 + p ( y ) g ( | T ( f 1 ) y T ( f 2 ) y | , | T ( f 3 ) y T ( f 2 ) y | ) d y = 0 + p ( y ) g ( | 1 T ( f 2 ) y 0 + K ( y , x ) f 2 ( x ) f 1 ( x ) f 2 ( x ) d x | , | 1 T ( f 2 ) y ) 0 + K ( y , x ) f 2 ( x ) f 3 ( x ) f 2 ( x ) d x | ) d y 0 + p ( y ) g ( 0 + K ( y , x ) f 2 ( x ) T ( f 2 ) y | f 1 ( x ) f 2 ( x ) | d x , 0 + K ( y , x ) f 2 ( x ) T ( f 2 ) y | f 3 ( x ) f 2 ( x ) | d x ) d y .
By using the Jensen integral inequality, we have
0 + p ( y ) g T ( f 1 ) y T ( f 2 ) y , T ( f 3 ) y T ( f 2 ) y d y 0 + p ( y ) 0 + K ( y , x ) f 2 ( x ) T ( f 2 ) y g f 1 ( x ) f 2 ( x ) , f 3 ( x ) f 2 ( x ) d x d y = 0 + g f 1 ( x ) f 2 ( x ) , f 3 ( x ) f 2 ( x ) f 2 ( x ) 0 + p ( y ) K ( y , x ) T ( f 2 ) y d y d x = 0 + ϕ ( x ) g f 1 ( x ) f 2 ( x ) , f 3 ( x ) f 2 ( x ) d x ,
where
ϕ ( x ) = f 2 ( x ) 0 + p ( y ) K ( y , x ) T ( f 2 ) y d y .
Hence
0 + p ( y ) g T ( f 1 ) y T ( f 2 ) y , T ( f 3 ) y T ( f 2 ) y d y 0 + ϕ ( y ) g f 1 ( y ) f 2 ( y ) , f 3 ( y ) f 2 ( y ) d y ,
where
ϕ ( y ) = f 2 ( y ) 0 + p ( x ) K ( x , y ) T ( f 2 ) x d x .

This completes the proof.

Proof of Theorem 2.4 From the hypotheses of Theorem 2.4 and in view of Hölder's inequality,

we obtain
| T ( f ) y | 0 y K ( y , x ) | f ( x ) | d x . ( 0 y ( K ( y , x ) ) μ d x ) 1 / μ ( 0 y | f ( x ) | v d x ) 1 / v M ( 0 y | f ( x ) | v d x ) 1 / v .
Now, let
z ( y ) = 0 y | f ( x ) | v d x .
Hence
z ( y ) = | f ( y ) | v .
Moreover, notes that
| T ( f ) y | M ( z ( y ) ) 1 / v .
Further, from the convexity of f(t1/v) it follows that the function t1-vf'(t) is non-decreasing, thus we have
0 + | T ( f ) y | ( 1 - v ) f | T ( f ) y | | f ( y ) | v d y 0 + M ( 1 - v ) z ( y ) ( 1 / v ) - 1 f M ( z ( y ) ) 1 / v z ( y ) d y = v M v 0 + f M ( z ( y ) ) 1 / v d M ( z ( y ) ) 1 / v = v M v f M ( z ( + ) ) 1 / v = v M v f M 0 + | f ( y ) | v d y 1 / v .

This completes the proof.

Declarations

Acknowledgements

The research of Chang-Jian Zhao was supported by the National Natural Science Foundation of China (10971205), and the research of Wing-Sum Cheung was partially supported by a HKU URC grant.

Authors’ Affiliations

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

References

  1. Opial Z: Sur une inégalité. Ann Polon Math 1960, 8: 29–32.MathSciNetGoogle Scholar
  2. Willett D: The existence-uniqueness theorem for an n th order linear ordinary differential equation. Am Math Monthly 1968, 75: 174–178. 10.2307/2315901MathSciNetView ArticleGoogle Scholar
  3. Hua LK: On an inequality of Opial. Sci China 1965, 14: 789–790.Google Scholar
  4. Godunova EK: Integral'nye neravenstva s proizvodnysi i proizvol'nymi vypuklymi funkcijami. Uc Zap Mosk Gos Ped In-ta im Lenina 1972, 460: 58–65.Google Scholar
  5. Rozanova GI: On an inequality of Maroni (Russian). Math Zametki 1967, 2: 221–224.Google Scholar
  6. Das KM: An inequality similar to Opial's inequality. Proc Am Math Soc 1969, 22: 258–261.Google Scholar
  7. Agarwal RP, Thandapani E: On some new integrodifferential inequalities. Anal sti Univ ''Al I Cuza'' din Iasi 1982, 28: 123–126.MathSciNetGoogle Scholar
  8. Yang GS: A note on inequality similar to Opial inequality. Tamkang J Math 1987, 18: 101–104.MathSciNetGoogle Scholar
  9. Agarwal RP, Lakshmikantham V: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, Singapore; 1993.View ArticleGoogle Scholar
  10. Bainov D, Simeonov P: Integral Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 1992.View ArticleGoogle Scholar
  11. Li JD: Opial-type integral inequalities involving several higher order derivatives. J Math Anal Appl 1992, 167: 98–100. 10.1016/0022-247X(92)90238-9MathSciNetView ArticleGoogle Scholar
  12. Cheung WS: On Opial-type inequalities in two variables. Aequationes Math 1989, 38: 236–244. 10.1007/BF01840008MathSciNetView ArticleGoogle Scholar
  13. Cheung WS: Some generalized Opial-type inequalities. J Math Anal Appl 1991, 162: 317–321. 10.1016/0022-247X(91)90152-PMathSciNetView ArticleGoogle Scholar
  14. Zhao CJ, Cheung WS: Sharp integral inequalities involving high-order partial derivatives. J Inequal Appl 2008, 2008: 1–10. Article ID 571417Google Scholar
  15. Agarwal RP, Pang PYH: Sharp opial-type inequalities in two variables. Appl Anal 1996, 56(3):227–242.MathSciNetGoogle Scholar
  16. Agarwal RP, Pang PYH: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Academic Publishers, Dordrecht; 1995.View ArticleGoogle Scholar
  17. Mitrinović DS, Pečarić JE: Generalizations of two inequalities of Godunova and Levin. Polish Acad Sci Math 1988, 36: 645–648.Google Scholar
  18. Godunova EK, Levin VI: O neikotoryh integralnyh nerabenstvah, soderzascih proizvodnye (Russian). Izv vuzov Matem 1969, 91: 20–24.Google Scholar

Copyright

© Zhao and Cheung; licensee Springer. 2012

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