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
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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
where this constantis 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].
References
Opial Z: Sur une inégalité. Ann Polon Math 1960, 8: 29–32.
Agarwal RP, Pang PYH: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Academic Publishers, Dordrecht; 1995.
Agarwal RP, Lakshmikantham V: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, Singapore 1993.
Bainov D, Simeonov P: Integral Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 1992.
Li JD: Opial-type integral inequalities involving several higher order derivatives. J Math Anal Appl 1992, 167: 98–100. 10.1016/0022-247X(92)90238-9
Mitrinovič DS, Pečarić JE, Fink AM: Inequalities involving Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht; 1991.
Cheung WS: On Opial-type inequalities in two variables. Aequationes Math 1989, 38: 236–244. 10.1007/BF01840008
Cheung WS: Some new Opial-type inequalities. Mathematika 1990, 37: 136–142. 10.1112/S0025579300012869
Cheung WS: Some generalized Opial-type inequalities. J Math Anal Appl 1991, 162: 317–321. 10.1016/0022-247X(91)90152-P
Cheung WS: Opial-type inequalities with m functions in n variables. Mathematika 1992, 39: 319–326. 10.1112/S0025579300015047
Cheung WS, Zhao DD, Pečarić JE: Opial-type inequalities for Differential Operators. Nonlinear Anal 2011, in press.
Godunova EK, Levin VI: On an inequality of Maroni. Mat Zametki 1967, 2: 221–224.
Mitrinovič DS: Analytic Inequalities. Springer-Verlag, Berlin, New York; 1970.
Pachpatte BG: On integral inequalities similar to Opial's inequality. Demonstratio Math 1989, 22: 21–27.
Pachpatte BG: On integral inequalities similar to Opial's inequality. Demonstratio Math 1989, 22: 21–27.
Pachpatte BG: Some inequalities similar to Opial's inequality. Demonstratio Math 1993, 26: 643–647.
Pachpatte BG: A note on generalized Opial type inequalities. Tamkang J Math 1993, 24: 229–235.
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.
Pečarić JE, Brnetić I: Note on generalization of Godunova-Levin-Opial inequality. Demonstratio Math 1997, 30: 545–549.
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.1997.5529
Rozanova GI: Integral inequalities with derivatives and with arbitrary convex functions. Moskov Gos Ped Inst Vcen Zap 1972, 460: 58–65.
Yang GS: Inequality of Opial-type in two variables. Tamkang J Math 1982, 13: 255–259.
Agarwal RP: Sharp Opial-type inequalities involving r -derivatives and their applications. Tohoku Math J 1995,47(4):567–593. 10.2748/tmj/1178225462
Agarwal RP, Pang PYH: Sharp opial-type inequalities in two variables. Appl Anal 1996,56(3):227–242. 10.1080/00036819508840324
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)00071-2
Karpuz B, Kaymakcalan B, Özkan UM: Some multi-dimenstonal Opial-type inequalities on time scales. J Math Ineq 2010,4(2):207–216.
Zhao CJ, Cheung WS: Sharp integral inequalities involving high-order partial derivatives. J Ineq Appl 2008, 2008: 10. Article ID 571417
Das KM: An inequality similar to Opial's inequality. Proc Amer Math Soc 1969, 22: 258–261.
Willett D: The existence-uniqueness theorem for an n -th order linear ordinary differential equation. Amer Math Monthly 1968, 75: 174–178. 10.2307/2315901
Beesack PR: On an integral inequality of Z. Opial Trans Amer Math Soc 1962, 104: 470–475. 10.1090/S0002-9947-1962-0139706-1
Agarwal RP, Thandapani E: On some new integrodifferential inequalities. Anal sti Univ "Al. I. Cuza" din Iasi 1982, 28: 123–126.
Yang GS: On a certain result of Z. Opial Proc Japan Acad 1966, 42: 78–83. 10.3792/pja/1195522120
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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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