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http://dx.doi.org/10.4134/BKMS.2012.49.6.1263

ON OPIAL INEQUALITIES INVOLVING HIGHER ORDER DERIVATIVES  

Zhao, Chang-Jian (Department of Mathematics China Jiliang University)
Cheung, Wing-Sum (Department of Mathematics The University of Hong Kong)
Publication Information
Bulletin of the Korean Mathematical Society / v.49, no.6, 2012 , pp. 1263-1274 More about this Journal
Abstract
In the present paper we establish some new Opial-type inequalities involving higher order partial derivatives. The results in special cases yield some of the recent results on Opial's inequality and provide new estimates on inequalities of this type.
Keywords
Opial's inequality; Opial-type inequalities; H$\ddot{o}$lder's inequality;
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1 B. G. Pachpatte, A note on generalized Opial type inequalities, Tamkang J. Math. 24 (1993), no. 2, 229-235.
2 J. E. Pecaric, An integral inequality, Analysis, geometry and groups: a Riemann legacy volume, 471-478, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1993.
3 J. E. Pecaric and I. Brnetic, Note on generalization of Godunova-Levin-Opial inequality, Demonstratio Math. 30 (1997), no. 3, 545-549.
4 J. E. Pecaric and I. Brnetic, Note on the generalization of the Godunova-Levin-Opial inequality in several independent variables, J. Math. Anal. Appl. 215 (1997), no. 1, 274-282.   DOI   ScienceOn
5 G. I. Rozanova, Integral inequalities with derivatives and with arbitrary convex functions, Moskov. Gos. Ped. Inst. Vcen. Zap. 460 (1972), 58-65.
6 D. Willett, The existence-uniqueness theorem for an n-th order linear ordinary differential equation, Amer. Math. Monthly 75 (1968), 174-178.   DOI   ScienceOn
7 G. S. Yang, Inequality of Opial-type in two variables, Tamkang J. Math. 13 (1982), no. 2, 255-259.
8 G. S. Yang, On a certain result of Z. Opial, Proc. Japan Acad. 42 (1966), 78-83.   DOI
9 C. J. Zhao and W. S. Cheung, Sharp integral inequalities involving high-order partial derivatives, J. Ineq. Appl. 2008 (2008), Article ID 571417, 10 pages.   DOI
10 R. P. Agarwal, Sharp Opial-type inequalities involving r-derivatives and their applications, Tohoku Math. J. 47 (1995), no. 4, 567-593.   DOI
11 R. P. Agarwal and V. Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, World Scientific, Singapore, 1993.
12 R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht, 1995.
13 R. P. Agarwal and P. Y. H. Pang, Sharp Opial-type inequalities in two variables, Appl. Anal. 56 (1995), no. 3-4, 227-242.   DOI   ScienceOn
14 R. P. Agarwal and E. Thandapani, On some new integro-differential inequalities, Anal. sti. Univ. "Al. I. Cuza" din Iasi 28 (1982), no. 1, 123-126.
15 H. Alzer, An Opial-type inequality involving higher-order derivatives of two functions, Appl. Math. Lett. 10 (1997), no. 4, 123-128.   DOI   ScienceOn
16 D. Bainov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1992.
17 P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470-475.   DOI   ScienceOn
18 W. S. Cheung, On Opial-type inequalities in two variables, Aequationes Math. 38 (1989), no. 2-3, 236-244.   DOI
19 W. S. Cheung, Some new Opial-type inequalities, Mathematika 37 (1990), no. 1, 136-142.   DOI
20 W. S. Cheung, Some generalized Opial-type inequalities, J. Math. Anal. Appl. 162 (1991), no. 2, 317-321.   DOI
21 E. K. Godunova and V. I. Levin, An inequality of Maroni, Mat. Zametki 2 (1967), 221-224.
22 W. S. Cheung, Opial-type inequalities with m functions in n variables, Mathematika 39 (1992), no. 2, 319-326.   DOI
23 W. S. Cheung, D. D. Zhao, and J. E. Pecaric, Opial-type inequalities for Differential Operators, to appear in Nonlinear Anal.
24 K. M. Das, An inequality similar to Opial's inequality, Proc. Amer. Math. Soc. 22 (1969), 258-261.
25 L. K. Hua, On an inequality of Opial, Sci. Sinica 14 (1965), 789-790.
26 B. Karpuz, B. Kaymakcalan, and U. M. Ozkan, Some multi-dimenstonal Opial-type inequalities on time scales, J. Math. Inequal. 4 (2010), no. 2, 207-216.
27 J. D. Li, Opial-type integral inequalities involving several higher order derivatives, J. Math. Anal. Appl. 167 (1992), no. 1, 98-100.   DOI
28 D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, Berlin, New York, 1970.
29 D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Inequalities involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
30 Z. Opial, Sur une inegalite, Ann. Polon. Math. 8 (1960), 29-32.   DOI
31 B. G. Pachpatte, On integral inequalities similar to Opial's inequality, Demonstratio Math. 22 (1989), no. 1, 21-27.
32 B. G. Pachpatte, On Opial-type integral inequalities, J. Math. Anal. Appl. 120 (1986), no. 2, 547-556.   DOI   ScienceOn
33 B. G. Pachpatte, On some new generalizations of Opial inequality, Demonstratio Math. 19 (1986), no. 2, 281-291.