Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T07:19:10.844Z Has data issue: false hasContentIssue false
  • English
  • Français

On Some Generalization of Inequalities of Opial, Yang and Shum

Published online by Cambridge University Press:  20 November 2018

Cheng-Shyong Lee
Cheng-Shyong Lee*
Affiliation:
Department of Mathematics, Evening School, Tamkang College of Arts and Science 5, Lane 199, King-Hwa street, Taipei, Taiwan (106), Republic of China
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1960, Z. Opial [20] proved the following interesting integral inequality:

Theorem A. If u is a continuously differentiable function on [0, b], and if u(0) = u(b) = 0, and u(x)>0 for x∊(0, b), then

1

where the constant b/4 is the best possible.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 1 , 01 March 1980 , pp. 71 - 80
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Beckenbach, E. F. and Bellman, R.. Inequalities, 2nd rev. ed; Ergebnisse der mathematik and ihrer Grenzgebiete, Heft 30 Springer-Verlag. N.Y. 1965.Google Scholar
2. Beesack, P. R., Hardy's inequality and its extensions, Pacific J. Math, 11 (1961), 39-62.Google Scholar
3. Beesack, P. R., On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470-475.Google Scholar
4. Beesack, P. R., Integral inequality involving a function and its derivative, Amer. Math. Monthly 78 (1971), 705-741.Google Scholar
5. Beesack, P. R. and Das, K. M., Extensions of OpiaVs inequality, Pacific J. Math. 26 (1968), 215-232.Google Scholar
6. Benson, D. C., Inequalities involving integrals of functions and their derivatives, J. Math. Anal. Appl. 17 (1967), 292-308.Google Scholar
7. Boyd, D. W. and Wong, J. S., An extension of OpiaVs inequality, J. Math. Anal. Appl. 19 (1967), 100-102.Google Scholar
8. Boyd, D. W., Best constants in inequalities related to OpiaVs inequality, J. Math. Anal. Appl. 25 (1969), 378-387.Google Scholar
9. Calvert, J., Some generalizations of OpiaVs inequality, Proc. Amer. Math. Soc. 18 (1967), 72-75.Google Scholar
1. Das, K. M., An inequality similar to OpiaVs inequality, Proc. Amer. Math. Soc. 22 (1969), 258-261.Google Scholar
11. Fink, A. M. and Jodict, Max Jr., A generalization of the Arithmetic-Geometric Mean's Inequality. Proc. Amer. Math. Soc. vol. 61 Number 2. 12. 1976.Google Scholar
12. Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, 2nd. ed. Cambridge Univ. Press. N.Y. 1952.Google Scholar
1. Hua, L. K., On an inequality of Opial, Sci. Sinica 14 (1965), 789-790.Google Scholar
14. Lee, C.-M., On a discrete analogue of inequalities of Opial and Yang. Canad. Math. Bull. 11 (1968), 73-77.Google Scholar
15. Levinson, N., On an inequality of Opial and Beesack, Proc. Amer. Math. Soc. 15 (1964), 565-566.Google Scholar
16. Mallows, C. L., An even simpler proof of OpiaVs inequality, Proc. Amer. Math. Soc. 16 (1965), 173.Google Scholar
17. Maroni, P. M., Sur Vinégalité d'opial -Beesack, C. R. Acad. Sci. Paris S?r. A264 (1967), A62-A64.Google Scholar
18. Mitrinovié, D. S. and Vasić, P. M., Analytic inequalities, Die, Grandlehren der Math. Wissenschaften, Band 165 Springer-Verlag, Berlin, 1970.Google Scholar
19. Olech, C., A simple proof of a certain result of Z. Opial, Ann. Polon, Math. 8 (1960), 61-63.Google Scholar
2. Opial, Z., Sur une inégalité, Ann. Polon, Math. 8 (1960), 29-32.Google Scholar
21. Pederson, R. N., On an inequality of Opial, Beesack and Levinson, Proc. Amer. Math. Soc. 16 (1965), 174.Google Scholar
22. Roydon, H. L., Real Analysis 2nd. the Macmillan Company, N.Y. Collier-Macmillan, Limited, London, 1972.Google Scholar
23. Shum, D. T., On integral inequalities related to Hardy's, Canad. Math. Bull. 14 (1971), 225-270.Google Scholar
24. Shum, D. T., A general and sharpened form of OpiaVs inequality, Canad. Math. Bull. vol. 17 (3), (1974), 385-389.Google Scholar
25. Shum, D. T., On a class of new inequalities, Trans. Amer. Math. Soc. vol. 204 (1975), 299-341.Google Scholar
26. Wong, J. S. W., A discrete analogue of Opia Vs inequality, Canad. Math. Bull. 10 (1967), 115-118.Google Scholar
27. Yang, G. S., On a certain result of Z. Opial, Proc. Japan Acad. 42 (1966), 78-83.Google Scholar