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On Some Generalization of Inequalities of Opial, Yang and Shum

  • Cheng-Shyong Lee (a1)

Extract

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.

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References

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1. Beckenbach, E. F. and Bellman, R.. Inequalities, 2nd rev. ed; Ergebnisse der mathematik and ihrer Grenzgebiete, Heft 30 Springer-Verlag. N.Y. 1965.
2. Beesack, P. R., Hardy's inequality and its extensions, Pacific J. Math, 11 (1961), 39-62.
3. Beesack, P. R., On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470-475.
4. Beesack, P. R., Integral inequality involving a function and its derivative, Amer. Math. Monthly 78 (1971), 705-741.
5. Beesack, P. R. and Das, K. M., Extensions of OpiaVs inequality, Pacific J. Math. 26 (1968), 215-232.
6. Benson, D. C., Inequalities involving integrals of functions and their derivatives, J. Math. Anal. Appl. 17 (1967), 292-308.
7. Boyd, D. W. and Wong, J. S., An extension of OpiaVs inequality, J. Math. Anal. Appl. 19 (1967), 100-102.
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9. Calvert, J., Some generalizations of OpiaVs inequality, Proc. Amer. Math. Soc. 18 (1967), 72-75.
1. Das, K. M., An inequality similar to OpiaVs inequality, Proc. Amer. Math. Soc. 22 (1969), 258-261.
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.
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1. Hua, L. K., On an inequality of Opial, Sci. Sinica 14 (1965), 789-790.
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22. Roydon, H. L., Real Analysis 2nd. the Macmillan Company, N.Y. Collier-Macmillan, Limited, London, 1972.
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25. Shum, D. T., On a class of new inequalities, Trans. Amer. Math. Soc. vol. 204 (1975), 299-341.
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