ON AN INTEGRAL INEQUALITY OF R. BELLMAN

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Published: Jun 1, 1991
DOI: https://doi.org/10.5556/j.tkjm.22.1991.4597
Keywords:
nonnegative concave functions Bellman’s inequality

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HORST ALZER
Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, Pretoria, Republic of South Africa.

Abstract




We prove: if $u$ and $v$ are non-negative, concave functions defined on $[0, 1]$ satisfying


\[\int_0^1 (u(x))^{2p} dx =\int_0^1 (v(x))^{2q} dx=1, \quad p>0, \quad q>0,\]


then


\[\int_0^1(u(x))^p (v(x))^q dx\ge\frac{2\sqrt{(2p+1)(2q+1)}}{(p+1)(q+1)}-1.\]




Article Details

How to Cite
ALZER, H. (1991). ON AN INTEGRAL INEQUALITY OF R. BELLMAN. Tamkang Journal of Mathematics, 22(2), 187–191. https://doi.org/10.5556/j.tkjm.22.1991.4597
Issue
Vol. 22 No. 2 (1991)
Section
Papers
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References

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R. Bellman, "Converses of Schwarz's inequality," Duke Math. J. 23(1956), 429-434.

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D. S. Mitrinovic, "Analytic Inequalities," Springer, New York, 1970.

C.-L. Wang, "An extension of a Bellman inequality," Utilitas Math. 8(1975), 251-256.