NOTE ON AN INTEGRAL INEQUALITY FOR CONCAVE FUNCTIONS

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HORST ALZER

Abstract




We prove: Let $p\in C^2[a, b]$ be non-negative and concave, and let $f\in C^2[a, b]$ with $f(a)=f(b)=0$. Then


\[ \left(\int_a^b p(x)(f'(x))^2 dx\right)^2\le \left(\int_a^b p(x)(f(x))^2 dx\right)\left(\int_a^b p(x)(f''(x))^2 dx\right) .\]





Moreover, we determine all cases of equality.





 




Article Details

How to Cite
ALZER, H. (1996). NOTE ON AN INTEGRAL INEQUALITY FOR CONCAVE FUNCTIONS. Tamkang Journal of Mathematics, 27(2), 161–163. https://doi.org/10.5556/j.tkjm.27.1996.4354
Section
Papers

References

J. L. Brenner and H. Alzer, "Integral inequalities for concave functions with applications to spec1·a1 functions," Proc. Roy. Soc. Edinburgh Sect.A, 118(1991), 173-192

G. H. Hardy, J. E. Littlewood and G. P6lya, Inequalities, Cambridge Univ. Press, 1934

L. -C. Shen, "Comments on an 止 inequality of A. K. Varma involving the first derivative of polynomials," Proc. Amer. Math. Soc. 111(1991), 955-959.