NOTE ON AN INTEGRAL INEQUALITY FOR CONCAVE FUNCTIONS

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.