Horst Alzer
Abstract
Open-access reader
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.
A significance statement is not available in the OpenAlex record.
A contribution statement is not available in the OpenAlex record.
Method details are not available in the OpenAlex metadata.
Findings are not separately available in the OpenAlex metadata.
Limitations are not available in the OpenAlex metadata.
Application details are not available in the OpenAlex metadata.
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.
Key concepts: Mathematics, Combinatorics