Open AccessNOTE ON AN INTEGRAL INEQUALITY FOR CONCAVE FUNCTIONS
Author(s) -
Horst Alzer
Publication year - 1996
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.27.1996.4354
Subject(s) - mathematics , combinatorics
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.