A refinement of Holder's integral inequality

Abstract

The purpose of this note is to show that there is monotonic continuous function $ p(t)$ such that

$$ \int_a^b \left(\prod_{i=1}^n f_i(x)\right) dx\le p(t)\le \prod_{i=1}^n  \left(\int_a^b f_i^{r_i}(x)dx \right)^{1\over r_i},$$

where $ f_1$, $ f_2,\ldots,f_n$ are real positive continuous functions on $ [a,b]$ and $ r_1$, $ r_2,\ldots,r_n$ are real positive numbers with $ \sum_{i=1}^n {1\over r_i}=1$.