Refined Hölder's inequality for measurable functions

Abstract

Let $\nu$ be a positive measure on a space $Y$ with $\nu(Y) \neq 0$ and let $f_j$ ($j = 1, 2, \dots, n$) be positive $\nu$-integrable functions on $Y$. For some positive real numbers $\alpha_j$ ($j = 1, 2, \dots, n$), $\beta_j$ ($j= 1, 2, \dots, k < n$) and a measurable subset $Y_1$ of $Y$, we have some inequalities. From these results, we refine Hölder's inequality.