The Improvement of Hölder’s Inequality with r -Conjugate Exponents Relative to the L p -Norm

Abstract

This paper investigates Hölder’s inequality under the condition of $r$ -conjugate exponents in the sense that ${\displaystyle {\sum }_{k=1}^{s}1/p_k}=1/r$ . Successively, we have, under $r$ -conjugate exponents relative to the $L^p$ -norm, investigated generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized $s$ -order Hölder’s inequality which is an expansion of the known Hölder’s inequality.