The converse of the Minkowski’s inequality theorem and its generalization

Abstract:

Let ($\Omega$, $\Sigma$, $\mu$) be a measure space with two sets $A,B \in \Sigma$ such that $0 < \mu (A) < 1 < \mu (B) < \infty$ , and let $\varphi :{{\mathbf {R}}_ + } \to {{\mathbf {R}}_ + }$ be bijective and ${\varphi ^{ - 1}}$ continuous at 0. We prove that if for all $\mu$-integrable step functions $x,y:\Omega \to {\mathbf {R}}$, \[ {\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |x + y|d\mu } } \right ) \leq {\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |x|d\mu } } \right ) + {\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |y|d\mu } } \right )\] then $\varphi (t) = \varphi (1){t^p}$ for some $p \geq 1$. In the case of normalized measure we prove a generalization of Minkowski’s inequality theorem. The suitable results for the reversed inequality are also presented.