A mixed Hölder and Minkowski inequality

Abstract:

Hölder’s inequality states that $\left \Vert x\right \Vert _{p}\left \Vert y\right \Vert _{q}-\left \langle x,y\right \rangle \ge 0$ for any $(x,y)\in \mathcal {L}^{p}(\Omega )\times \mathcal {L}^{q}(\Omega )$ with $1/p+1/q=1$. In the same situation we prove the following stronger chains of inequalities, where $z=y|y|^{q-2}$: \[ \left \Vert x\right \Vert _{p}\left \Vert y \right \Vert _{q}-\left \langle x,y\right \rangle \ge (1/p)\big [\big (\left \Vert x \right \Vert _{p}+\left \Vert z\right \Vert _{p}\big )^{p} -\left \Vert x+z\right \Vert _{p}^{p}\big ]\ge 0 \quad \text {if }p\in (1,2], \] \[ 0\le \left \Vert x\right \Vert _{p}\left \Vert y\right \Vert _{q}-\left \langle x,y\right \rangle \le (1/p)\big [\big (\left \Vert x \right \Vert _{p}+\left \Vert z\right \Vert _{p}\big )^{p} -\left \Vert x+z\right \Vert _{p}^{p}\big ] \quad \text {if }p\ge 2.\] A similar result holds for complex valued functions with Re$(\left \langle x,y\right \rangle )$ substituting for $\left \langle x,y\right \rangle$. We obtain these inequalities from some stronger (though slightly more involved) ones.