A mixed Hölder and Minkowski inequality

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 $z=y|y|^{q-2}$:

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.