On the monotone properties of general affine surface areas under the Steiner symmetrization

In this paper, we prove that, if functions (concave) $\varphi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\varphi }$ affine surface area is monotone increasing, while the $L_{\psi }$ affine surface area is monotone decreasing under the Steiner symmetrization. Consequently, we can prove related affine isoperimetric inequalities, under certain conditions on $\varphi$ and $\psi$, without assuming that the convex body involved has centroid (or the Santal'{o} point) at the origin.