Schatten p-norm inequalities related to a characterization of inner product spaces

Let $A_1,\mathrm{.}\mathrm{.}\mathrm{.}{A}_n$ be operators acting on a separable complex Hilbert space such that ${\sum }_{i=1}^n{A}_i=0$. It is shown that if $A_1,\mathrm{.}\mathrm{.}\mathrm{.}{A}_n$ belong to a Schatten $p$-class, for some $p>0$, then 2^{p/2}n^{p-1} \sum_{i=1}^n |A_i|^p_p \leq \sum_{i,j=1}^n|A_i\pm A_j|^p_p for $0<p\le 2$, and the reverse inequality holds for $2\le p<\mathrm{\infty }$. Moreover, \sum_{i,j=1}^n|A_i\pm A_j|^2_p \leq 2n^{2/p} \sum_{i=1}^n |A_i|^2_p for $0<p\le 2$, and the reverse inequality holds for $2\le p<\mathrm{\infty }$. These inequalities are related to a characterization of inner product spaces due to E.R. Lorch.