  Let $\mathcal{C}_p$ be the Schatten $p$-class for $p>0$. Generalizations of
the parallelogram law for the Schatten 2-norms have been given in the following
form: If $\mathbf{A}=\{A_1,A_2,...,A_n\}$ and $\mathbf{B}=\{B_1,B_2,...,B_n\}$
are two sets of operators in $\mathcal{C}_2$, then
$$\sum_{i,j=1}^n\|A_i-A_j\|_2^2 + \sum_{i,j=1}^n\|B_i-B_j\|_2^2 =
2\sum_{i,j=1}^n\|A_i-B_j\|_2^2 - 2\Norm{\sum_{i=1}^n(A_i-B_i)}_2^2.$$ In this
paper, we give generalizations of this as pairs of inequalities for Schatten
$p$-norms, which hold for certain values of $p$ and reduce to the equality
above for $p=2$. Moreover, we present some related inequalities for three sets
of operators.
