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, ldots,A_n } and mathbf{B}= {B_1,B_2, ldots,B_n } are two sets of operators of mathcal{C}_2 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.

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