Schatten p-norm inequalities related to an extended operator parallelogram law

Let CpCp be the Schatten p  -class for p>0p>0. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if A={A1,A2,…,An}A={A1,A2,…,An} and B={B1,B2,…,Bn}B={B1,B2,…,Bn} are two sets of operators in, then C2C2∑i,j=1n‖Ai-Aj‖22+∑i,j=1n‖Bi-Bj‖22=2∑i,j=1n‖Ai-Bj‖22-2∑i=1n(Ai-Bi)22.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=2p=2. Moreover, we present some related inequalities for three sets of operators.