Let A and B be positive operators 0 < q ? 1. In this paper, we shall show that if Aq?0 (A?0/2B?0A?0/2) q?0/?0+?0 (B?0/2A?0B?0/2) Bq?0 hold for fixed ?0 > 0. Then the following inequalities hold: Aq1? (A?/2B?A?/2) q1?/?+? (B?/2A?B?/2) Bq1? all ?0, q1 q. Also, a normality of class p-A(s, t) s 0, t p Moreover, T or T* belongs to p-wA(s, some 1 S is an operator which W(S) ST = T*S, then self-...