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