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Let H and K be bounded positive operators on a Hilbert space, and assume that H is nonsingular. Based on Pedersen and Takesaki's research on the operator equation K=THT, Furuta and Bach gave deep discussion on the equation K=T12(T12H1nT12)nT12 where n is a natural number. As a continuation, this paper is to consider the equation Kp=Hδ2T12(T12Hδ+rT12)p−δδ+rT12Hδ2 where p>0, r>0 and p⩾δ>−r. As applications, we prove that the inclusion relations among class wA(p,r) operators are strict and show a generalization of Aluthge's result.