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For positive operatorsAandBwithAinvertible it is shown that(AB2A)½≤A2implies(AB2A)¾≤ABA. The inequalities in the title for0≤B≤Aare then derived as a conquence.doi:10.1155/s0161171295000263 fatcat:k3sslhmz5veutfpxxrinjrwfoy