The well known Young and Heinz inequalities for positive real numbers have been generalized to positive semidefinite Hermitian matrices (see, for example, R. Merris and S. Pierce [Proc. Am. Math. Soc. 31, 437–440 (1972; Zbl 0209.06404)]). In the present paper the authors use similar methods to prove reverse inequalities. The main lemma from which their results follow is: If \(Q\) is an \(n\times n\) positive definite Hermitian matrix, and \(0\) \(\leq\nu \leq1\), then
\[ r_{0}(I-Q^{1/2})^{2}+Q^{1-\nu}\leq\nu I+(1-\nu)Q\leq R_{0}(I-Q^{1/2} )^{2}+Q^{1-\nu} \]
where \(r_{0}=\min(\nu,1-\nu)\) and \(R_{0}=\max(\nu,1-\nu).\) (The left hand inequality is essentially a strengthened form of the Young inequality, and the right hand inequality is the so-called reverse inequality.)