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An inequality for matrices that interpolates between the Cauchy-Schwarz and the arithmetic-geometric mean inequalities for unitarily invariant norms has been obtained by Audenaert. Alakhrass obtained a related result to Audenaert's inequality using a log-convex function g defined on [0,1]. Very recently, Zou obtained an inequality for matrices that unifies Hölder's inequality and the arithmetic-geometric mean inequality for unitarily invariant norms. A generalized version of Zou's inequality for unitarily invariant norms is given, and an alternative proof of Audenaert's inequality using a refined version of Alakhrass's function is presented.