If A, B are positive operators on a Hilbert space it was shown by T. Furuta [Proc. Am. Math. Soc. 101, 85-88 (1987; Zbl 0721.47023)] that for ,
for , , and
The author of the present note gives a proof of this based on the theory of means of operators. The method is to prove the inequality for a restricted range of parameter values and to prove a reduction lemma which allows these to change in a prescribed manner.