  In 2003 Luo proved an inequality relating the Wigner-Yanase information and
the $SLD$-information. In this paper we prove that Luo's inequality is a
particular case of a general inequality which holds for any regular quantum
Fisher information. Moreover we show that this general inequality is a
consequence of the Kubo-Ando inequality that states that any matrix mean is
bigger than the harmonic mean and smaller than the arithmetic mean.
