A modified Heinz’s inequality

Concerning the Heinz’s inequality,Chan and Kwong [N.N. Chan,M.K. Kwong,Hermitian matrix inequalities and a conjecture,Amer.Math.Monthly 92 (1985) 533–541] conjectured that A ⩾ B ⩾ O will imply and Furuta [T. Furuta,A ⩾ B ⩾ O assures for r ⩾ 0,p ⩾ 0,q ⩾ 1 with (1 + 2r)q ⩾ p + 2r,Proc.Amer.Math.Soc.101 (1987),85–88] gave its affirmative answer as follows:If A ⩾ B ⩾ O, then ,for .And,in [K. Tanahashi,The Furuta inequality with negative powers,Proc.Amer.Math.Soc.127 (1999) 1683–1692],Tanahashi studied the same inequality for the invertible case.In this paper,we shall determine the region of γ such that the operator inequality (AγAαAγ)β ⩾ (AγBαAγ)β holds for any bounded linear operators A and B on a Hilbert space H such as A ⩾ B ⩾ bI (some b > 0) and for any given α and β such as α > 0 and β > 0.It is easily seen that the inequalities (AγAαAγ)β ⩾ (AγBαAγ)β and(BγAαBγ)β ⩾ (BγBαBγ)β are equivalent.