Abstract :
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 greater-or-equal, slanted B greater-or-equal, slanted O will imply image and Furuta [T. Furuta,A greater-or-equal, slanted B greater-or-equal, slanted O assuresimage for r greater-or-equal, slanted 0,p greater-or-equal, slanted 0,q greater-or-equal, slanted 1 with (1 + 2r)q greater-or-equal, slanted p + 2r,Proc.Amer.Math.Soc.101 (1987),85–88] gave its affirmative answer as follows:If A greater-or-equal, slanted B greater-or-equal, slanted O, then image,for image.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.
