Some Multiplicative Inequalities for Heinz Operator Mean

Abstract

In this paper we obtain some new multiplicative inequalities for Heinz operator mean. 1. Introduction Throughout this paper A; B are positive invertible operators on a complex Hilbert space (H; h ; i) : We use the following notations for operators and 2 [0; 1] Ar B := (1 )A+ B; the weighted operator arithmetic mean, and A] B := A 1=2 A BA 1=2 A; the weighted operator geometric mean [14]. When = 1 2 we write ArB and A]B for brevity, respectively. De ne the Heinz operator mean by H (A;B) := 1 2 (A] B +A]1 B) : The following interpolatory inequality is obvious (1.1) A]B H (A;B) ArB for any 2 [0; 1]: We recall that Specht’s ratio is de ned by [16] (1.2) S (h) := 8> >: h 1 h 1 e ln h 1 h 1 if h 2 (0; 1) [ (1;1) ; 1 if h = 1: It is well known that limh!1 S (h) = 1; S (h) = S 1 h > 1 for h > 0; h 6= 1. The function is decreasing on (0; 1) and increasing on (1;1) : The following result provides an upper and lower bound for the Heinz mean in terms of the operator geometric mean A]B : Theorem 1 (Dragomir, 2015 [6]). Assume that A; B are positive invertible operators and the constants M > m > 0 are such that (1.3) mA B MA: 1991 Mathematics Subject Classi cation. 47A63, 47A30, 15A60, 26D15, 26D10.