Abstract
Let A, B be invertible positive operators on a Hilbert space H. We present some improved reverses of Young type inequalities, in particular, $$ (1-\nu)^{2\nu}(A\nabla B)+(1-\nu)^{2(1-\nu)}H_{2\nu}(A,B) \geq2(1-\nu ) (A\sharp B) $$ and $$ (1-\nu)^{2\nu}H_{2\nu}(A,B)+(1-\nu)^{2(1-\nu)}(A\nabla B) \geq2(1-\nu ) (A\sharp B), $$ where $0\leq\upsilon\leq\frac{1}{2}$ . We also give some new inequalities involving the Heinz mean for the Hilbert-Schmidt norm.
Highlights
Let H be a Hilbert space and let Bh(H) be the semi-space of all bounded linear self-adjoint operators on H
The theory of operator means for positive operators on a Hilbert space was initiated by Ando and established by him and Kubo in connection with Lowners theory for the operator monotone functions [ ]
3 Conclusions In the present paper we got reverses of the scalar Young type inequality and using them we obtained the reverses of Young type inequalities for operators
Summary
Let H be a Hilbert space and let Bh(H) be the semi-space of all bounded linear self-adjoint operators on H. Let B(H) and B(H)+, respectively, denote the set of all bounded linear operators on a complex Hilbert space H and set of all positive operators in Bh(H). The set of all positive invertible operators is denoted by B(H)++. A and B are invertible, the ν-harmonic mean of A and B, denoted by A!νB is defined as. The operator version of the Heron means is denoted by. To obtain inequalities for bounded self-adjoint operators on Hilbert space, we shall use the following monotonicity property for operator functions: If X ∈ Bh(H) with a spectrum Sp(X) and f , g are continuous real-valued functions on. Zhao et al [ ] gave an inequality for the Heinz and Heron means as follows: If A and B are two positive and invertible operators, .
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