Some refinements of young type inequality for positive linear map

Abstract

Abstract We obtain a refined Young type inequality in this paper. The conclusion is presented as follows: Let A, B ∈ B(𝓗) be two positive operators and p ∈ [0, 1], then A ♯ p B + G ∗ ( A ♯ p B ) G ≤ A ∇ p B − 2 r ( A ∇ B − A ♯ B ) , $$\begin{array}{} \displaystyle A\sharp_p B+G^*(A\sharp_p B)G\le A\nabla_p B-2r(A\nabla B-A\sharp B), \end{array}$$ where r = min{p, 1 – p}, G = L ( 2 p ) 2 $\begin{array}{} \displaystyle \frac{\sqrt{L(2p)}}{2} \end{array}$ A –1 S(A|B), L(t) is periodic with period one and L(t) = t 2 2 1 − t t 2 t $\begin{array}{} \displaystyle \frac{t^2}{2}\left( \frac{1-t}{t} \right)^{2t} \end{array}$ for t ∈ [0, 1]. Moreover, we give the s-th powering of two inequalities related to the above one with s > 0 which refines Lin’s work. In the mean time, we present an inequality involving Hilbert-Schmidt norm.