Some refinements of young type inequality for positive linear map

作者：Chaojun Yang, Yaxin Gao, Fangyan Lu · 发表于：Mathematica Slovaca · 年份：2019 · DOI：10.1515/ms-2017-0277 · 被引用次数：11 · 研究领域：Mathematical Inequalities and Applications、Analytic and geometric function theory、Holomorphic and Operator Theory

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.