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♯pB+G∗(A♯pB)G≤A∇pB−2r(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(2p)2 $\\begin{array}{} \\displaystyle \\frac{\\sqrt{L(2p)}}{2} \\end{array}$ A–1S(A|B), L(t) is periodic with period one and L(t) = t221−tt2t $\\begin{array}{} \\displaystyle \\frac{t^2}{2}\\left( \\frac{1-t}{t} \\right)^{2t} \\end{array}$ 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.