Improved Young and Heinz operator inequalities for unitarily invariant norms

Abstract

Abstract In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, X ∈ Mn (ℂ): | | | A ν X B 1 − ν + A 1 − ν X B ν 2 | | | ≤ ( 4 r 0 − 1 ) | | | A 1 2 X B 1 2 | | | + 2 ( 1 − 2 r 0 ) | | | ( 1 − α ) A 1 2 X B 1 2 + α ( A X + X B 2 ) | | | , $$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where 1 4 ≤ ν ≤ 3 4 , α ∈ [ 1 2 , ∞ ) $\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty ) \end{array}$ and r 0 = min{ν, 1 – ν}.