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):
$$\begin{array}{}
\begin{split}
\displaystyle
\Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big|
\leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}|||
\\
&+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}}
+\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|,
\end{split}
\end{array}$$
where
$\begin{array}{}
\displaystyle
\frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty )
\end{array}$ and r0 = min{ν, 1 – ν}.
Unitarily invariant norm inequalities for matrix means
