Better bounds on the numerical radii of Hilbert space operators

Abstract Kittaneh proved that if A is a bounded linear operator on a complex Hilbert space, then 1 4 ‖ | A | 2 + | A ⁎ | 2 ‖ ≤ ω 2 ( A ) , where ω ( ⋅ ) and ‖ ⋅ ‖ are the numerical radius and the usual operator norm, and | A | = ( A ⁎ A ) 1 / 2 . In this paper, we show that 1 4 ‖ | A | 2 + | A ⁎ | 2 ‖ ≤ 1 2 ω 2 ( A ) + 1 8 ‖ ( A + A ⁎ ) ( A − A ⁎ ) ‖ ≤ ω 2 ( A ) . Meanwhile, we give an improvement of the norm inequality presented by Bhatia and Kittaneh for the positive operators.