Better bounds on the numerical radii of Hilbert space operators

Kittaneh proved that if A is a bounded linear operator on a complex Hilbert space, then 1/4 parallel to vertical bar A vertical bar(2) + vertical bar A*vertical bar(2)parallel to <= omega(2) (A), where omega(.) and parallel to.parallel to are the numerical radius and the usual operator norm, and vertical bar A vertical bar = (A*A)(1/2). In this paper, we show that 1/4 parallel to vertical bar A vertical bar(2) + vertical bar A vertical bar(2)parallel to <= 1/2 omega(2) (A)+1/8 parallel to(A+A*) (A-A*)parallel to <=omega(2) (A) Meanwhile, we give an improvement of the norm inequality presented by Bhatia and Kittaneh for the positive operators. (C) 2020 Elsevier Inc. All rights reserved.