On the inequality w(AB)≤c||A||w(B) where A is a positive operator

Abstract

Abu-Omar and Kittaneh [Numerical radius inequalities for products of Hilbert space operators, J. Operator Theory 72(2) (2014), 521--527], wonder what is the smallest constant c such that w(AB)≤c||A||w(B) for all bounded linear operators A, B on a complex Hilbert space with A is positive. Here, w(.) stands for the numerical radius. In this paper, we prove that c=3sqrt{3}/4.