Author’s abstract: It is shown that if \(A\) is a bounded linear operator on a complex Hilbert space, then \[ w(A)\leq{1\over2}(\| A\|+\| A^2\|^{1/2}), \] where \(w(A)\) and \(\| A\|\) are the numerical radius and the usual operator norm of \(A\), respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.