Numerical radius and spectral radius inequalities with an estimation for roots of a polynomial

Suppose A is a bounded linear operator defined on a complex Hilbert space. Among other numerical radius inequalities, it is proved (by using the Aluthge transform $${\widetilde{A}}$$ A ~ of A) that $$\begin{aligned} w^2(A)\le & {} \frac{1}{2} {\left( \left\| A{\widetilde{A}}A^* \right\| \left\| {\widetilde{A}} \right\| \right) ^{1/2} } + \frac{1}{4} \Big \Vert A^*A+AA^* \Big \Vert , \end{aligned}$$ w 2 ( A ) ≤ 1 2 A A ~ A ∗ A ~ 1 / 2 + 1 4 ‖ A ∗ A + A A ∗ ‖ , where w(A) is the numerical radius of A. This numerical radius bound improves the well known existing bound $$\begin{aligned} w(A) \le \frac{1}{2} \left( \Vert A \Vert + {\left\| A^2\right\| ^{1/2}} \right) . \end{aligned}$$ w ( A ) ≤ 1 2 ‖ A ‖ + A 2 1 / 2 . Additionally, we explore the spectral radius bounds of the sum, product and commutator of bounded linear operators. Furthermore, by using the spectral radius bound for the sum of two operators, we provide an estimation for the roots of a complex polynomial.