Numerical radius inequalities and estimation of zeros of polynomials

Abstract: Let $A$ be a bounded linear operator defined on a complex Hilbert space and let $|A|=(A^*A)^{1/2}$ be the positive square root of $A$. Among other refinements of the well known numerical radius inequality $w^2(A)\leq \frac12 |A^A+AA^|$, we show that \begin{eqnarray*} w^2(A)&\leq&\frac{1}{4} w^2 \left(|A|+i|A^|\right)+\frac{1}{8}\left||A|^2+|A^|^2\right |+\frac{1}{4}w\left(|A||A^*|\right) &\leq& \frac12 |A^A+AA^|. \end{eqnarray*} Also, we develop inequalities involving numerical radius and spectral radius for the sum of the product operators, from which we derive the following inequalities $$ w^p(A) \leq \frac{1}{\sqrt{2} } w(|A|^p+i|A^*|^p )\leq |A|^p$$ for all $p\geq 1.$ Further, we derive new bounds for the zeros of complex polynomials.