Numerical radius inequalities and estimation of zeros of polynomials

Let $A$ be a bounded linear operator defined on a complex Hilbert space and let $\mathrm{\mid }A\mathrm{\mid }=(A^{\ast }A{)}^{1\mathrm{/}2}$ be the positive square root of $A$. Among other refinements of the well known numerical radius inequality $w^2(A)\le \frac{1}{2}\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel }$, 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 $$w^p(A)\le \frac{1}{\sqrt{2}}w(\mathrm{\mid }A{\mathrm{\mid }}^p+i\mathrm{\mid }{A}^{\ast }{\mathrm{\mid }}^p)\le \mathrm{\parallel }A{\mathrm{\parallel }}^p$$ for all $p\ge 1.$ Further, we derive new bounds for the zeros of complex polynomials.