  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.
