Proper improvement of well-known numerical radius inequalities and their applications

New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space $\mathcal{H}$ are given. In particular, it is established that if $T$ is a bounded linear operator on a Hilbert space $\mathcal{H}$ then $$w^2(T)\leq \min_{0\leq \alpha \leq 1} \left \| \alpha T^*T +(1-\alpha)TT^* \right \|,$$ where $w(T)$ is the numerical radius of $T.$ The inequalities obtained here are non-trivial improvement of the well-known numerical radius inequalities. As an application we estimate bounds for the zeros of a complex monic polynomial.