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 operator then $$\begin{aligned} w^2(T)\le \min _{0\le \alpha \le 1} \left\| T^*T +(1-\alpha )TT^* \right\| , \end{aligned}$$ where w(T) T. The obtained here non-trivial improvement well-known inequalities. As an application we estimate bounds zeros monic polynomial.