New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space H are given. In particular, it is established that if T is a bounded linear operator on a Hilbert space H then w2(T)≤min0≤α≤1∥αT∗T+(1-α)TT∗∥,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.