Abstract
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 $$\begin{aligned} w^2(T)\le \min _{0\le \alpha \le 1} \left\| \alpha T^*T +(1-\alpha )TT^* \right\| , \end{aligned}$$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.
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