Improvement of inequalities related to powers of the numerical radius.

We presented some improvements of the inequalities involving the numerical radius powers for products and sums of the operators investigated in the Hilbert space. We generalized and improved numerical radius inequalities with a generalization of the mixed Schwarz inequality. Among other things, with the help of a fraction and its power, as well as the introduction of ξ, we provided a very good improvement for the ω^r(E), for E ∈ B(H_s). [ABSTRACT FROM AUTHOR]