Some Refinements of Numerical Radius Inequalities

We propose some refinements for the second inequality in 1 2 A ≤ w A ≤ A , where A ∈ B ( H ). In particular, if A is hyponormal, then, by refining the Young inequality with the Kantorovich constant K K (⋅, ⋅), we show that w A ≤ 1 2 inf x = 1 ζ x A + + A ∗ ≤ 1 2 A + A ∗ , where ζ x = K A x x A ∗ x x 2 r , r = min λ 1 − λ , and 0 ≤ λ ≤ 1. We also give a reverse for the classical numerical radius power inequality w ( A n ) ≤ w n ( A ) for any operator A ∈ B ( H ) case where n = 2 .