An operator T is called (α,β)-normal (0 ≤ α ≤ 1 ≤ β) if α²T*T ≤ TT* ≤ β²T*T.
In this paper, we establish various inequalities between the operator norm and
its numerical radius of (α,β)-normal operators in Hilbert spaces. For this
purpose, we employ some classical inequalities for vectors in inner product
spaces.