Some generalized numerical radius inequalities for Hilbert space operators

Abstract: We generalize several inequalities involving powers of the numerical radius for product of two operators acting on a Hilbert space. For any $A, B, X\in \mathbb{B}(\mathscr{H})$ such that $A,B$ are positive, we establish some numerical radius inequalities for $A^\alpha XB^\alpha$ and $A^\alpha X B^{1-\alpha}\,\,(0 \leq \alpha \leq 1)$ and Heinz means under mild conditions.