Numerical radius bounds for certain operators

We provide sharp bounds for the numerical radius of bounded linear operators defined on a complex Hilbert space. We also provide sharp bounds for the numerical radius of \(A^{\alpha }XB^{1-\alpha }\), \(A^{\alpha }XB^{\alpha }\) and the Heinz means of operators, where A, B, X are bounded linear operators with \(A,B\ge 0\) and \(0\le \alpha \le 1.\) Further, we study the A-numerical radius inequalities for semi-Hilbertian space operators. We prove that \(w_A(T) \le \left( 1-\frac{1}{2^{n-1}}\right) ^{1/n} \Vert T\Vert _A\) when \(AT^n=0\) for some least positive integer n. Some equalities for the A-numerical radius inequalities are also studied.