Weighted numerical radius inequalities for operator and operator matrices

The concepts of weighted numerical radius has been defined in recent times. In this article, we obtain several upper bound for weighted numerical radius of operators and $2\times 2$ operator matrices which generalize and improves some well known famous inequality for classical numerical radius. We also obtain an upper bound for the weighted numerical radius of the Aluthge transformation, $\tilde{T}$ of an operator $T\in \mathcal{B}(\mathcal{H}),$ where $\tilde{T}=\mathrm{\mid }T{\mathrm{\mid }}^{1\mathrm{/}2}U\mathrm{\mid }T{\mathrm{\mid }}^{1\mathrm{/}2}$ and $T=U\mathrm{\mid }T\mathrm{\mid }$ be the canonical polar decomposition of $T\mathrm{.}$