Let A be a positive bounded operator on a Hilbert space \({\mathcal {H}}\) . The semi-inner product \(\left\langle x,y\right\rangle _{A}:=\left\langle Ax,y\right\rangle \) , x, y \(\in \) \({\mathcal {H}}\) , induces a semi-norm \(\left\| .\right\| _{A}\) on \({\mathcal {H}}\) . Let \(\omega _{A}\left( T\right) \) denote the A-numerical radius of an operator T in semi-Hilbertian space \(\left( {\mathcal {H}},\left\| .\right\| _{A}\right) \) . Our aim in this work is to give new inequalities of A-numerical radius of operators in semi-Hilbertian spaces. These inequalities improve and generalize some earlier related inequalities. In particular, we show some new improvements for the inequality: \(\begin{aligned} \frac{1}{4}\left\| T^{\sharp _{A}}T+TT^{ _{A}}\right\| _{A}\le \omega _{A}^{2}\left( T\right) \le \frac{1}{2}\left\| T^{\sharp _{A}}T+TT^{\sharp _{A}}\right\| _{A}\text {.} \end{aligned}\) Some other related results are also obtained.
