  Several new improvements of the $A$-numerical radius inequalities for
operators acting on a semi-Hilbert space, i.e., a space generated by a positive
operator $A$, are proved. In particular, among other inequalities, we show that
\begin{align*} \frac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A
\leq\frac{1}{4}\Big(2\omega_A^2(T)+\gamma(T)\Big) \leq \omega_A^2(T),
\end{align*} where
$$\gamma(T)=\sqrt{\left(\|\Re_A(T)\|_A^2-\|\Im_A(T)\|_A^2\right)^2+4\|\Re_A(T)\Im_A(T)\|_A^2}.$$
Here $\omega_A(X)$ and $\|X\|_A$ denote respectively the $A$-numerical radius
and the $A$-seminorm of an operator $X$. Also,
$\Re_A(T):=\frac{T+T^{\sharp_A}}{2}$ and $\Im_A(T):=\frac{T-T^{\sharp_A}}{2i}$,
where $T^{\sharp_A}$ is a distinguished $A$-adjoint operator of $T$. Further,
some new refinements of the triangle inequality related to $\|\cdot\|_A$ are
established.
