  Let $r_A(T)$ denote the $A$-spectral radius of an operator $T$ which is
bounded with respect to the seminorm induced by a positive operator $A$ on a
complex Hilbert space $\mathcal{H}$. In this paper, we aim to establish some
$A$-spectral radius inequalities for products, sums and commutators of
$A$-bounded operators. Moreover, under suitable conditions on $T$ and $A$ we
show that
  \begin{equation*} r_A\left( \sum_{k=0}^{+\infty}c_{k}T^{k}\right) \leq
\sum_{k=0}^{+\infty}|c_{k}|\left[r_A(T)\right]^{k},
  \end{equation*} where $c_k$ are complex numbers for all $k\in \mathbb{N}$.
