  In this paper we consider the limit cycles of the planar system
$$\frac{d}{dt}(x,y)=\mathbf X_n+\mathbf X_m, $$ where $\mathbf X_n$ and
$\mathbf X_m$ are quasi-homogeneous vector fields of degree $n$ and $m$
respectively. We prove that under a new hypothesis, the maximal number of limit
cycles of the system is $1$. We also show that our result can be applied to
some systems when the previous results are invalid. The proof is based on the
investigations for the Abel equation and the generalized-polar equation
associated with the system, respectively. Usually these two kinds of equations
need to be dealt with separately, and for both equations, an efficient approach
to estimate the number of periodic solutions is constructing suitable auxiliary
functions. In the present paper we develop a formula on the divergence, which
allows us to construct an auxiliary function of one equation with the auxiliary
function of the other equation, and vice versa.
