Abstract: In this paper we consider a class of polynomial planar system with two small parameters, $\varepsilon$ and $\lambda$, satisfying $0<\varepsilon\ll\ lambda\ll1$. The corresponding first order Melnikov function $M_1$ with respect to $\varepsilon$ depends on $\lambda$ so that it has an expansion of the form $M_1(h,\lambda)=\sum\limits_{k=0}^\infty M_{1k}(h)\lambda^k.$ Assume that $M_{1k'}(h)$ is the first non-zero coefficient in the expansion. Then by estimating the number of zeros of $M_{1k'}(h)$, we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for $0<\varepsilon\ll\lambda\ll1$, when $k'=0$ or $1$. In addition, for each $k\in \mathbb{N}$, an upper bound of the maximal number of zeros of $M_{1k}(h)$, taking into account their multiplicities, is presented.

Key words: Limit cycle, Melnikov function, integrable system