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In this article, we derive an inequality of {\L}ojasiewicz-Siciak type forcertain sets arising in the context of the complex dynamics in dimension 1.More precisely, if we denote by $dist$ the euclidian distance in $\mathbb{C}$,we show that the Green function $G_K$ of the filled Julia set $K$ of apolynomial such that $\mathring{K}\neq \emptyset$ satisfies the so-called {\L}Scondition $\displaystyle G_A\geq c\cdot dist(\cdot, K)^{c'}$ in a neighborhoodof $K$, for some constants $c,c'>0$. Relatively few examples of compact setssatisfying the {\L}S condition are known. Our result highlights an interestingclass of compact sets fulfilling this condition. The fact that filled Juliasets satisfy the {\L}S condition may seem surprising, since they are in generalvery irregular. In order to prove our main result, we define and study the setof obstruction points to the {\L}S condition. We also prove, in dimension$n\geq 1$, that for a polynomially convex and L-regular compact set of nonempty interior, these obstruction points are rare, in a sense which will bespecified.

{\L}S condition for filled Julia sets in $\mathbb{C}$