  Under a potential-theoretical hypothesis named $f$-Property with $f$
satisfying $\displaystyle\int_t^\infty \dfrac{da}{a f(a)}<\infty$, we show that
the Kobayashi metric $K(z,X)$ on a weakly pseudoconvex domain $\Om$, satisfies
the estimate $K(z,X)\ge Cg(\delta_\Om(x)^{-1})|X|$ for any $X\in T^{1,0}\Om$
where $(g(t))^{-1}$ denotes the above integral and $\delta_\Om(z)$ is the
distance from $z$ to $b\Om$.
