Lojasiewicz inequalities in a certain class of smooth functions

Let $f$ be a germ of a smooth function at the orirgin in ${\text{RR}}^n\mathrm{.}$ We show that if $f$ is Kouchnirenko’s nondegenerate and satisfies the so called Kamimoto–Nose condition then it admits the Łojasiewicz inequalities. We compute the Łojasiewicz exponents for some special cases. In particular, if $f$ is a germ of a smooth convex Kouchnirenko’s nondegenerate function and satisfies the Kamimoto–Nose condition, then all its Łojasiewicz exponents can be expressed very simply in terms of its Newton polyhedron.