Markov's inequality in the o-minimal structure of convergent generalized power series

We prove that the UPC (uniformly polynomially cuspidal) condition holds in an o-minimal structure generated by some convergent generalized power series. The result is obtained by geometric and analytic methods. We get automatically some consequences. First of all, we obtain a new large class of sets with cusps satisfying Markov's inequality (the class is strictly larger than the class of compact, fat and globally subanalytic sets). It may be useful from the point of view of approximation theory (multivariate polynomial inequalities) and constructive function theory. Our main result finds also practical application in complex analysis - it gives new examples of sets with the HCP property (Holder continuity property of the Siciak extremal function).