On the Best Exponent in Markov Inequality

Let E be a compact set preserving the Markov inequality and m(E) be its best exponent i.e., m(E) is the infimum of all possible exponents in this inequality on E. It is known that $\alpha (E) \le \frac1{m(E)}$ where α(E) is the best exponent in Hölder continuity property of the (pluri)complex Green function (with pole at infinity) of E. We show that if E???? ^N (or ? ^N ) with N?≥?2 then the Markov inequality need not be fulfilled with m(E). We also construct a set E????² such that the Markov inequality holds at the tip of exponential cusps composing E but for the whole set E we have m(E)?=?∞. Moreover, we prove that sup m(E)?=?∞ where the supremum is taken over all compact sets E???? preserving the Markov inequality. Finally, we prove that if E is a Markov set in ? then its image F(E) under a holomorphic mapping F is a Markov set too. More precisely, we prove that $m(F(E))\leq m(E)\cdot \Big(1+ \max\limits_{ \partial E\cap\{F'(t)=0\}}\textrm{ord}_t F'\Big)$ .