Abstract
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 ⊂ ℂN (or ℝN) with N ≥ 2 then the Markov inequality need not be fulfilled with m(E). We also construct a set E ⊂ ℝ2 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 ⊂ ℝ preserving the Markov inequality. Finally, we prove that if E is a Markov set in ℂ then its image F(E) under a holomorphic mapping F is a Markov set too. More precisely, we prove that \(m(F(E))\leq m(E)\cdot \Big(1+ \max\limits_{ \partial E\cap\{F'(t)=0\}}\textrm{ord}_t F'\Big)\).
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Baran, M., Białas-Cież, L. & Milówka, B. On the Best Exponent in Markov Inequality. Potential Anal 38, 635–651 (2013). https://doi.org/10.1007/s11118-012-9290-0
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DOI: https://doi.org/10.1007/s11118-012-9290-0