For each non empty compact set \(K\subset\mathbb C^N\) let \(V_K\) denote its Siciak extremal function, \(V_K(z):=\sup\{\varphi(z): \varphi\) is plurisubharmonic on \(\mathbb C^N\), \(\sup\{\varphi(w)-\log(1+|w|): w\in\mathbb C^N\}<+\infty\), and \(\varphi\leq0\) on \(K\}\), \(z\in\mathbb C^N\). We say that \(K\) has the HCP property if there exist \(\omega, \mu>0\) such that \(V_K(z)\leq\omega(\operatorname{dist}(z,K))^\mu\), for each \(z\in\mathbb C^N\) with \(\operatorname{dist}(z,K)\leq1\). We say that \(K\) satisfies Markov’s inequality if there exist \(\varepsilon, C>0\) such that for each polynomial \(Q\) of \(N\) variables we have \(\|\frac{\partial^{|\alpha|}Q}{\partial z_1^{\alpha_1}\cdots\partial z_N^{\alpha_N}}\|_K\leq C(\deg Q)^\varepsilon)^{|\alpha|} \|Q\|_K\), \(\alpha\in\mathbb N_0^N\). Let \(h:U\longrightarrow\mathbb C^{N'}\) be holomorphic, where \(U\) is open in \(\mathbb C^N\). Let \(U_\ast\) denote the union of those connected components \(S\) of \(U\) such that \(\operatorname{rank} d_\zeta h=N'\) for some \(\zeta\in S\). We say that \(h\) is nondenegerate if \(U_\ast=U\). The main results of the paper are the following theorems:
– Assume that \(K\) has the HCP property and \(\widehat K\subset U\). Then the following conditions are equivalent: (i) \(h(K)\) has the HCP property; (ii) \(V_{h(K)}\) is continuous; (iii) \(h(K)\subset (h(K\cap U_\ast))\widehat{\;}\).
– Assume that \(h\) is nondegenerate, \(K\) has the HCP property, and \(\widehat K\subset U\). Then \(h(K)\) has the HCP property.
– Assume that \(h\) is nondegenerate, \(K\) satisfies Markov’s inequality, \(\widehat K\subset U\), and \(h(K)\) is nonpluripolar. Then \(h(K)\) satisfies Markov’s inequality.