Łojasiewicz-type inequalities in complex analysis
Let \(K \subseteq \mathbb C^N\) be a non-empty compact set. The Siciak extremal function \(\Phi_K\) is defined by \[ \Phi_K(z) := \sup \Big\{ |p(z)|^{1/\deg p} : p \in \mathbb C[z],\, \deg p >0,\, \sup_{z \in K} |p(z)| \leq 1\Big\}, \quad z \in \mathbb C^N. \] It is an important tool in approximation theory, complex analysis and pluripotential theory. The compact set \(K \subseteq \mathbb C^N\) is said to satisfy the Łojasiewicz-Siciak (ŁS) condition if there exist \(\eta, \kappa >0\) and an open neighborhood \(U\) of \(K\) in \(\mathbb C^N\) such that \[ \Phi_K(z) \geq 1 + \eta \, \text{dist} (z,K)^\kappa \quad \text{ for all } z \in U. \] In particular, if \(K\) satisfies the (ŁS) condition then it is polynomially convex. In the paper under review the author provides conditions under which the (ŁS) condition is preserved by taking preimages or images of holomorphic mappings. As a consequence many new examples of sets having the (ŁS) property are given. The paper continues previous work on the (ŁS) condition of same the author started in [J. Math. Anal. Appl. 430, No. 2, 755--776 (2015; Zbl 1331.32017)] using techniques from subanalytic geometry, most notably the Łojasiewicz inequality.
scientific article; zbMATH DE number 6946525
scientific article