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Inequalities for products of polynomials. II. (English) Zbl 1217.30003

Suppose that \(\{p_k(z)\}_{k=1}^m\) is a finite sequence of polynomials with complex coefficients and that \(p(z)\) is their product. Let \(n\) denote the degree of \(p\). For a compact subset \(E\) of the complex plane, consider the question of finding the smallest constant \(M_E > 0\) such that \(\prod_{k=1}^m \|p_k\|_E \leq M_E^n \|p\|_E\).
This paper continues the study of the questions considered in the first paper of this series [Math. Scand. 104, No. 1, 147–160 (2009; Zbl 1163.30009)] concerning how \(M_E\) varies with \(E\). In Section 2, the authors consider the class of sets \(E\) that are invariant under the cyclic group of rotations generated by a rotation through the angle \(2\pi/k\) about a fixed point. Under a certain further restriction they show that in this case, \(M_E\) is maximized for the set \(S_k\) consisting of the rays of the unit circle with one endpoint at \(0\) and the other at the \(k\)th roots of unity. If \(E\) is further assumed to be convex, then again, under a certain further restriction, they show that \(M_E\) is maximized for \(P_k\), the regular polygon with vertices at the \(k\)th roots of unity. They conjecture that these results hold without the additional hypotheses that are needed here to prove them.
In Section 3, they consider the question of how much the constant \(M_E\) improves (if at all) when the number of factors \(m\) is fixed. Let \(B_m(E)\) denote the best constant in the case of \(m\) factors. The reviewer [Bull. Lond. Math. Soc. 26, No. 5, 449–454 (1994; Zbl 0820.30005)] determined \(B_m(E)\) when \(E\) is the unit disk; in this case, \(B_m(E) < M_E\) for all \(m\). Here the authors determine the \(B_m(E)\) when \(E\) has positive logarithmic capacity. It can happen that \(B_m(E) = M_E\): this depends on the size of the set of maximizing points for the function \(d_E(z) = \max_{t \in E} |z - t|\). However, they show that if \(E\) is bounded by finitely many closed \(C^1\)-smooth Jordan curves, then \(B_m(E) < M_E\) for all \(m\).

MSC:

30C10 Polynomials and rational functions of one complex variable
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