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A sharp version of Mahler’s inequality for products of polynomials. (English) Zbl 0929.30005

The authors show that for any \(\alpha=(\alpha_1,\dots,\alpha_n)\in \mathbb R^n_+\) and any set \(\{a_j\in\mathbb C\), \(1\leq j\leq n\}\) one has \[ \prod_{j=1}^n\| | z-a_j| ^{\alpha_j}\| \leq(2^{d_n(\alpha)}/\rho_n(\alpha)) \| \prod_{j=1}^n| z-a_j| ^{\alpha_j}\| , \] where \(d_n(\alpha)=\alpha_1+\dots+\alpha_n\), which becomes equality iff \(\{a_j\), \(1\leq j\leq n\}\) is a solution of the minimization problem: \(\rho_n(\alpha):=\min_{| a_j| =1}\| \prod_{j=1}^n | z-a_j| ^{\alpha_j}\| .\) Here \(\| f\| :=\max_{| z| =1}| f(z)| \). It is well known that for \(\alpha^*= (1,\dots,1)\) we have \(\rho_n(\alpha^*)=2\) with \(z^n+1\) being the unique (up to a rotation) extremal polynomial for \(\rho_n(\alpha^*)\). Hence, if \(p_1,\dots,p_n\) are complex polynomials such that their product \(p\) is of degree \(n\), the authors obtain the inequality \[ \| p_1\| \cdots\| p_m\| \leq 2^{n-1}\| p\| , \] where the constant \(2^{n-1}\) is optimal, which is attained iff \(p(z)=z^n+\rho\), with \(| \rho| =1\) and \(m=n\). In particular, this improves a well-known inequality of K. Mahler [Mathematica, London 7, 98-100 (1960; Zbl 0099.25003)] established with the constant \(2^n\) and answers a question of Sarantopoulos [personal communication, 1996] who conjectured the constant \(2^{n-1}\) in Mahler’s inequality.

MSC:

30C10 Polynomials and rational functions of one complex variable
11C08 Polynomials in number theory

Citations:

Zbl 0099.25003
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