Landau’s inequality via Hadamard’s. (English) Zbl 0827.30004

For a complex polynomial \(P(z) = a_d \prod^d_{i = 1} (z - \alpha_i) = a_d z^d + \cdots + a_0\) Mahler’s measure \(M(P)\) is defined by \(M(P) = |a_d|\prod_{i = 1}^d \max \{1, |\alpha_i|\}\). It has been shown by E. Landau [Bull. Soc. Math. Fr. 33, 251-261 (1905; JFM 36.0467.01)=Collected Works 2, 180- 190 (Thales (1905)] that \(M(P)\) does not exceed \((|a_0 |^2 + \cdots + |a_d |^2)^{1/2}\). Other proofs of this inequality have been given by K. Mahler [Mathematika, London 7, 98-100 (1960; Zbl 0099.25003)], J. V. Gonçalves [Univ. Lisboa Rev. Fac. Ci., II. Ser. A 1, 167-171 (1950; Zbl 0039.01205)] and M. Mignotte [Math. Comput. 28, 1153-1157 (1974; Zbl 0299.12101)].
The authors provide two new short proofs, both utilizing Hadamard’s inequality for determinants. They also obtain a simple proof of Jensen’s formula for polynomials (which in this case is actually due to C. G. J. Jacobi [J. Reine Angew. Math. 2, 1-8 (1827; JFM 36.0467.01)]).
Remark of the reviewer: The title of Landau’s paper is misspelled in the references: Petrovic should be replaced by Petrovich.


30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
12D99 Real and complex fields
12D05 Polynomials in real and complex fields: factorization
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