Landau's inequality via Hadamard's
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 \textit{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 \textit{K. Mahler} [Mathematika, London 7, 98-100 (1960; Zbl 0099.25003)], \textit{J. V. Gonçalves} [Univ. Lisboa Rev. Fac. Ci., II. Ser. A 1, 167-171 (1950; Zbl 0039.01205)] and \textit{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 \textit{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.
scientific article; zbMATH DE number 762068