Distances between critical points and midpoints of zeros of hyperbolic polynomials

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• 作者：Dimitar K. Dimitrov ; Vladimir P. Kostov
• 关键词：Hyperbolic polynomial ; Strictly hyperbolic polynomial ; Zero ; Midpoint ; Critical point ; Entire function ; Laguerre– ; Pó ; lya class
• 刊名：Bulletin des Sciences Mathematiques
• 出版年：2010
• 出版时间：March 2010
• 年：2010
• 卷：134
• 期：2
• 页码：196-206
• 全文大小：154 K

文摘

Let p(x) be a polynomial of degree n with only real zeros x₁x₂x_n. Consider their midpoints z_k=(x_k+x_k+1)/2 and the zeros ξ₁ξ₂ξ_n−1 of p^′(z). Motivated by a question posed by D. Farmer and R. Rhoades, we compare the smallest and largest distances between consecutive ξ_k to the ones between consecutive z_k. The corresponding problem for zeros and critical points of entire functions of order one from the Laguerre–Pólya class is also discussed.