On the Behavior of Gegenbauer Polynomials in the Complex Plane

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• 作者：Geno Nikolov (1)
  Alexander Alexandrov (1)
  
• 关键词：Primary 30C10 ; 30A10 ; Secondary 33C45 ; 41A17 ; Gegenbauer polynomials ; Jensen inequalities ; Markov inequality ; Duffin and Schaeffer type inequalities
• 刊名：Results in Mathematics
• 出版年：2012
• 出版时间：4 - December 2012
• 年：2012
• 卷：62
• 期：3
• 页码：415-428
• 全文大小：236KB
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• 作者单位：Geno Nikolov (1)
  Alexander Alexandrov (1)
  
  1. Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1164, Sofia, Bulgaria
  
• ISSN：1420-9012

文摘

It is well-known that the squared modulus of every function f from the Laguerre–Polya class ${\mathcal{L}-\mathcal{P}}$ of entire functions obeys a MacLaurin series representation $$|f(x+i y)|^2=\sum_{k=0}^{\infty} L_k(f;x)\,y^{2k}, \quad x,y\in\mathbb{R}$$ , which reduces to a finite sum when f is a polynomial having only real zeros. The coefficients {L k } are representable as non-linear differential operators acting on f, and by a classical result of Jensen L k (f;x)??0 for ${f\in \mathcal{L}-\mathcal{P}}$ and ${x\in \mathbb{R}}$ . Here, we prove a conjecture formulated by the first-named author in 2005, which states that for ${f=P_n^{(\lambda)} }$ , the n-th Gegenbauer polynomial, the functions ${\{L_k(f;x)\}_{k=1}^{n}}$ are monotone decreasing on the negative semi-axis and monotone increasing on the positive semi-axis. This result pertains to certain polynomial inequalities in the spirit of the celebrated refinement of Markov’s inequality, found by R. J. Duffin and A. C. Schaeffer in 1941.