NDLI: On the Behavior of Gegenbauer Polynomials in the Complex Plane

Content Provider	Springer Nature Link
Author	Alexandrov, Alexander Nikolov, Ge
Copyright Year	2012
Abstract	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.
Ending Page	428
Page Count	14
Starting Page	415
File Format	PDF
ISSN	14226383
e-ISSN	14209012
Journal	Results in Mathematics
Issue Number	3
Volume Number	62
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2012-08-19
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Inequalities in approximation (Bernstein, Jackson, Nikol'skiÄ-type inequalities) Gegenbauer polynomials Jensen inequalities Markov inequality Inequalities in the complex domain Polynomials Mathematics Duffin and Schaeffer type inequalities
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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