[Paper Review] Discrete analogues of the Laguerre inequalities and a conjecture of I. Krasikov

George Csordás, Matthew Chasse|arXiv (Cornell University)|May 27, 2010

Mathematical functions and polynomials17 references3 citations

TL;DR

This paper proves I. Krasikov's conjecture on discrete Laguerre-type inequalities for polynomials with real zeros and mesh size at least 1, establishing that a finite difference version of the classical Laguerre inequality holds. The result is extended to transcendental entire functions in the Laguerre-Pólya class of order less than 2 or minimal type of order 2, under a similar mesh condition, using uniform convergence of polynomial sequences and Hadamard factorization.

ABSTRACT

A conjecture of I. Krasikov is proved. Several discrete analogues of classical polynomial inequalities are derived, along with results which allow extensions to a class of transcendental entire functions in the Laguerre-Pólya class.

Motivation & Objective

• To prove I. Krasikov's conjecture on a discrete analogue of the Laguerre inequality for polynomials with real zeros and minimal mesh size 1.
• To extend the discrete inequality to a class of transcendental entire functions in the Laguerre-Pólya class.
• To establish conditions under which the finite difference inequality remains non-negative for functions with zero spacing constraints.
• To explore the connection between zero spacing, convergence of polynomial sequences, and the validity of discrete inequalities.
• To formulate and motivate further conjectures on the extension of the inequality to broader classes of entire functions.

Proposed method

• Prove the conjecture using finite difference operators applied to polynomials with real zeros and mesh size μ(p) ≥ 1.
• Define the finite difference functional $ f_n(x,1,p) = (n-1)[p(x+1)-p(x-1)]^2 - 4n p(x)[p(x+1)-2p(x)+p(x-1)] $, showing it is non-negative under the mesh condition.
• Use uniform convergence of polynomial sequences with increasing zero spacing to approximate the exponential function and extend results to transcendental functions.
• Apply the Hadamard factorization theorem to represent functions in the Laguerre-Pólya class and approximate them by partial products.
• Leverage Montel’s theorem and normal families to extract convergent subsequences of polynomials with controlled zero spacing.
• Use limit arguments to pass from polynomial inequalities to the corresponding inequalities for entire functions in the Laguerre-Pólya class.

Experimental results

Research questions

• RQ1 Does the discrete Laguerre-type inequality proposed by I. Krasikov hold for all polynomials with real zeros and mesh size at least 1?
• RQ2 Can the discrete inequality be extended from polynomials to transcendental entire functions in the Laguerre-Pólya class?
• RQ3 What conditions on zero spacing (mesh size) ensure the validity of the finite difference inequality for entire functions?
• RQ4 Is the inequality $ f_{ ext{infty}}(x,1, varphi) \geq 0 $ valid for all $ \varphi \in \mathcal{L}\text{-}\mathcal{P} $ with $ \mu_{\infty}(\varphi) \geq 1 $?
• RQ5 Can the convergence of polynomial sequences with controlled zero spacing be used to extend classical inequalities to the entire function setting?

Key findings

• The conjecture of I. Krasikov is proven: for any polynomial $ p(x) \in \mathcal{L}\text{-}\mathcal{P}_n $ with $ \mu(p) \geq 1 $, the inequality $ (n-1)[p(x+1)-p(x-1)]^2 - 4n p(x)[p(x+1)-2p(x)+p(x-1)] \geq 0 $ holds for all real $ x $.
• The inequality is extended to transcendental entire functions $ \varphi \in \mathcal{L}\text{-}\mathcal{P} $ of order $ \rho < 2 $, or of minimal type with $ \rho = 2 $, provided the infinite mesh $ \mu_{\infty}(\varphi) \geq 1 $.
• A sequence of polynomials with increasing zero spacing converges uniformly on compact subsets of $ \mathbb{C} $ to $ e^x $, enabling the extension of the inequality to the exponential function and beyond.
• The limit of finite difference functionals applied to approximating polynomials converges to the corresponding functional for the limit function, preserving non-negativity.
• The proof relies on the uniform boundedness of logarithmic derivatives and normal families, ensuring convergence of subsequences to entire functions in the Laguerre-Pólya class.
• A conjecture is formulated that the inequality holds for all $ \varphi \in \mathcal{L}\text{-}\mathcal{P} $ with $ \mu_{\infty}(\varphi) \geq 1 $, suggesting a broader validity beyond the current conditions.