The Turán and Laguerre inequalities for quasi-polynomial-like functions

2023/10/20 by Krystian Gajdzica, Gajdzica, Krystian
Mathematics · #11P84 #Analytic and geometric function theory #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Number Theory (math.NT) #Primary 11P82 #Secondary 05A17

paper · pdf · doi:10.48550/arxiv.2310.13814

openalex publication_date 2023/10/20 · openalex created_date 2023/10/25 · openalex updated_date 2026/07/28

Abstract

This paper deals with both the higher order Turán inequalities and the Laguerre inequalities for quasi-polynomial-like functions -- that are expressions of the form f(n)=c_l(n)n^l+⋯+c_d(n)n^d+o(n^d), where d,l∈ℕ and d\leqslant l. A natural example of such a function is the A-partition function p_A(n), which enumerates the number of partitions of n with parts in the fixed finite multiset A=\a₁,a₂,…,a_k\ of positive integers. For an arbitrary positive integer d, we present efficient criteria for both the order d Turán inequality and the dth Laguarre inequality for quasi-polynomial-like functions. In particular, we apply these results to deduce non-trivial analogues for p_A(n).

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