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

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2024年4月21日

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Krystian Gajdzica

摘要

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 $f(n)=c_l(n)n^l+\cdots +c_d(n)n^d+o(n^d)$ $f (n) = c_{l} (n) n^{l} + \dots + c_{d} (n) n^{d} + o (n^{d})$ , where $d,l\in \mathbb {N}$ $d, l \in N$ and $d\leqslant l$ $d ⩽ l$ . A natural example of such a function is the A-partition function $p_{A}(n)$ $p_{A} (n)$ , which enumerates the number of partitions of n with parts in the fixed finite multiset $A=\{a_1,a_2,\ldots ,a_k\}$ $A = {a_{1}, a_{2}, \dots, a_{k}}$ 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 $p_A(n)$ $p_{A} (n)$ .