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

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 $f(n)=c_l(n)n^l+\cdots +c_d(n)n^d+o(n^d)$, where $d,l\in \mathbb{N}$ and $d⩽l$. 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 $A=\{a_1,a_2,\dots ,a_k\}$ of positive integers. For an arbitrary positive integer $d$, we present efficient criteria for both the order $d$ Turán inequality and the $d$th Laguarre inequality for quasi-polynomial-like functions. In particular, we apply these results to deduce non-trivial analogues for $p_A(n)$.