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)=cₗ(n)nˡ+⋯ +cd(n)nᵈ+o(nᵈ) f ( n ) = c l ( n ) n l + ⋯ + c d ( n ) n d + o ( n d ) , where d,l∈ N d , l ∈ N and d l d ⩽ l . A natural example of such a function is the A -partition function pA(n) p A ( n ) , which enumerates the number of partitions of n with parts in the fixed finite multiset A=₁,a₂,… ,aₖ\ A = { a 1 , a 2 , … , 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 pA(n) p A ( n ) .