Inequalities for the sequences with unified form

Abstract We consider a class of sequence $c(n)$ with the following asymptotic form: $$ \begin{align*}c(n) \sim \frac C{n^\kappa} \exp\left(\sum_{\lambda\in\mathcal S}A_\lambda n^\lambda \right) \sum_{\mu\in\mathcal T} \frac{\beta_\mu}{n^\mu} \qquad (n \to \infty).\end{align*} $$ We give criteria for the Turán inequality of any order, the double Turán inequality, and the Laguerre inequality of any order of $c(n)$ for sufficiently large n . We also give the companion inequalities for the Turán inequality and the Laguerre inequality of any order for $c(n)$ . As applications, we will show that the numbers of commuting $\ell $ -tuples in $S_n$ , the partition without sequence, the plane partition, the partition into k -gonal numbers, the finite-dimensional representations of groups $\mathfrak {su}(3)$ and $\mathfrak {so}(5),$ and the coefficients of infinite product generating functions asymptotically satisfy these inequalities. Some of them settle open problems proposed by Bringmann, Franke, and Heim.