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Inequalities for the sequences with unified form
- Part of
-
- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 20 November 2025, pp. 1-20
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- Article
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We consider a class of sequence
$c(n)$ with the following asymptotic form: We give criteria for the Turán inequality of any order, the double Turán inequality, and the Laguerre inequality of any order of
$$ \begin{align*}c(n) \sim \frac C{n^\kappa} \exp\left(\sum_{\lambda\in\mathcal S}A_\lambda n^\lambda \right) \sum_{\mu\in\mathcal T} \frac{\beta_\mu}{n^\mu} \qquad (n \to \infty).\end{align*} $$
$c(n)$ 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.
