New Zealand Journal of Mathematics

Bounds on the second Hankel determinant and Toeplitz determinants of logarithmic coefficients for a class of analytic functions

Varesha Sharma — 2026-05-27

In this paper, we establish sharp initial bounds for the second Hankel determinant and the second Toeplitz determinant of logarithmic coefficients for the class $\mathcal{R}_{car}$, which consists of functions $f$ that satisfy a specific subordination relationship with a function in the open unit disk $\mathbb{D}$.

The behaviour under iteration of a class of spatially-discretised quadratic maps

Jonathan Hoseana — 2026-05-27

Potentially drastic changes in the dynamical behaviour of iterated maps due to rounding errors in computer arithmetic have motivated the study of spatially-discretised maps, whereby the dynamical behaviour of a map is compared to that of a variant obtained by composing the map with a spatial discretisation operator, such as the floor function. In this paper, we study the dynamical behaviour of the spatially-discretised quadratic map $x\mapsto \left\lfloor \lambda x^2 \right\rfloor$, for all values of $\lambda\in\mathbb{R}$. Specifically, we prove that the map possesses at most three non-zero fixed points, and that every orbit of the map either diverges or becomes eventually constant at one of the existing fixed points.

A new proof of an identity concerning $5$-core partitions

Russelle Guadalupe — 2026-05-27

Let $a_5(n)$ be the number of partitions of $n$ that are $5$-cores. We provide a new elementary proof of an identity involving $a_5(n)$ due to Baruah and Berndt by employing $2$-dissection formulas and identities involving the Ramanujan's parameter $k(q)$ due to Cooper, Chern, and Tang.