Inequalities related to the coefficients of the $j$-function

TL;DR

This paper establishes bounds and inequalities for the Fourier coefficients of the $j$-function, extending beyond their known asymptotic behavior to provide more precise estimates.

Contribution

It provides new upper and lower bounds for the Fourier coefficients of the $j$-function, enabling the derivation of related inequalities.

Findings

Derived explicit bounds for $c(n)$

Established inequalities involving $c(n)$

Extended understanding of $j$-function coefficients

Abstract

In recent years, the log-concavity or log-convexity of combinatorial sequences and their root sequences, higher order Tur{\'a}n inequalities, and Laguerre inequalities of order two have been widely studied. However, the research of the Fourier coefficient $c(n)$ of the $j$-function is limited to its asymptotic form. In this paper, we give the appropriate upper and lower bounds of $c(n)$ to establish the inequalities associated with it.