Proof of a conjecture of Granath on optimal bounds of the Landau constants

Abstract

We study the asymptotic expansion for the Landau constants $G_n$, \begin{equation*} \pi G_{n}\sim \ln(16N)+\gamma+\sum^{\infty}_{k=1}\frac{\alpha_k}{N^k} ~~\mbox{as} ~ n\rightarrow\infty, \end{equation*} where $N=n+1$, and $\gamma$ is Euler's constant. We show that the signs of the coefficients $\alpha_{k}$ demonstrate a periodic behavior such that $(-1)^{\frac {l(l+1)} 2} \alpha_{l+1}< 0$ for all $l$. We further prove a conjecture of Granath which states that $(-1)^{\frac {l(l+1)} 2} \varepsilon_l(N)<0$ for $l=0,1,2,\cdots$ and $n=0,1,2,\cdots$, $\varepsilon_l(N)$ being the error due to truncation at the $l$-th order term. Consequently, we also obtain the sharp bounds up to arbitrary orders of the form \begin{equation*} \ln(16N)+\gamma+\sum_{k=1}^{p}\frac{\alpha_{k}}{N^{k}}<\pi G_{n}<\ln(16N)+\gamma+\sum_{k=1}^{q}\frac{\alpha_{k}}{N^{k}} \end{equation*} for all $n=0,1,2\cdots$, all $p=4s+1,\; 4s+2$ and $q=4m,\; 4m+3$, with $s=0,1,2,\cdots$ and $m=0, 1, 2,\cdots$.