The Weighted Davenport constant of a group and a related extremal problem II

For a finite abelian group $G$ with $\exp(G)=n$ and an integer $k\ge 2$, Balachandran and Mazumdar \cite{BM} introduced the extremal function $\fD_G(k)$ which is defined to be $\min\{|A|: \emptyset \neq A\subseteq[1,n-1]\textrm{\ with\ }D_A(G)\le k\}$ (and $\infty$ if there is no such $A$), where $D_A(G)$ denotes the $A$-weighted Davenport constant of the group $G$. Denoting $\fD_G(k)$ by $\fD(p,k)$ when $G=\bF_p$ (for $p$ prime), it is known (\cite{BM}) that $p^{1/k}-1\le \fD(p,k)\le O_k(p\log p)^{1/k}$ holds for each $k\ge 2$ and $p$ sufficiently large, and that for $k=2,4$, we have the sharper bound $\fD(p,k)\le O(p^{1/k})$. It was furthermore conjectured that $\fD(p,k)=\Theta(p^{1/k})$. In this short paper we prove that $\fD(p,k)\le 4^{k^2}p^{1/k}$ for sufficiently large primes $p$.