  Let $G$ be a finite abelian group. We show that its Davenport constant $D(G)$
satisfies $D(G)\leq \exp(G)+\frac{|G|}{\exp(G)}-1$, provided that
$\exp(G)\geq\sqrt{|G|}$, and $D(G)\leq 2\sqrt{|G|}-1$, if $\exp(G)<\sqrt{|G|}$.
This proves a conjecture by Balasubramanian and the first named author.
