  Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number}
is defined to be the minimum number of vertices $n$ such that any $k$-edge
coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$
or a monochromatic copy of $G$. In this paper, we obtain general upper and
lower bounds on the Gallai-Ramsey numbers for double stars $S(n,m)$, where
$S(n,m)$ is the graph obtained from the union of two stars $K_{1,n}$ and
$K_{1,m}$ by adding an edge between their centers. We also provide the sharp
result in some cases.
