  The functional equations of the Riemann zeta function, the Hurwitz zeta
function, and the Lerch zeta function have been well known for a long time and
there are great importance when studying these zeta-functions. For example,
fundamental properties of the upper bounds, the distribution of zeros, the
zero-free regions in the Riemann zeta function start from functional equations.
  In this paper, we prove a functional equations of the Barnes double
zeta-function $ \zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty
\sum_{n=0}^\infty (\alpha+vm+wn)^{-s} $. Also, applying this functional
equation and the Phragm\'en-Lindel\"of convexity principle, we obtain some
upper bounds for $ \zeta_2(\sigma + it, \alpha ; v, w) \ (0\leq \sigma \leq 2)
$ with respect to $ t $ as $ t \rightarrow \infty $.
