Approximate functional equation and upper bounds for the Barnes double zeta-function

As one of the asymptotic formulas of the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In this paper, we prove an approximate functional equation 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 approximate functional equation and the van der Corput method, we obtain upper bounds for $\zeta_2(1/2 + it, \alpha ; v, w)$ and $\zeta_2(3/2 + it, \alpha ; v, w)$ with respect to $t$ as $t \rightarrow \infty$.