In this article, the bounds on the Q-factor, a quantity inversely proportional to bandwidth, are derived and investigated for narrowband phased array antennas. Arrays in free space and above a ground plane are considered. The Q-factor bound is determined by solving a minimization problem over the electric current density. The support of these current densities is on an element-enclosing region, and the bound holds for lossless antenna elements enclosed in this region. The Q-factor minimization problem is formulated as a quadratically constrained quadratic optimization problem that is solved either by a semi-definite relaxation or an eigenvalue-based method. We illustrate numerically how these bounds can be used to determine tradeoff relations between the Q-factor and other design specifications: element form factor, size, efficiency, scanning capabilities, and polarization purity.