This paper addresses the problem of testing simple hypotheses about the mean of
a bivariate normal distribution with identity covariance matrix against restricted
alternatives. The LRTs and their power functions for such types of hypotheses are
derived. Furthermore, through some elementary calculus, it is shown that the power
function of the LRT satisfies certain monotonicity and symmetry properties. We treat
two cases, the case of one-sided alternatives restricted to some closed convex cone,
and the case of two-sided alternatives restricted to a two-sided cone