To date, it is not known whether Davis' inequality holds for two- parameter martingales M, i.e. whether the ``pure'' norms induced by the supremum of the modulus of M and the square root of the sum of squared two-parameter increments are equivalent. There are two relevant ``mixed'' norms associated with quantities described in the following way. Fix one parameter. Take the square root of the sum of squared increments of M as a martingale in the other direction. Then take the supremum of the resulting submartingale over the previously fixed parameter. We show by establishing inequalities of the Davis type, that these are equivalent to the pure square sum norm and extend them to the continuous parameter case and to more general Orlicz norms.