  Heisenberg's uncertainty principle is formulated for a set of generalized
measurements within the framework of majorization theory, resulting in a
partial uncertainty order on probability vectors that is stronger than those
based on quasi-entropic measures. The theorem that results from this
formulation guarantees that the uncertainty of the results of a set of
generalized measurements without a common eigenstate has an inviolable lower
bound which depends on the measurement set but not the state. A corollary to
this theorem yields a parallel formulation of the uncertainty principle for
generalized measurements based on quasi-entropic measures. Optimal majorization
bounds for two and three mutually unbiased bases in two dimensions are
calculated. Similarly, the leading term of the majorization bound for position
and momentum measurements is calculated which provides a strong statement of
Heisenberg's uncertainty principle in direct operational terms. Another theorem
provides a majorization condition for the least uncertain generalized
measurement of a given state with interesting physical implications.
