Bounded uncertainty relations provide the minimum value of the uncertainty
assuming some additional information on the state. We derive analytically an
uncertainty relation bounded by a pair of constraints, those of purity and
Gaussianity. In a limiting case this uncertainty relation reproduces the
purity-bounded derived by V I Man'ko and V V Dodonov and the
Gaussianity-bounded one [Phys. Rev. A 86, 030102R (2012)].Comment: Major changes in the presentation of the results but also in the
proofs which have become more compact. Submitted to Journal of Physics
