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 Manko and V V Dodonov and the Gaussianity-bounded one [Phys. Rev. A 86, 030102R (2012)].