  New sum and product uncertainty relations, containing variances of three or
four observables, but not containing explicitly their covariances, are derived.
One of consequences is the new inequality, giving a nonzero lower bound for the
product of two variances in the case of zero mean value of the commutator
between the related operators. Moreover, explicit examples show that in some
cases this new bound can be better than the known Robertson--Schr\"odinger one.
