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The Schr{\"o}dinger inequality is known to underlie the Kennard-Robertsoninequality, which is the standard expression of quantum uncertainty for theproduct of variances of two observables $A$ and $B$, in the sense that thelatter is derived from the former. In this paper we point out that, albeit moresubtle, there is yet another inequality which underlies the Schr{\"o}dingerinequality in the same sense. The key component of this observation is the useof the weak-value operator $A_{\rm w}(B)$ introduced in our previous works(named after Aharonov's weak value), which was shown to act as the proxyoperator for $A$ when $B$ is measured. The lower bound of our novel inequalitysupplements that of the Schr{\"o}dinger inequality by a term representing thediscord between $A_{\rm w}(B)$ and $A$. In addition, the decomposition of theSchr{\"o}dinger inequality, which was also obtained in our previous works bymaking use the weak-value operator, is examined more closely to analyze itsstructure and the minimal uncertainty states. Our results are exemplified withsome elementary spin 1 and 3/2 models as well as the familiar case of $A$ and$B$ being the position and momentum of a particle.

Uncertainty Relations of Variances in View of the Weak Value