Conditions for Simple Bounds on Elementary-Excitation Observables Given a Single Measured Observable

Bounds are found for the values of an observable which are consistent with the measured value of a second observable and satisfy the conditions that $f_0\mathrm{}\left(e\right)\mathrm{}$, $f_1\mathrm{}\left(e\right)\mathrm{}$, and $Y\left(e\right)=\frac{f_0^{\prime }\mathrm{}\left(e\right)\mathrm{}}{f_1^{\prime }\mathrm{}\left(e\right)\mathrm{}}$ are well behaved and have nonzero slope on the open interval $e_L \mathrm{to} e_U$. $f_0\mathrm{}\left(e\right)\mathrm{}$ and $f_1\mathrm{}\left(e\right)\mathrm{}$ are, respectively, the contributions to the undetermined and determined observable from an elementary excitation state of energy $e$.