  Background: Ultra-low $Q$-value $\beta$-decays are interesting processes to
study with potential applications to nuclear $\beta$-decay theory and neutrino
physics. While a number of potential ultra-low $Q$-value $\beta$-decay
candidates exist, improved mass measurements are necessary to determine which
are energetically allowed. Method: Penning trap mass spectrometry was used to
determine the atomic mass of $^{89}$Y and $^{139}$La, from which $\beta$-decay
$Q$-values for $^{89}$Sr and $^{139}$Ba were obtained to determine if there
could be an ultra-low $Q$-value decay branch in the $\beta$-decay of $^{89}$Sr
$\rightarrow$ $^{89}$Y or $^{139}$Ba $\rightarrow$ $^{139}$La. Results: The
$^{89}$Sr $\rightarrow$ $^{89}$Y and $^{139}$Ba $\rightarrow$ $^{139}$La
$\beta$-decay $Q$-values were measured to be $Q_{\rm{Sr}}$ = 1502.20(0.35) keV
and $Q_{\rm{Ba}}$ = 2308.37(68) keV. These were compared to energies of excited
states in $^{89}$Y at 1507.4(1) keV, and in $^{139}$La at 2310(19) keV and
2313(1) keV to determine $Q$-values of -5.20(37) keV for the potential
ultra-low $\beta$-decay branch of $^{89}$Sr and -1.6(19.0) keV and -4.6(1.2)
keV for those of $^{139}$Ba. Conclusion: The potential ultra-low $Q$-value
decay branch of $^{89}$Sr to the $^{89}$Y (3/2$^-$, 1507.4 keV) state is
energetically forbidden and has been ruled out. The potential ultra-low
$Q$-value decay branch of $^{139}$Ba to the 2313 keV state in $^{139}$La with
unknown J$^{\pi}$ has also been ruled out at the 4$\sigma$ level, while more
precise energy level data is needed for the $^{139}$La (1/2$^+$, 2310 keV)
state to determine if an ultra-low $Q$-value $\beta$-decay branch to this state
is energetically allowed.
