  We use the HII galaxies $L - \sigma$ relation and the resulting Hubble
expansion cosmological probe of a sample of just 25 high-$z$ (up to $z \sim
2.33$) HII galaxies, in a joint likelihood analysis with other well tested
cosmological probes (CMB, BAOs) in an attempt to constrain the dark energy
equation of state (EoS). The constraints, although still weak, are in excellent
agreement with those of a similar joint analysis using the well established
SNIa Hubble expansion probe. Interestingly, even with the current small number
of available high redshift HII galaxies, the HII/BAO/CMB joint analysis gives a
13% improvement of the quintessence dark energy cosmological constraints
compared to the BAO/CMB joint analysis. We have further performed extensive
Monte Carlo simulations, with a realistic redshift sampling, to explore the
extent to which the use of the $L - \sigma$ relation, observed in HII galaxies,
can constrain effectively the parameter space of the dark energy EoS. The
simulations predict substantial improvement in the constraints when increasing
the sample of high-$z$ HII galaxies to 500, a goal that can be achieved in
reasonable observing times with existing large telescopes and state-of-the-art
instrumentation.
