We use HII starburst galaxy apparent magnitude measurements to constrain cosmological parameters in six cosmological models. A joint analysis of HII galaxy, quasar angular size, baryon acoustic oscillations peak length scale, and Hubble parameter measurements result in relatively model-independent and restrictive estimates of the current values of the non-relativistic matter density parameter $Omega_{rm m_0}$ and the Hubble constant $H_0$. These estimates favor a 2.0$sigma$ to 3.4$sigma$ (depending on cosmological model) lower $H_0$ than what is measured from the local expansion rate. The combined data are consistent with dark energy being a cosmological constant and with flat spatial hypersurfaces, but do not strongly rule out mild dark energy dynamics or slightly non-flat spatial geometries.
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