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We explore cosmological constraints on the sum of the three active neutrinomasses $M_{\nu}$ in the context of dynamical dark energy (DDE) models withequation of state (EoS) parametrized as a function of redshift $z$ by$w(z)=w_0+w_a\,z/(1+z)$, and satisfying $w(z)\geq-1$ for all $z$. We perform aBayesian analysis and show that, within these models, the bounds on $M_{\nu}$\textit{do not degrade} with respect to those obtained in the $\Lambda$CDMcase; in fact the bounds are slightly tighter, despite the enlarged parameterspace. We explain our results based on the observation that, for fixed choicesof $w_0\,,w_a$ such that $w(z)\geq-1$ (but not $w=-1$ for all $z$), the upperlimit on $M_{\nu}$ is tighter than the $\Lambda$CDM limit because of thewell-known degeneracy between $w$ and $M_{\nu}$. The Bayesian analysis we havecarried out then integrates over the possible values of $w_0$-$w_a$ such that$w(z)\geq-1$, all of which correspond to tighter limits on $M_{\nu}$ than the$\Lambda$CDM limit. We find a 95\% confidence level (C.L.) upper bound of$M_{\nu}<0.13\,\mathrm{eV}$. This bound can be compared with$M_{\nu}<0.16\,\mathrm{eV}$ at 95\%~C.L., obtained within the $\Lambda$CDMmodel, and $M_{\nu}<0.41\,\mathrm{eV}$ at 95\%~C.L., obtained in a DDE modelwith arbitrary EoS (which allows values of $w < -1$). Contrary to the resultsderived for DDE models with arbitrary EoS, we find that a dark energy componentwith $w(z)\geq-1$ is unable to alleviate the tension between high-redshiftobservables and direct measurements of the Hubble constant $H_0$. Finally, inlight of the results of this analysis, we also discuss the implications for DDEmodels of a possible determination of the neutrino mass hierarchy by laboratorysearches. (abstract abridged)

Constraints on the sum of the neutrino masses in dynamical dark energy models with $w(z) \geq -1$ are tighter than those obtained in $\Lambda$CDM