Consequences of $mu$-$tau$ reflection symmetry for $3+1$ neutrino mixing

We investigate the consequences of $mu-tau$ reflection symmetry in presence of a light sterile neutrino for the $3+1$ neutrino mixing scheme. We discuss the implications of total $mu-tau$ reflection symmetry as well partial $mu-tau$ reflection symmetry. For the total $mu-tau$ reflection symmetry we find values of $theta_{23}$ and $delta$ remains confined near $pi/4$ and $pm pi/2$ respectively. The current allowed region for $theta_{23}$ and $delta$ in case of inverted hierarchy lies outside the area preferred by the total $mu-tau$ reflection symmetry. However, interesting predictions on the neutrino mixing angles and Dirac CP violating phases are obtained considering partial $mu-tau$ reflection symmetry. We obtain predictive correlations between the neutrino mixing angle $theta_{23}$ and Dirac CP phase $delta$ and study the testability of these correlations at the future long baseline experiment DUNE. We find that while the imposition of $mu-tau$ reflection symmetry in the first column admit both normal and inverted neutrino mass hierarchy, demanding $mu-tau$ reflection symmetry for the second column excludes the inverted hierarchy. Interestingly, the sterile mixing angle $theta_{34}$ gets tightly constrained considering the $mu-tau$ reflection symmetry in the fourth column. We also study consequences of $mu-tau$ reflection symmetry for the Majorana phases and neutrinoless double beta decay.