Functional Determinant on Pseudo-Einstein 3-manifolds

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,\theta )$, we establish an expression for the difference of determinants of the Paneitz type operators $A_{\theta }$, related to the problem of prescribing the $Q^{\mathrm{\prime }}$-curvature, under the conformal change $\theta \mapsto e^w\theta$ with $w\in \mathrm{\P }$ the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four dimensional Riemannian manifolds established in \cite{BO2}. We also provide an existence result of maximizers for the scaling invariant functional determinant as in \cite{CY}.