  The QED$_3$-Gross-Neveu model is a (2+1)-dimensional U(1) gauge theory
involving Dirac fermions and a critical real scalar field. This theory has
recently been argued to represent a dual description of the deconfined quantum
critical point between Neel and valence bond solid orders in frustrated quantum
magnets. We study the critical behavior of the QED$_3$-Gross-Neveu model by
means of an epsilon expansion around the upper critical space-time dimension of
$D_c^+=4$ up to the three-loop order. Estimates for critical exponents in 2+1
dimensions are obtained by evaluating the different Pade approximants of their
series expansion in epsilon. We find that these estimates, within the spread of
the Pade approximants, satisfy a nontrivial scaling relation which follows from
the emergent SO(5) symmetry implied by the duality conjecture. We also
construct explicit evidence for the equivalence between the QED$_3$-Gross-Neveu
model and a corresponding critical four-fermion gauge theory that was
previously studied within the 1/N expansion in space-time dimensions 2<D<4.
