  In this article, we prove the existence and regularity of a smooth solution
for a supersonic-sonic patch arising in a modified Frankl problem in the study
of three-dimensional axisymmetric steady isentropic relativistic transonic
flows over a symmetric airfoil. We consider a general convex equation of state
which makes this problem complicated as well as interesting in the context of
the general theory for transonic flows. Such type of patches appear in many
transonic flows over an airfoil and flow near the nozzle throat. Here the main
difficulty is the coupling of nonhomogeneous terms due to axisymmetry and the
sonic degeneracy for the relativistic flow. However, using the well-received
characteristic decompositions of angle variables and a partial hodograph
transformation we prove the existence and regularity of solution in the partial
hodograph plane first. Further, by using an inverse transformation we construct
a smooth solution in the physical plane and discuss the uniform regularity of
solution up to the associated sonic curve. Finally, we also discuss the uniform
regularity of the sonic curve.
