This paper studies first the differential inequalities that make it possible
to build a global theory of pseudo-holomorphic functions in the case of one or
several complex variables. In the case of one complex dimension, we prove that
the differential inequalities describing pseudo-holomorphic functions can be
used to define a one-real-dimensional manifold (by the vanishing of a function
with nonzero gradient), which is here a 1-parameter family of plane curves. On
studying the associated envelopes, such a parameter can be eliminated by
solving two nonlinear partial differential equations. The classical
differential geometry of curves can be therefore exploited to get a novel
perspective on the equations describing the global theory of pseudo-holomorphic
functions.Comment: 25 page
