Nonlinear asymptotic mean value characterizations of holomorphic functions

We first prove a characterization of holomorphic functions in terms of a suitable mean value property. We then build up upon this result to obtain some (nonlinear) asymptotic characterizations for complex-valued solutions of certain nonlinear systems, which have to do with the classical Cauchy-Riemann equations. From these asymptotic characterizations, we derive suitable asymptotic mean value properties, which are used to construct appropriate (vectorial) dynamical programming principles. The aim is to construct approximation schemes for the so-called contact solutions, recently introduced by N. Kat- zourakis, of the nonlinear systems here considered.