The authors introduce a new concept to study multi-valued solutions of differential equations.
Let and be two domains in and be a meromorphic function in . Denote by the number of domains which maps conformally and one-to-one onto . The number coincides with the number of simple islands of covering surfaces over the domain in Ahlfor’s theory of covering surfaces and can also be considered as a characteristics of the function in an arbitrary domain describing the global behavior of functions .
Finally, the authors derive an estimate on this number for a meromorphic solution of an equation of the form
where , , is an analytic function in the variables and .