Differentia Subordinations for Non-analytic Functions

In paper [1], Petru T. Mocanu has obtained sufficient conditions for a function in the classes C1(U), respectively C2(U) to be univalent and to map U onto a domain which is starlike (with respect to origin), respectively convex. Those conditions are similar to those in the analytic case. In paper [2], Petru T. Mocanu has obtained sufficient conditions of univalency for complex functions in the class C1 which are also similar to those in the analytic case. Having those papers as inspiration, we have tried to introduce the notion of subordination for non-analytic functions of classes C1 and C2 following the classical theory of differential subordination for analytic functions introduced by S.S. Miller and P.T. Mocanu in papers [3] and [4] and developed in the book [5]. Let Ω be any set in the complex plane C, let p be a non-analytic function in the unit disc U, p ∈ C2(U) and let ψ(r, s, t; z) : C3×U → C. In article [6] we have considered the problem of determining properties of the function p, non-analytic in the unit disc U, such that p satisfies the differential subordination.