A NEW BOUNDARY APPROACH FOR DIFFERENTIAL SUBORDINATIONS

In this paper we introduce and examine the differential subordination of the form p(z) + zp'(z)phi(p(z), zp'(z)) < h(z), z is an element of D := {z is an element of C : |z | < 1}, where h is a convex univalent function with 0 is an element of partial derivative h(D). Proofs of main results are based on the original lemmas on convex functions and functions starlike with respect to the boundary point, and offer a new approach in the theory. In particular, the above differential subordination leads to generalizations of the well-known Briot-Bouquet differential subordination. An application to the differential subordination related to the harmonic mean is demonstrated. The main results are applicable also to some classical subclasses of the class of normalized analytic functions in D. Relevant applications are indicated.