ABSTRACT

In two articles in 1978 [ 182] and 1981 [ 183] the authors extended these

enriched this new field. In this monograph we will describe some of these

new results and their applications.

A differential inequality of the form (2.1-1) does not have a direct analog for complex-valued functions, i.e. we cannot merely replace the real-valued

function f(t) in (2.1-1) with a complex-valued function f(z). However, the

first inclusion relation of (2.1-2) does have a natural complex analog such as

where

analogously to (2.1-2) we can ask if there is a "smallest" set A c C such

that

There are two other problems associated with (2.1-3). Given Q and A, do

these three problems, for this very elementary example, we see some of the

ideas used in developing the theory of differential subordinations. These problems will be generalized and their solutions will be described in this text.

Let Q and A be any sets in C, let p be analytic in the unit disk U with