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