Firstly, we begin by proving the subordination theorem involving the integral operator defined by (1.1). For this purpose, we need the following class of admissible functions.
Definition 2.1 Let Ω be a set in ℂ, , and . The class of admissible functions consists of those functions that satisfy the admissibility condition
whenever
and
for , , and .
Theorem 2.1 Let . If satisfies
(2.1)
then
Proof Define the function in by
(2.2)
From (2.2) with the relation (1.3), we get
(2.3)
Further computations show that
(2.4)
Define the transformation from to ℂ by
(2.5)
Let
(2.6)
Using equations (2.2), (2.3) and (2.4), from (2.6), we obtain
(2.7)
Hence, (2.1) becomes
Note that
and so the admissibility condition for is equivalent to the admissibility condition for . Therefore, by Theorem 1.1, or
which evidently completes the proof of Theorem 2.1. □
If is a simply connected domain, then for some conformal mapping h of onto Ω. In this case, the class is written as . The following result is an immediate consequence of Theorem 2.1.
Theorem 2.2 Let . If satisfies
(2.8)
then
Our next result is an extension of Theorem 2.1 to the case where the behavior of q on is not known.
Corollary 2.3 Let and q be univalent in with . Let for some where . If satisfies
then
Proof Theorem 2.1 yields . The result is now deduced from . □
Theorem 2.4 Let h and q be univalent in with and set and . Let satisfy one of the following conditions:
-
(1)
for some , or
-
(2)
there exists such that for all .
If satisfies (2.8), then
Proof The proof is similar to that of [[1], Theorem 2.3d] and so is omitted. □
The next theorem yields the best dominant of the differential subordination (2.7).
Theorem 2.5 Let h be univalent in and let . Suppose that the differential equation
(2.9)
has a solution q with and satisfies one of the following conditions:
-
(1)
and ,
-
(2)
q is univalent in and for some , or
-
(3)
q is univalent in and there exists such that for all .
If satisfies (2.8) and
is analytic in , then
and q is the best dominant.
Proof Following the same arguments as in [[1], Theorem 2.3e], we deduce that q is a dominant from Theorem 2.2 and Theorem 2.4. Since q satisfies (2.9), it is also a solution of (2.8) and therefore q will be dominated by all dominants. Hence, q is the best dominant. □
In the particular case , , and in view of Definition 2.1, the class of admissible functions , denoted by , is described below.
Definition 2.2 Let Ω be a set in ℂ, , and . The class of admissible functions consists of those functions such that
(2.10)
whenever , , , and .
Corollary 2.6 Let . If satisfies
then
In the special case , the class is simply denoted by .
Corollary 2.7 Let . If satisfies
then
Corollary 2.8 Let , and let be an analytic function in with for . If satisfies
then
Proof This follows from Corollary 2.6 by taking and , where . To use Corollary 2.6, we need to show that , that is, the admissible condition (2.10) is satisfied. This follows since
for , , , and . Hence, by Corollary 2.6, we deduce the required results. □