1. Introduction
We begin by introducing the important classes of functions considered in this article. Let
denote the class of functions analytic in the unit disk
, and let
A common problem in complex analysis is to determine the range of a function
from a differential inclusion or containment relation involving several of the derivatives of
p. Let
and
be sets in
, and
D be a differential operator such that
is an analytic function defined on
. A natural question is to ask what conditions on
D,
and
are needed so that
In this case, we have a
differential inclusion ⇒
function inclusion. There are many papers of this type that deal with special differential inclusions implying an inclusion for the image of the function
p. Similarly, there are many papers that deal with special differential containments and corresponding containments for the image of the function
p of the form
In this case, we have a
differential containment ⇒
function containment. Both sets of papers have resulted in many applications in complex analysis. See the monographs [
1,
2] for many results, applications and extensive bibliographies of results such as (1) and (2).
An open question to consider is to combine the two concepts in (1) and (2) and determine conditions on
D,
and
so that the mixed problem of differential inclusions implies a function containment of the form
In this case, we have a differential inclusion ⇒ function containment.
In a recent article [
3] the authors have extended results described in (1) to systems of two simultaneous second-order differential operators in two complex-valued functions. It is our intention to do the same with (3).
2. Definitions
We first indicate the forms of the two simultaneous second-order analytic differential operators that we will consider in this article.
Definition 1. Let and let be analytic in for . For and we define the second-order differential operators , for , byThroughout this article we will assume that is analytic in . Let
and
be sets in
and
be the second-order differential operators defined in (4), for
. The analogue of (3) that we will consider in this article deals with
two simultaneous differential inclusions implying function containments of the following form
In many cases, the containments on the right-sides of (5) can be written in terms of superordinations. We recall those definitions. Let f and F be members of . The function f is said to be subordinate to F (or F is superordinate to f), written , if there exists a function w analytic in U with and , such that . If, in addition, F is univalent, then if and only if and .
If
p and
q in (5) are univalent, and
and
are simply connected domains, then it is possible to rephrase the right-side of (5) in terms of superordination. If
is a simply connected domain containing the point
and
, then there is a conformal mapping
of
U onto
such that
, and if
is a simply connected domain containing the point
and
, then there is a conformal mapping
of
U onto
such that
. In this case, (5) can be rewritten as
We shall refer to the left sides of (5) and (6) as a System of Simultaneous Differential Inclusions (SSDI).
There are three basic pairs of elements in (5) and (6): the differential operators , the sets , and the sets (or functions ). If two of these elements are given, one would hope to find conditions on the third.
Our aim in this article is to solve a system of such simultaneous differential inclusions—analogous to solving a system of simultaneous differential equations in the real-plane. We restrict our development to systems consisting of two second-order differential inclusions in two unknown functions. The results presented here can be extended in a natural way to their corresponding third-order cases. We begin by introducing some important definitions.
Definition 2. Let be sets in and let be the analytic differential operators defined in (4) for . If and satisfy the SSDIthen p and q are called Solutions of the SSDI.
We will show that certain SSDI’s have solutions, and that these solutions have particular properties such as those given on the right-sides of (5) and (6).
Example 1. Let and and consider the SSDI given by It is easy to check that the univalent functions are Solutions of the SSDI given in (8).
Example 2. Let , the right half plane for . Let and and consider the SSDI given by It is clear that this SSDI has no solutions since there are no analytic functions p and q that can satisfy this system at .
Definition 3. The set of analytic functions as given in (6) is called aset of subordinants of the Solutions of the SSDI(6) or more simply aset of subordinantsif and for all p and q satisfying the left-side of (6). A set of subordinants that satisfies and for all subordinants of (6) is called aset of best subordinantsof (6). Please note that the set of best subordinants is unique up to a rotation ofU.
It is our intent to show that for certain types of SSDI we can obtain corresponding sets of subordinants and best subordinants of the system.
The analogue of the
best subordinants in Definition 3 for the SSDI (5) would be finding the
largest inclusion sets and
such that
3. Admissibility and a Fundamental Theorem
For the development of the theory we need to the consider the following class of univalent functions defined on the closed unit disc.
Definition 4. Let denote the set of functions g that are analytic and univalent on the set , whereand are such that Min for . The subclass of for which is denoted by . As a simple example of a member of the class , consider the function . For this function we have and Min for and hence .
The following lemma [
1] (p. 22) and [
4] has played a key role in many results involving the theory of differential subordinations and will also play a key role in this article.
Lemma 1 (Miller/Mocanu Lemma.).
Let with and , and let . If there exist points and (p) such that , and , then there exists an m, where such thatWe first define a special class of differential operators needed to solve a SSDI.
Definition 5. Let be analytic in , and with corresponding sets as given in Definition 4 for . Let be a subset of and let and be positive integers.The Set of Admissible Differential Operators consists of those pairs of differential operators , with as given in Definition 1, for , which satisfy the twoadmissibility conditions
when
,
,
,
,
and
.
when
,
,
,
,
and
.
In the special case when , we denote the set of operators by ψ. In the special case when and are simply connected domains and and are conformal maps of onto and respectively, we denote the set by .
In the case of first-order differential operators the admissibility conditions (9) and (10), with
for
simplify to
when
,
and
.
when
,
and
.
A closer look at conditions (9) and (10) [or (11) and (12)] indicate that there are different conditions on each of the operators and in the pair . An operator pair may not be in the Set of Admissible Operators as given by Definition 5, but the pair may be in the Set of Admissible Operators. We will see a case of this in Examples 3 and 4. In Example 3 we show that the pair is not in the Set of Admissible Operators, while in Example 4 we show that the pair is in the Set of Admissible Operators.
Example 3. Let , the right-half complex plane, and let and satisfy the SSDI We will show that this pair
with the functions
is not in the Set of Admissible Operators. Writing (13) in standard form we see that the functions
and
are of the form
We need to show that this pair of operators does not satisfy
In order for this last statement to be true, according to condition (9) of the first part of Definition 5, requires showing that
when
,
and
. This condition is equivalent to requiring that
Since this is not satisfied when , condition (14) cannot be satisfied and the pair of differential operators given in (13) is not in the Set of Admissible Operators.
We next interchange the differential operators in Example 3 to obtain an appropriate pair of operators.
Example 4. Let , the right-half complex plane, and let and satisfy the SSDI We will show that this pair
, with the functions
, is in the Set of Admissible Operators. Writing (15) in standard form we see that the functions
and
are of the form
We need to show that
. According to Definition 5, we need to show that
when
,
and
. This follows since
Hence is in the Set of Admissible Operators.
The following theorem is a foundation result for the theory of Second-Order SSDI.
Theorem 1. Let and be sets in , let and be analytic in , let , , and let . If and satisfy the SSDIthen and , and are a set of subordinants of (16). Proof. (a) For the first implication, if we assume
, then by Lemma 1 there exist points
,
and
such that
,
,
Using these results in (9) of Definition 5 we conclude
Since this contradicts the first part of (16) we must have .
(b) For the second implication, if we assume
, then by Lemma 1 there exist points
,
and
such that
,
,
Using these results in (10) of Definition 5 we obtain
Since this contradicts the second part of (16) we must have . □
As a result of the above theorem we can obtain subordinants of a SSDI of the form (16) by merely checking that the operators and satisfy the admissibility conditions (9) and (10) [or (11) and (12)] of Definition 5. This simple algebraic check yields subordinants of various SSDI that would be very difficult to obtain directly.
In the following two examples we use Theorem 1 to find subordinants of a SSDI.
Example 5. LetU, , and supposefor . It is our intention to prove that The differential operators in (17) are of the formwith We will use Theorem 1 to prove (18) with
. We only need to show that the pair of operators
, as given in (19), satisfy the admissibility conditions of Definition 5, namely that
. According to Definition 5 and (19) this requires showing that
when
and
. This simplifies to the conditions that
which are true because of the conditions on the four variables. Hence by Theorem 1 we conclude that
which proves (18).
Example 6. Let and be analytic in , with . Let , and supposefor . It is our intention to prove that The differential operators in (20) are of the form
with
We will use Theorem 1 to prove that if
p and
q satisfy (20) then they have subordinants
and
respectively given by
. We need to show that the pair
as given in (19) is in the Set of Admissible Operators, i.e., that
. We need to show that
when
,
and
. This follows since
Hence by Theorem 1 we conclude that
if
p and
q satisfy (20), then
and
.
The definition of the pair of operators , and their dependency on the conditions that and indicates that Theorem 1 depends very heavily on the functions and behaving very nicely on the boundary of U. If this is not the case or if their behavior on the boundary is unknown, it may still be possible to obtain a variant of the theorem by the following limiting process.
Theorem 2. Let and be analytic in , let be a subset of and let and be univalent on , with and . Let and for . Let for and suppose there exists such that for all . If and have the properties that and are analytic in andthen and . Proof. If we replace
z by
in
,
,
,
,
and
we obtain
for
. If we set
and
we obtain
for
. Since
we can apply Theorem 1 to conclude that
and
for
. If we now let
, we obtain the results
and
. □