Differential Subordinations for Nonanalytic Functions

Abstract

In the paper by Mocanu (1980), Mocanu has obtained sufficient conditions for a function in the classes , respectively, and to be univalent and to map onto a domain which is starlike (with respect to origin), respectively, and convex. Those conditions are similar to those in the analytic case. In the paper by Mocanu (1981), Mocanu has obtained sufficient conditions of univalency for complex functions in the class which are also similar to those in the analytic case. Having those papers as inspiration, we try to introduce the notion of subordination for nonanalytic functions of classes and following the classical theory of differential subordination for analytic functions introduced by Miller and Mocanu in their papers (1978 and 1981) and developed in their book (2000). Let be any set in the complex plane , let be a nonanalytic function in the unit disc , and let . In this paper, we consider the problem of determining properties of the function , nonanalytic in the unit disc , such that satisfies the differential subordination .

1. Introduction and Preliminaries

Let be the unit disc of the complex plane with and , and let be the class of analytic functions on . We also consider the class of nonanalytic functions of classes and , respectively.

Definition 1 (see [1, 2]). Let be a complex function defined in the unit disc .
For , we put
We say that the function belongs to the class or , if the functions and of the real variables and have continuous first-order or second-order partial derivatives in , respectively.
For , we denote where

Remark 2. (1) It is obvious that the linear differential operators and verify the usual rules of differential calculus. For instance,  (2) We also have the following useful formulas:  (3) If , then and .
Hence the constant functions for the operators and are the functions of the forms and , respectively.
(4) If , we have  Therefore,
(5) The Jacobian of the function is given by  If , for , then it is well-known that is a locally homeomorphic function preserving the orientation.
(6) Consider
Let  and let denote the class of analytic functions in .

2. Main Results

We will give the definition of subordination for nonanalytic functions of classes and following the classical theory of differential subordination introduced by S. S. Miller and P. T. Mocanu in papers [3, 4] and developed in their book [5].

Definition 3. Let and be two nonanalytic functions with and . One says that the function is subordinated to the function , denoted by or , if there exists a function nonanalytic in , with and , , such that ,.

Property 1. If , then and .

Proof. From Definition 3 we have and and using Lemma C (Section 4) we have , .
Then
For the case when the function is nonanalytic and injective with the following theorem holds.

Theorem 4. Let and be two nonanalytic functions in with and and injective in .
Then if and only if and .

Proof. The first part of Theorem 4 is given by Property 1.
Let us assume and .
We let and since is injective, there exists the inverse function . Then function is a nonanalytic function in with and verifies evidently the conditions , , . From , we obtain which gives that .
We next present the general form of the method of differential subordinations (or the method of admissible functions) for nonanalytic functions.
Let and be any sets in , let be nonanalytic in the unit disc , , , , and let , .
This theory deals with generalizations of the following implication:

Remark 5. If , then relation is equivalent to which is the general form of the implications from the classical theory of subordinations.
Related to condition , we state three problems that characterize the theory of differential subordinations for nonanalytic functions in the unit disc.

Problem 1. Given and , find conditions on such that holds. We call such a an admissible function.

Problem 2. Given and , find such that holds. Furthermore, find the “smallest” such .

Problem 3. Given and , find such that holds. Furthermore, find the “largest” such .
If either or in is a simply connected domain, then can be rewritten in terms of subordination.
Let be a simply connected domain containing the point and . Let , , be a nonanalytic function which is a conformal mapping of onto such that .
In this case, can be rewritten as
If is also a simply connected domain and , then there is a function , , nonanalytic, which is a conformal mapping of onto , such that . If, in addition, the function is a nonanalytic function in , , then can be rewritten as
This last result leads us to some important definitions.

Definition 6. Let be a nonanalytic function in , , and let be a nonanalytic function in , .
If is a nonanalytic function in , , and satisfies the (second-order) differential subordination then is called a solution of the differential subordination.
The nonanalytic function is called a dominant of the solutions of the differential subordination or more simply a dominant, if for all satisfying . A dominant that satisfies for all dominants of is said to be the best dominant of . Note that the best dominant is unique up to a rotation of .
Let be a set in and suppose that is replaced by
Although this is a differential inclusion, we will also refer to as a (second-order) differential subordination and use the same definitions of solution, dominant, and best dominant as given above.
In the case when and in are simply connected domains, we have seen that can be rewritten in terms of subordinations, such as given in . Using this and Definition 6, we can restate Problems 1–3 as follows.
Problem  1. Given and two nonanalytic and injective functions in , , , find a class of admissible functions such that holds.
Problem  2. Given the differential subordination , find a dominant . Moreover, find the best dominant.
Problem  3. Given , , nonanalytic function in , and dominant , , nonanalytic function in , find the largest class of nonanalytic and injective functions in , , such that holds.

Definition 7. One lets denote the set of functions nonanalytic and injective on with , where
Moreover, we assume that , for .
The set is called exception set.
The functions , , and are in ; hence, is a nonempty set.

3. The Class of Admissible Functions for Nonanalytic Functions

Definition 8. Let , let be a nonanalytic function, , , and let be a natural number, . One lets denote the class of functions which satisfy the condition where , , and .
The set is called the class of admissible functions and condition is called admissibility condition.

Remark 9. (1) If , the admissibility condition becomes  where , , and .
(2) If , the admissibility condition becomes where , , and .
(3) We let .

Remark 10. If , then the admissibility condition becomes the well-known admissibility condition for analytic functions , when , , and

4. Fundamental Lemmas

In order to prove the fundamental theorems, we must first prove some auxiliary theorems.

Lemma A. Let be a nonconstant continuous injective function inside a circle of center and arbitrary radius, with . Then, inside this circle, there exist points and such that

Proof. If function is continuous inside the circle , then function is also continuous inside this circle. Then there exists a neighbourhood such that is a bounded function. Then there exist two points and inside where function has a smallest value and a highest value, and , respectively, and . Then, for any , we have

Lemma B (maximum of modulus). Let be a nonconstant, continuous, injective function inside a circle . Then cannot attain its maximum value inside the circle, meaning

Proof. Assume that there exists a point inside such that . We have since if the function would be constant . If function is continuous inside , then from Lemma A we have that there exists a point inside such that . This is impossible since we have assumed that is the maximum value. Hence, belongs to the border of the circle.

Lemma C (the first part of Schwarz’s lemma). Let be a continuous function in and let the following conditions hold: and ; then , for all .

Proof. Let the function Since functions and are continuous in , we have that is continuous in .
From , we obtain that is continuous in .
Let with . Then function is continuous in . Using Lemma B, we have
If we let , we obtain which implies .

Remark 11. Lemmas A, B, and C hold also for some continuous, nonanalytic functions.
We assume that Lemma A and Lemma B also hold without the condition that function is injective, but this condition helped in giving the strict inequalities.

Lemma 12 (I. S. Jack, S. S. Miller, and P. T. Mocanu). Let with and let be a nonanalytic function in , continuous on with and (or ).
If then there exists a number , , such that(i),(ii).

Proof. Let , and . Then By differentiating (32) with respect to , we obtain
In [1], the author proved that if , then
Using (34) in (33), we obtain For , (35) becomes Since and is a point of maximum for , we have and .
From (36), we have and we deduce We show that . In order to prove this, we define the function Since functions and are continuous in , function is continuous in . From the definition of function , and hence function is continuous in .
Using Lemma B, we deduce Using Lemma C, we have , , since If we take , , in (42), we obtain and, from this, We calculate Since , we deduce that and hence . In order to prove inequality (ii), we differentiate relation (35) with respect to and we obtain Using and , (47) becomes For , (48) becomes Since and is a point of maximum for , we have and , and from (49) we have and we get Since , (51) becomes Relation (52) can be written in the form and we deduce and hence

Remark 13. If then we have(i);(ii).

Lemma 14 (S. S. Miller and P. T. Mocanu). Let with , and let be a nonanalytic function or with , .
If there exist points and such that and , where , then there exists a real number , , such that

Proof. Let the function For , we have From the hypothesis, we have that and, using this in (58), we obtain Using relation (59), we get From the maximum of modulus theorem, we have that On the other hand, , since . Using Lemma 12, we have Using (57), we obtain and we have If , and , then (65) becomes By differentiating relation (66) with respect to , we obtain Using , we have Using (68) in (67), we have Let Using (70) in (69), we obtain For , (71) becomes Using (62) in (72), we obtain By differentiating relation (69), with respect to , we obtain Using , we have which is equivalent to From (72), for and , we have From (59), we have and we obtain which is equivalent to Using (62), we have Using (63), we obtain Since , from (71), we obtain

Remark 15. If , then .

Lemma 16 (S. S. Miller and P. T. Mocanu). Let be a nonanalytic function, , with , let be a nonanalytic function with and , and let be a natural number with . If is not subordinate to , written as , then there exist points and and a number such that and(i);(ii).

Proof. From the hypothesis, we have that and since , we have .
Let .
Since , for , we have and . We get that there exists a point such that . From this, we obtain that there exists such that .
We conclude that there exist and such that and . From Lemma 14, we have which concludes the proof.

5. Fundamental Theorems

Theorem 17. Let , a nonanalytic function, with and . If function is a nonanalytic function, with , and verifies the condition then .

Proof. We assume that . From Lemma 16, we have that there exist points and and a number such that the following conditions are satisfied:(i);(ii).
Using these conditions with , , , and in Definition 8, we obtain
Since (85) contradicts (84), we have that the assumption made is false; hence , .

Remark 18. Theorem 17 also holds if (84) is replaced by for any nonanalytic function which maps onto .

Remark 19. In the hypothesis of Theorem 17, we have assumed that the behaviour of is known on the border of . If we do not know the behaviour of on the border of , then we may also prove that using the following limit procedure.

Theorem 20. Let , let be a nonanalytic function, , injective in , with , and let for some , where . If is a nonanalytic function, with , then implies that .

Proof. Since , we have that the function is injective on ; hence and . The function is an admissible function and so, from Theorem 17, we have that On the other hand, implies that From (89) and (90), we obtain which gives .

Remark 21. In the case when , is a simply connected domain and is a nonanalytic function, , and is injective in ; if we assume that , then, by letting , from Theorem 17, we obtain the following result.

Theorem 22. Let be a nonanalytic function, injective in , and with ; let be a nonanalytic function, injective in , , , and . If is a nonanalytic function, injective in , with , and function is a nonanalytic function, and , then implies that

This result can be extended for the case when the behaviour of on is not known.

Theorem 23. Let and be nonanalytic functions, injective in , with and with , .
We let and . Let function be nonanalytic in , , with and satisfying one of the following conditions:(i) for some or(ii)there exists a certain such that for all .
If function is nonanalytic in , , and function is nonanalytic and injective in , then implies .

Proof. (i) From Theorem 17, we have . On the other hand, for . From , we have that , .
(ii) If we let , then
By applying Theorem 17 and Remark 18 with , we obtain for all .
By letting , we obtain .

Example 24. Let the function be nonanalytic in ; then

Proof. Let the function be a nonanalytic function in , . In order to prove implication, it is enough to prove that . For that, it suffices to show that satisfies the admissibility condition.
Let , , , and .
In order to check the admissibility condition, we calculate For , since ,
We check the admissibility condition: since . Hence, .
From Theorem 17, we have .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.