Abstract
New classes of univalent harmonic functions are introduced. We give sufficient coefficient conditions for these classes. These coefficient conditions are shown to be also necessary if certain restrictions are imposed on the coefficients of these harmonic functions. By using extreme points theory we also obtain coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. Radii of convexity and starlikeness of the classes are also considered.
1. Introduction
A continuous complex-valued function defined in a simply connected domain is said to be harmonic in if and are real harmonic in . There is a close interrelation between analytic functions and harmonic functions. For real harmonic functions and there exist analytic functions and so that and . Therefore, every complex-valued function harmonic in with can be uniquely represented aswhere and are analytic functions in with . Then we call the analytic part and the coanalytic part of . It is easy to verify that the Jacobian of is given by The mapping is locally univalent if in . A result of Lewy [1] shows that the converse is true for harmonic mappings. Therefore, is locally univalent and sense-preserving if and only if
Let denote the class of harmonic functions in the unit disc , where , and by we denote the class of function normalized by . Then we may express the analytic functions and defined by (1) as that is,By we denote the class of functions which are univalent and sense-preserving in , and by we denote the class of functions for which the coanalytic part vanishes.
We say that a function is subordinate to a function and write (or simply ), if there exists a complex-valued function which maps into oneself with , such that In particular, if is univalent in , we have the following equivalence:
For functions of the formby we denote the Hadamard product or convolution of and , defined by
Firstly, Clunie and Sheil-Small [2] and Sheil-Small [3] studied together with some of its geometric subclasses. In particular, they investigated harmonic starlike functions in and harmonic convex functions in , which are defined as follows. We say that is said to be harmonic starlike functions in if is a starlike domain with respect to the origin. Likewise is said to be harmonic convex functions in if is a convex domain.
In particular, we have that is harmonic starlike function if or equivalentlywhere
Let , Motivated by Janowski [4] we define the following classes of functions.
Let denote the class of functions such thatAlso, by we denote the class of functions such that Moreover, let us define We should notice here that Janowski [4] introduced the classes and . The classes and were investigated by Jahangiri [5, 6]. The classes and are the classes of functions which are starlike in or convex in , respectively, for all . It is easy to verify (e.g., see [5]) that
In the paper we obtain some necessary and sufficient conditions for defined classes of functions. Some topological properties, radii of convexity and starlikeness, and extreme points of the classes are also considered. By using extreme points theory we obtain coefficients estimates, distortion theorems, and integral mean inequalities for the classes of functions.
2. Necessary and Sufficient Conditions
Theorem 1. Let . Then if and only if where
Proof. Let be of form (1). Then if and only if it satisfies (13) or equivalentlySince we have Thus, conditions (17) and (19) are equivalent, and the proof is completed.
Putting in Theorem 1 we obtain the following corollary.
Corollary 2. Let . Then if and only if where
Theorem 3. If a function of form (5) satisfies the conditionwhere then .
Proof. It is clear that the theorem is true for the function . Let be a function of form (5) and let there exist such that or . Since by (24) we have Thus, by (3) function is locally univalent and sense-preserving in . Moreover, if , , then Therefore, by (27) we have This leads to the univalence of ; that is, . Therefore, if and only if there exists a complex-valued function , , such that or equivalentlyThus, it is sufficient to prove that Indeed, letting we have whence .
Motivated by Silverman [7] we denote by the class of functions of form (5) such that , ; that is,Moreover, let us define Now, we show that condition (24) is also the sufficient condition for a function to be in class .
Theorem 4. Let be a function of form (35). Then if and only if condition (24) holds true.
Proof. In view of Theorem 3 we need only to show that each function satisfies the coefficient inequality (24). If , then it satisfies (32) or equivalently Therefore, putting we obtainIt is clear that the denominator of the left hand side cannot vanish for . Moreover, it is positive for , and in consequence for . Thus, by (38) we haveThe sequence of partial sums associated with the series is nondecreasing sequence. Moreover, by (39) it is bounded by . Hence, the sequence is convergent and which yields assertion (24).
By using Theorems 1–4 and definition of the class we have the following three corollaries.
Corollary 5. Let . Then if and only if where is defined by (18).
Corollary 6. If a function of form (5) satisfies the conditionthen .
Corollary 7. Let be a function of form (35). Then if and only if condition (42) holds true.
Similar to Theorem 4 and Corollary 6 we can obtain the following two results.
Theorem 8. Let be a function of form (35). Then if and only if
Corollary 9. Let be a function of form (35). Then if and only if
Thus we have the following.
Corollary 10. Consider
3. Radii of Starlikeness and Convexity
We say that a function is starlike of order in ifAnalogously, we say that a function is convex of order in if It is easy to verify that for a function condition (46) is equivalent to or equivalently
Let be a subclass of the class . We define the radius of starlikeness and the radius of convexity for the class by
Theorem 11. The radius of starlikeness of order for the class , , is given bywhere and are defined by (25).
Proof. Let be of form (35). Then, for we have Thus, condition (49) is true if and only ifBy Theorem 4, we havewhere and are defined by (25). Since , condition (53) is true if that is, if (or and It follows that function is starlike of order in the disk , where isThe radii of starlikeness and of functions of form (75) are given by Therefore, the radius given by (57) cannot be larger. Thus we have (51).
The following result may be proved in much the same way as Theorem 11.
Theorem 12. Let and be defined by (25). Then
4. Topological Properties
We consider the usual topology on defined by a metric in which a sequence in converges to if and only if it converges to uniformly on each compact subset of . It follows from the theorems of Weierstrass and Montel that this topological space is complete.
Let be a subclass of the class . A function is called an extreme point of if the condition implies . We will use the notation to denote the set of all extreme points of . It is clear that .
We say that is locally uniformly bounded if for each , , there is a real constant so that
We say that a class is convex if Moreover, we define the closed convex hull of as the intersection of all closed convex subsets of that contain . We denote the closed convex hull of by .
A real-valued functional is called convex on a convex class if
The Krein-Milman theorem (see [8]) is fundamental in the theory of extreme points. In particular, it implies the following lemma.
Lemma 13. If is a nonempty compact subclass of the class , then is nonempty and .
Due to Hallenbeck and MacGregor [9, pp. 45] (see also [10]) we prove the following result.
Lemma 14. Let be a nonempty compact convex subclass of the class and let be a real-valued, continuous, and convex functional on . Then
Proof. Since the functional is continuous on the compact set there exists Therefore, the set is nonempty compact subclass of . Hence, by Lemma 13 we find that has an extreme point . Suppose that where and . Then and we must have ; that is, . Since is an extreme point of we have and, in consequence, . Thus, we conclude that there exists , and the proof is complete.
Since is a complete metric space, Montel’s theorem (see [11]) implies the following lemma.
Lemma 15. A class is compact if and only if is closed and locally uniformly bounded.
Theorem 16. The class is convex and compact subset of .
Proof. Let be functions of the formand let . Since and by Theorem 4 we have the function belongs to the class . Hence, the class is convex. Furthermore, for , we haveThus, we conclude that the class is locally uniformly bounded. By Lemma 15, we only need to show that it is closed; that is, if and , then . Let and be given by (68) and (5), respectively. Using Theorem 4 we have Since , we conclude that and as . The sequence of partial sums associated with the series is nondecreasing sequence. Moreover, by (72) it is bounded by . Therefore, the sequence is convergent and This gives condition (24), and, in consequence, , which completes the proof.
Theorem 17. Consider where
Proof. Suppose that , and where are functions of form (8). Then, by (24) we have , and, in consequence, for and for . It follows that , and consequently . Similarly, we verify that the functions of form (75) are the extreme points of the class . Now, suppose that and is not of form (75). Then there exists such that If , then, putting we have that , and Thus, . Similarly, if , then, putting we have that , and It follows that , and the proof is completed.
The following result may be proved in much the same way as Theorems 16 and 17.
Theorem 18. The class is convex and compact subset of . Moreover, where
5. Applications
It is clear that for the locally uniformly bounded class we haveThus, by Theorems 16, 17, and 18 we have the following two corollaries.
Corollary 19. Consider where are defined by (75).
Corollary 20. Consider where are defined by (83).
For each fixed value of , , the following real-valued functionals are continuous and convex on :
Therefore, by Lemma 14 and Theorems 17 and 18 we have the corollaries listed below.
Corollary 21. Let be a function of form (35). Thenwhere are defined by (25). The result is sharp. The functions of form (75) are the extremal functions.
Corollary 22. Let be a function of form (35). Then where are defined by (25). The result is sharp. The functions of form (83) are the extremal functions.
Corollary 23. Let , . Then The result is sharp. The function of form (75) is the extremal function.
Corollary 24. Let and . Then The result is sharp. The function of form (83) is the extremal function.
Due to Littlewood [12] we obtain the integral means inequalities for functions from the classes
Lemma 25 (see [12]). Let . If , then
Let and be defined by (75) and let . Since and , by Lemma 25 we have Thus, we have the following lemma.
Lemma 26. Let , . Then where and are defined by (75).
By Lemmas 14 and 26 and Theorem 17 we have the following corollary.
Corollary 27. Let . If , then where is the function defined by (75).
In a similar manner, we can prove Corollary 23 below.
Corollary 28. Let . If , then where is the function defined by (83).
The following covering results follow from Corollaries 23 and 24.
Corollary 29. If , then
Corollary 30. If , then
Remark 31. By choosing the parameters in the defined classes of functions we can obtain new and also well-known results (see, e.g., [5–7, 10]).
6. Conclusion
In the paper new classes of univalent harmonic functions are introduced. Necessary and sufficient conditions for defined classes of functions are obtained. Some topological properties, radii of convexity and starlikeness, and extreme points of the classes are also considered. By using extreme points theory we obtained coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work is supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, University of Rzeszów.