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ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 179856, 11 pages
A Brief Study of Certain Class of Harmonic Functions of Bazilevič Type
1Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, PMB 4000, Ogbomosho, Nigeria
2Department of Mathematics and Informatics, 1 Decembrie 1918 University of Alba Iulia, 5 Gabriel Bethlen Street, 510009 Alba Iulia, Romania
Received 24 March 2013; Accepted 18 April 2013
Academic Editors: R. Avery, Y. Han, G. L. Karakostas, and C. Zhu
Copyright © 2013 A. T. Oladipo and D. Breaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We define and investigate a new subclass of Bazilevič type harmonic univalent functions using a linear operator. We investigated the harmonic structures in terms of its coefficient conditions, extreme points, distortion bounds, convolution, and convex combination. So, also, we discussed the subordination properties for the functions in this class.
Let denote the usual class of analytic functions of the form which are analytic in the unit disk and normalized with and . Also, we denote the subclass of consisting of analytic and univalent functions in the unit disk by .
Here, we recall some definitions and concepts of classes of analytic functions. Let . Then, if and only if This class is called starlike class of analytic function.
Also, let . Then, if and only if This class is called convex class of analytic function. The above two classes have been repeatedly investigated by various authors like [1–4] just to mention but few, as the literatures littered everywhere.
The theory of analytic functions has wide application in many physical problem: problems as in heat conduction, electrostatic potential and fluid flows, and theory of fractals constitute practical examples. The concern of this work is the study of a particular family of analytic functions defined in a given domain by certain geometric conditions which are useful in the above problems.
Let , and let be univalent in . If is analytic in and satisfies the differential subordination , then is called a solution of the differential subordination. The univalent function is called a dominant of the solution of the differential subordination, . If and are univalent in and satisfy the differential superordination , then is called a solution of the differential superordination. An analytic function is called subordinate of the solution of the differential superordination if . For details (see [5–7]).
Sălăgean  introduced the following differential operator: From (1), we can write that Using binomial expansion on (5), we have We then define the class of analytic functions of fractional power as where (is real, and it is principal determination only).
Thus, we obtain the differential operator Let us also define the function by where is the Pochhammer symbol defined by
Corresponding to the function , we defined a linear operator Or equivalently
Remark 1. For , operator (11) reduces to Carlson-shaffer operator , and also for different values of , it imposes the Saitoh operator  and recently the Mahzoon-Lotha . for , poses the Salagean derivative operator.
Now, let be the class of functions containing the operator (11) and satisfying the relation For , , , and , we obtain the well-known subclass Also, for , , and , we have For , , and we have the following subclass which contain Carlson-Shaffer operator: The starting point in the study of functions defined in (13) is the discovery in 1995 by Russian Mathematician Bazilevič  of functions in defined by where and . The number and are real, and all powers are meant as principal determinant only. The family of functions in (17) became known as Bazilevič functions and is, in this work, denoted by . Except that, Bazilevič showed that each function is univalent in , very little is known regarding the family as a whole. However, with some simplifications, it may be possible to understand and investigate the family. Indeed, it is easy to verify that, with special choices of the parameters and and the function , the family cracks down to some well-known subclasses of univalent functions.
For instance, if we take , we have On differentiation, the expression (18) yields Or equivalently The subclasses of Bazilevič functions satisfying (19) are called Bazilevič functions of type and are denoted by (see ). In 1973, Noonan  gave a plausible description of functions of the class as those functions in for which each , and the tangent to the curve never turns back on itself as much as radian. If , the class reduces to the family of close-to-convex functions; that is, If we decide to choose in (21), we have which implies that is starlike. Furthermore, if we replace by in (22), we obtain which shows that is convex. Moreover, if in (20), then we have the family  of functions satisfying The various subfamilies of Bazilevič functions are being studied repeatedly by many authors; the literatures in this direction littered everywhere (see Bernard's Bibliography of Schlich functions ).
In 1992, Abduhalim  introduced a generalization of functions satisfying (24) as where the parameter and the operator are defined as earlier. He denoted this class of functions by . It is easily seen that his generalization has extraneously included analytic functions satisfying which are largely nonunivalent in the unit disk. By proving the inclusion Abdulhalim was able to show that for all , each function of the class is univalent in .
Notable contributors like MacGregor, [15, 16], Noonan , Singh , Thomas , Tuan and Anh , Yamaguchi , and Opoola  had earlier considered various special cases of the parameters and of (25) and established many interesting properties of function in those particular cases.
In some general sense, it is possible to further improve work on the function defined by the geometric condition (25). Therefore, we intend to investigate this family from the viewpoints of subordination and harmonic univalent functions and determine coefficient inequalities, extreme points, distortion bounds, convolution, and convex combination.
2. Subordination Results
The objective of this section is to find the sufficient conditions of functions belonging to the class .
For this purpose, the following Lemmas will be necessary.
Lemma 2 (see ). Let be a complex number. Let () be a univalent function in such that If , (), satisfies the differential subordination then and is the best dominant.
Lemma 3 (see ). Let be analytic in with . If attains its maximum value on the circle at a point , then where is a real number and .
Lemma 4 (see ). If satisfies then
Now, we begin our main results as the following.
Theorem 5. Let ( is real), , and be a complex number such that If the subordination holds, then .
Proof. Suppose that Then, simple computations give Thus, in the view of Lemma 2, we have .
For , , . We have the following.
Corollary 6. Let be a complex number such that If the subordination Holds, then .
For , , , and , we have the following
Corollary 7. Let be a complex number such that If the subordination holds, then .
For , , we have the following.
Corollary 8. Let be a complex number such that If the subordination holds, then .
For , , , we have the following.
Corollary 9. Let be a complex number such that If the subordination holds, then .
Theorem 10. Let the functions take the form (7) and satisfy Then, .
Proof. Let be defined by Then, is analytic in , and since , then . Also, it follows that Now, let us proceed to prove that . Suppose that there exists a point such that Then, using Lemma 3 and letting and , yields that Thus, we have which contradicts the hypothesis (46). Therefore, we conclude that for all and where ( is real). This completes the proof of the theorem.
Letting , , and in Theorem 10, we have the following.
Corollary 11. Let the function take the form (1) and satisfy Then,
Corollary 12. Let the function take the form (1) and satisfy Then, With various special choices of the parameters involved, many existing and new subclasses of Bazilevič functions could be derived.
3. Harmonic Structure of Bazilevič Type
In this section, the authors wish to have a look into the Bazilevič type harmonic univalent functions.
A continuous complex-valued function defined in a simply connected domain is said to be harmonic in . In any simply connected domain, we can write where and are analytic in . We call the analytic part and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and sense preserving in is that Denote by the class of functions of the form (57) that are harmonic univalent and sense-preserving in the disk . The subclasses of harmonic functions have been studied by some authors for different purposes with different properties (see [24–26]). But unfortunately, it is becoming very difficult to see the literatures on Bazilevič-type harmonic univalent function, and this may be likely associated with the problem index always poses. This paper is designed to address this issue.
In this work, we may express the analytic functions and as Therefore, We define our linear operator as given in (11) such that where We let be the family of harmonic functions of the form (57) such that where , ( is real), , and is earlier defined in (11).
Furthermore, let the subclass consist of harmonic functions so that and are of the form The authors in this work wish to study the Bazilevič-type harmonic univalent functions defined by linear operator in which has positive coefficients. We claim that our results are quite new and not explored in the literatures.
Assigning specific values to , , , , in the subclass , we obtain the following subclasses which may be the expected results by using definition of earlier authors of subclasses of Bazilevič functions such as classes studied by Abduhalim , Yamaguchi , Macgregor , and Singh , just to mention but few.
We first prove a sufficient condition for the function in .
Theorem 13. Let , where and are as earlier defined if where , , , and ; is real then is sense-preserving, harmonic univalent in , and .
Proof. If , then
which proves the univalence. Note that is sense-preserving in . This is because
By (63) and (64), we have Using the fact that if and only if , it suffices to show that That is, This last expression is nonnegative by (66), and so the proof is complete.
The harmonic function where , ( is real), , , and shows that the coefficient bound given by (66) is sharp. The functions of the form (72) are in because In the following theorem, it is shown that the condition (66) is also necessary for functions , where and are as earlier defined.
Theorem 14. Let . Then, if and only if
Proof. Since , we only need to prove the “only if” part of the theorem. To this end, for functions of the form (64), we notice that the condition is equivalent to The above required condition (75) must hold for all values of in . Upon clearing the values of on the positive real axis, where , we must have and the proof is complete.
Theorem 15. Let , where and are as given earlier. Then, if and only if where , (), where , , .
In particular, the extreme points of are and .
Proof. For functions , where and are as earlier defined, we have
and so .
Conversely, suppose that . Setting and , therefore, can be written as as required.
Our next result is on distortion bounds for the functions in the class .
Theorem 16. Let . Then, for , one has
Proof. We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted.
Let . Taking the absolute value of , we obtain for . This shows that the bound given in Theorem 16 is sharp.
The following covering results follow from the left-hand inequality in Theorem 18.
Corollary 17. If function , where and are as given earlier in , then For harmonic functions The convolution of and is given by Using this definition, one shows that the class is closed under convolution.
Theorem 18. For , let , and let . Then, .
Proof. Let the functions
and be in . Then, the convolution is given by (89).
We wish to show that the coefficients of satisfy the required condition given in Theorem 14. For , , and . Now, for the convolution function , we obtain Therefore .
Next, we show that the class is closed under convex combinations of its members.
Let functions be defined, for , by
Theorem 19. Let the functions defined by (92) be in the class for every . Then, the functions defined by are also in the class , where .
Proof. According to the definition of , we can write Furthermore, since are in for every (,), then by (66), we have Hence, the theorem is proved.
Corollary 20. The class is closed under convex linear combination.
Proof. Let the functions () defined by (92) be in the class . Then, the function defined by