A Brief Study of Certain Class of Harmonic Functions of BazileviI Type

f (z) } > μ, (0 ≤ μ < 1, z ∈ U) . (3) 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 γ : C2 → C, and let φ be univalent in U. If p is analytic in U and satisfies the differential subordination


Introduction
Let denote the usual class of analytic functions of the form which are analytic in the unit disk = { : | | < 1} and normalized with (0) = 0 and (0) − 1 = 0. Also, we denote the subclass of consisting of analytic and univalent functions ( ) in the unit disk by .
This class is called convex class of analytic function. The above two classes have been repeatedly investigated by various authors like [1][2][3][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 : 2 → , and let be univalent in . If is analytic in and satisfies the differential subordination Φ( ( ), ( )) ≺ ( ), 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 Φ( ( ), ( )) are univalent in and satisfy the differential superordination ( ) ≺ Φ( ( ), ( )), 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][6][7]).

ISRN Mathematical Analysis
Sȃlȃgean [8] 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 > 0 (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 = 0, = 1 operator (11) reduces to Carlson-shaffer operator [3], and also for different values of , it imposes the Saitoh operator [7] and recently the Mahzoon-Lotha [9]. for = , = 1 poses the Salagean derivative operator. Now, let ( ) be the class of functions containing the operator (11) and satisfying the relation For = , = 1, = 1, and = 0, we obtain the well-known subclass Also, for = 0, = , and = 1, we have For = 0, = 1, and = 1 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 [10] of functions in defined by where ∈ and ∈ * . The number > 0 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 = 0, we have On differentiation, the expression (18) yields ISRN Mathematical Analysis 3 Or equivalently The subclasses of Bazilevič functions satisfying (19) are called Bazilevič functions of type and are denoted by ( ) (see [11]). In 1973, Noonan [12] gave a plausible description of functions of the class ( ) as those functions in for which each < 1, and the tangent to the curve ( ) = { ( exp ) , 0 ≤ < 2 } never turns back on itself as much as radian. If = 1, 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 1 ( ) [11] 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 [13]).
In 1992, Abduhalim [14] 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 [12], Singh [11], Thomas [17], Tuan and Anh [18], Yamaguchi [19], and Opoola [20] 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.

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 4 (see [23]). If ∈ satisfies Now, we begin our main results as the following.
Proof. Suppose that Then, simple computations give Thus, in the view of Lemma 2, we have ( ) ∈ ( , , ).
For = 1, = , we have the following.

Corollary 11. Let the function take the form (1) and satisfy
Then,

Harmonic Structure of BazileviI Type
In this section, the authors wish to have a look into the Bazilevič type harmonic univalent functions. A continuous complex-valued function = + V defined in a simply connected domain is said to be harmonic in . In any simply connected domain, we can write where ℎ and are analytic in . We call ℎ 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][25][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 6 ISRN Mathematical Analysis index always poses. This paper is designed to address this issue.
In this work, we may express the analytic functions ℎ 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 > 0, ( is real), ∈ 0 , and ( , ) is earlier defined in (11). Furthermore, let the subclass ( , , ) consist of harmonic functions so that ℎ 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 ℎ 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 [14], Yamaguchi [19], Macgregor [15], and Singh [11], just to mention but few.
We first prove a sufficient condition for the function in ( , , ).
Proof. If 1 ̸ = 2 , then which proves the univalence. Note that is sense-preserving in . This is because By (63) This last expression is nonnegative by (66), and so the proof is complete.
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 0 ≤ = < 1, we must have and the proof is complete.

ISRN Mathematical Analysis 9
Our next result is on distortion bounds for the functions in the class ( , , ).
The following covering results follow from the left-hand inequality in Theorem 18.  (89) Using this definition, one shows that the class ( , , ) is closed under convolution.