A Subclass of Harmonic Univalent Functions with Varying Arguments Defined by Generalized Derivative Operator
Making use of the generalized derivative operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc and are related to uniformly convex functions. We investigate the coefficient bounds, neighborhood, and extreme points for this generalized class of functions.
A continuous complex-valued function defined in a simply connected complex domain is said to be harmonic in if both and are real harmonic in . Such functions can be expressed as 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 for all in (see ). Let be the class of functions of the form (1.1) that are harmonic univalent and sense preserving in the unit disk for which . Then for , we may express the analytic functions and as In 1984, Clunie and Sheil-Small  investigated the class as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on and its subclasses. Now we will introduce a generalized derivative operator for given by (1.2). For fixed positive natural and , where We note that by specializing the parameters, especially when , reduces to which is introduced by Salagean in .
Now we will introduce the following definition.
Definition 1.1. For , let denote the subfamily of starlike harmonic functions of the form (1.1) such that
for a suitable real and where .
We also let where is the class of harmonic functions with varying arguments introduced by Jahangiri and Silverman  consisting of functions of the form (1.1) in for which there exists a real number such that where and . The same class introduced in  with different differential operator.
In this paper, we obtain a sufficient coefficient condition for functions given by (1.2) to be in the class . It is shown that this coefficient condition is necessary also for functions belonging to the class . Further, extreme points for functions in are also obtained.
2. Main Result
We begin deriving a sufficient coefficient condition for the functions belonging to the class . This result is contained in the following.
Theorem 2.1. Let given by (1.2). Furthermore, let where , then .
Proof. We first show that if the inequality (2.1) holds for the coefficients of , then the required condition (1.5) is satisfied. Using (1.3) and (1.5), we can write
In view of the simple assertion that if and only if , it is sufficies to show that
Substituting for and the appropriate expressions in (2.4), we get by virtue of the inequality (2.1). This implies that .
Now we obtain the necessary and sufficient condition for function be given with condition (1.6).
Theorem 2.2. Let be given by (1.2). Then if and only if where .
Proof. Since , we only need to prove the necessary part of the theorem. Assume that , then by virtue of (1.3) to (1.5), we obtain
The above inequality is equivalent to
This condition must hold for all values of , such that . Upon choosing according to (1.6) and noting that , the above inequality reduces to
If (2.6) does not hold, then the numerator in (2.9) is negative for sufficiently close to 1. Therefore, there exists a point in for which the quotient in (2.9) is negative. This contradicts our assumption that . We thus conclude that it is both necessary and sufficient that the coefficient bound inequality (2.6) holds true when . This completes the proof of Theorem 2.2.
Theorem 2.3. The closed convex hull of (denoted by clco is By setting and , then for fixed, the extreme points for clco are where and .
Proof. Any function in clco may be expressed as
where the coefficients satisfy the inequality (2.1). Set , , , for . Writing , , and ; , we get
In particular, setting
We see that extreme points of clco .
To see that is not in extreme point, note that may be written as a convex linear combination of functions in clco .
To see that is not an extreme point if both and , we will show that it can then also be expressed as a convex linear combinations of functions in clco . Without loss of generality, assume . Choose small enough so that . Set and . We then see that both are in clco and that The extremal coefficient bounds show that functions of the form (2.11) are the extreme points for clco , and so the proof is complete.
In our case, let us define the generalized -neighborhood of to be the set
Theorem 2.4. Let be given by (1.2). If satisfies the conditions where , and then .
The work here was supported by UKM-ST-06-FRGS0244-2010.
Y. Avici and E. Zotkiewicz, “On harmonic univalent mappings,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 44, pp. 1–7, 1990.View at: Google Scholar