Research Article | Open Access
Huda Aldweby, Maslina Darus, "A Subclass of Harmonic Univalent Functions Associated with -Analogue of Dziok-Srivastava Operator", International Scholarly Research Notices, vol. 2013, Article ID 382312, 6 pages, 2013. https://doi.org/10.1155/2013/382312
A Subclass of Harmonic Univalent Functions Associated with -Analogue of Dziok-Srivastava Operator
We study a class of complex-valued harmonic univalent functions using a generalized operator involving basic hypergeometric function. Precisely, we give a necessary and sufficient coefficient condition for functions in this class. Distortion bounds, extreme points, and neighborhood of such functions are also considered.
Let be the open unit disc, and let denote the class of functions which are complex valued, harmonic, univalent, and sense preserving in normalized by . Each 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 in (see ). In , there is a more comprehensive study on harmonic univalent functions. Thus, for , we may write Note that reduces to , the class of normalized analytic univalent functions, if the coanalytic part of is identically zero.
The study of basic hypergeometric series (also called -hypergeometric series) essentially started in 1748 when Euler considered the infinite product . In the literature, we were told that the development of these functions was much slower until, in 1878, Heine converted a simple observation that which returns the theory of basic hypergeometric series to the famous theory of Gauss’s hypergeometric series. The importance of basic hypergeometric functions is due to their application in deriving -analogue of well-known functions, such as -analogues of the exponential, gamma, and beta functions. In this paper, we define a class of starlike harmonic functions using basic hypergeometric functions and investigate its properties like coefficient condition, distortion theorem, and extreme points.
For complex parameters , , , , , , we define the basic hypergeometric function by where denote the set of positive integers and is the -shifted factorial defined by We note that where is the well-known generalized hypergeometric function. By the ratio test, one observes that for and the series defined in (2) converges absolutely in so that it represented an analytic function in . For more mathematical background of basic hypergeometric functions, one may refer to [3, 4].
Corresponding to the function , consider The authors  defined the linear operator by where stands for convolution and To make the notation simple, we write We define the operator (8) of harmonic function given by (1) as
Definition 1. For , let denote the subfamily of starlike harmonic functions of the form (1) such that Following , a function is said to be in the class if of the form (1) satisfies the condition that and if there exists a real number such that By specializing the parameters of , we obtain different classes of starlike harmonic functions, for example,(i)for ,  is the class of sense-preserving harmonic univalent functions which are starlike of order in ; that is, ;(ii)for , , and ,  is the class of starlike harmonic univalent functions with , where is the Ruscheweyh derivative (see ); (iii)for , and ,  is the class of starlike harmonic univalent functions with , where is the Dziok-Srivastava operator (see ).
2. Main Results
In our first theorem, we introduce a sufficient coefficient bound for harmonic functions in .
Proof. To prove that , we only need to show that if (15) holds, then the required condition (12) is satisfied. For (12), we can write Using the fact that if and only if , it suffices to show that Substituting for and in (15) yields The last expression is nonnegative by (15), and so, .
Now, we obtain the necessary and sufficient conditions for given by (14).
Proof. Since , we only to prove the only if part of the theorem. So that for functions , we notice that the condition is equivalent to
The previous condition must hold for all values of in . Upon choosing according to (14), we must have If condition (19) does not hold, then the numerator in (22) is negative for sufficiently close to 1. Hence, there exist in (0,1) for which the quotient of (22) is negative. This contradicts the fact that , and this completes the proof.
The following theorem gives the distortion bounds for functions in which yield a covering result for this class.
Theorem 4. If , then where
Proof. We will only prove the right hand inequality. The proof for the left hand inequality is similar.
let . Taking the absolute value of , we obtain That is,
Corollary 5. Let be of the form (1) so that . Then, Next, one determines the extreme points of closed convex hull of denoted by .
Theorem 6. Set For fixed, the extreme points for are where and .
Proof. Any function may be expressed as
where the coefficients satisfy the inequality (15). Set , , , for . Writing , and , we have
In particular, set
Therefore, the extreme points of are contained in . To see that is not an extreme point, note that may be written as a convex linear combination of functions in as follows: If both and , we will show that it can also be expressed as a convex linear combination of functions in . Without loss of generality, assume that . Choose small enough so that . Set and . We then see that both are in and note that The extremal coefficient bound shows that the functions of the form (29) are extreme points for , and so the proof is complete.
We define the --neighborhood of a function as follows:
In our case, let us define the generalized --neighborhood of to be the set
Theorem 7. Let be given by (1). If satisfies the conditions then .
Proof. Let satisfy (39) and let belong to . We have Hence,
The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02 and UKM-DLP-2011-050.
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