A New Subclass of Harmonic Univalent Functions Associated with Dziok-Srivastava Operator
The purpose of the present paper is to study a certain subclass of harmonic univalent functions associated with Dziok-Srivastava operator. We obtain coefficient conditions, distortion bounds, and extreme points for the above class of harmonic univalent functions belonging to this class and discuss a class preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combination. The results obtained for the class reduced to the corresponding results for several known classes in the literature are briefly indicated.
A continuous complex-valued function defined in a simply connected domain , is said to be harmonic in if both and are harmonic in . In any simply connected domain we can write , where and are analytic in . A necessary and sufficient condition for to be locally univalent and sense preserving in is that . See Clunie and Sheil-Small .
Denote by the class of functions that are harmonic univalent and sense preserving in the unit disk for which . Then for , we may express the analytic function and as
Note that , is reduced to the class of normalized analytic univalent functions if the coanalytic part of is identically zero.
For more basic results on harmonic univalent functions, one may refer to the following introductory text book by Duren  (see also [3–5]).
For and , , the generalized hypergeometric functions are defined by where is the Pochhammer symbol defined by
Corresponding to the function
The Dziok-Srivastava operator [6, 7] is defined for by
where * stands for convolution of two power series.
To make the notation simple, we write
Special cases of the Dziok-Srivastava operator includes the Hohlov operator , the Carlson-Shaffer operator , the Ruscheweyh derivative operator , and the Srivastava-Owa fractional derivative operators ([11–13]).
We define the Dziok-Srivastava operator of the harmonic functions given by (1.1) as
Recently, Porwal [14, Chapter 5] defined the subclass consisting of harmonic univalent functions satisfying the following condition:
He proved that if is given by (1.1) and if
For the class of is reduced to the class studied by Uralegaddi et al. .
Generalizing the class , we let denote the family of functions of form (1.1) which satisfy the condition where and .
Further, let be the subclass of consisting of functions of the form
In this paper, we give a sufficient condition for , given by (1.1) to be in , and it is shown that this condition is also necessary for functions in . We then obtain distortion theorem, extreme points, convolution conditions, and convex combinations and discuss a class preserving integral operator for functions in .
2. Main Results
First, we give a sufficient coefficient bound for the class .
Theorem 2.1. If is given by (1.1) and if then .
It suffices to show that
We have where denotes .
The last expression is bounded above by 1, if which is equivalent to
But (2.6) is true by hypothesis.
Hence and the theorem is proved.
In the following theorem, it is proved that the condition (2.1) is also necessary for functions that are given by (1.11).
Theorem 2.2. A function is in , if and only if
Proof. Since , we only need to prove the “only if” part of the theorem. For this we show that if the condition (2.8) does not hold.
Note that a necessary and sufficient condition for given by (1.9) is in if is equivalent to
The above condition must hold for all values of , . Upon choosing the values of on the positive real axis where , we must have
If the condition (2.8) does not hold then the numerator of (2.11) is negative for and sufficiently close to 1. Thus there exists a in (0,1) for which the quotient in (2.11) is negative. This contradicts the required condition for and so the proof is complete.
Next, we determine the extreme points of the closed convex hulls of , denoted by .
Theorem 2.3. , if and only if where , and . In particular the extreme points of are and .
Proof. For functions of the form (2.12), we have
and so .
Conversely, suppose that . Set , and , .
Then note that by Theorem 2.2, and . We define , and by Theorem 2.2, .
Consequently, we obtain as required.
Theorem 2.4. If , then
Proof. We only prove the right hand inequality. The proof for left hand inequality is similar and will be omitted. Let . Taking the absolute value of , we have
For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form We define the convolution of two harmonic functions and as
Using this definition, we show that the class is closed under convolution.
Theorem 2.5. For , let and . Then
Proof. Let be in , and be in .
Then the convolution is given by (2.19). We wish to show that the coefficients of satisfy the required condition given in Theorem 2.2. For we note that and . Now, for the convolution function , we obtain
Theorem 2.6. The class is closed under convex combination.
Proof. For , let , where is given by
Then by Theorem 2.2,
For , , the convex combination of may be written as
Then by (2.23),
This is the condition required by Theorem 2.2 and so .
3. A Family of Class Preserving Integral Operator
Let be defined by (1.1); then defined by the relation
Theorem 3.1. Let be given by (1.11) and , then is defined by (3.1) also belong to .
Proof. Let be in , then by Theorem 2.2, we have
By definition of , we have
Now we have
The first author is thankful to University Grants Commission, New Delhi (Government of India) for supporting this paper financially under research project (no. 8-2 (223)/2011 (MRP/Sc/NRCB)). The authors are also thankful to the reviewers for giving fruitful suggestions and comments.
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