Abstract
Here in this paper, we are using the concepts of -calculus operator theory associated with harmonic functions and define the -Noor integral operator for harmonic functions We investigate a new class of harmonic functions . In this class, we prove a necessary and sufficient convolution condition for the functions and also we proved that this sufficient coefficient condition is sense preserving and univalent in the class . It is proved that this coefficient condition is necessary for the functions in its subclass . By using this necessary and sufficient coefficient condition, we obtained results based on the convexity and compactness and results on the radii of -starlikeness and -convexity of order in the class . Also we obtained extreme points for the functions in the class
1. Introduction and Definitions
A complex-valued function is said to be harmonic in in open unit disc if both and are real valued harmonic functions in Also the complex-valued harmonic function can also be expressed as where and are analytic in In particular, is called analytic part, and is called coanalytic part of The Jacobian of the function is given by
It is known (see [1]) that every harmonic function to be locally univalent and sense preserving in if and only if in which is equivalent to in such that
For detail (see [2]). Let indicates the class of harmonic functions in Also let denoted by the family of harmonic functions which have the series expansion of the form: where and are analytic functions with the following series expansion:
The series defined in (3) and (4) are convergent in the open unit disc Also let represents all functions (say ) which are univalent analytic in and satisfy the condition
Further, let denotes the class of all harmonic functions which are sense preserving and univalent in . The class reduces to the class if coanalytic part of is zero.
Clunie and Small [3] and Small [4] studied the class along with some of their subfamilies. Particularly, they explored and studied the families of starlike harmonic and convex harmonic functions in which are given as follows: where
In [5], Dziok introduced a new family , and of Janowski harmonic functions and defined by where is given by (5). We can see that
The convolution of two functions , is defined by where
Similarly, the convolution of two harmonic functions and is defined by
The function subordinate to a function and write if there exists a complex-valued function which map into itself such that and In particular, if is univalent in then we have the following equivalence:
In the nineteen century, several mathematician has been using -calculus operator theory in various area of science, such that fractional calculus, -difference equation, optimal control, -integral equations, and geometric function theory (GFT). In 1908, Jackson [6] introduced the -derivative and -integral operator and discussed some of their applications. In the year 1990, Ismail et al. [7] gave the idea of -extension of class of -starlike functions by implementing the -calculus theory. Kanas and Raducanu [8] used -calculus operator theory and introduced the -Ruscheweyh differential operator for analytic functions. Zhang et. al [9] introduced a generalized conic domain by using the basic concepts of -calculus and studied new subclass of -starlike functions. Arif et al. defined -Noor integral operator [10] by using the concept of convolution and used it to investigated some new subclasses of analytic functions. Further, in article [11], Khan et al. discussed some applications of -derivative operator for multivalent functions, while coefficient estimates for a certain family of analytic functions involving a -derivative operator were discussed by Raza et al. [12]. Recently, Srivastava et. al published few articles in which they implemented basic concepts of -calculus operator theory and studied class of -starlike functions from different aspects (see [13–16]). Additionally, a recently published article by Srivastava [17] is very suitable for researchers to work on this topic. For more recently, Khan et al. [18, 19] used the concepts of -calculus operator theory to define some new subclasses of analytic functions. Also for more detail, we may refer to [20–25].
For, the -derivative operator of is defined as follows:
Making use of (3) and (14), and for , we have
For more detail (see [26, 27]).
Definition 1 (see [10, 28]). The -Noor integral operator for the analytic function is defined by where Thus, we have where Clearly,
Remark 2. When then -Noor integral operator reduces to Noor integral operator (see [29]).
First of all Jahangiri [30] applied certain -calculus operators to complex harmonic functions and obtained some useful results, while Porwal and Gupta discussed some application of -calculus to harmonic univalent functions in [31]. Recently Arif et al. [27] introduced some new families of harmonic functions associated with the symmetric circular region. For some more recent investigation about harmonic univalent functions, we may refer to [32, 33]. By taking the motivation from the article Arif et al. [27], we define the -Noor integral operator for the harmonic function
Definition 3. Let the -Noor integral operator of order for the harmonic function be defined as where and is given by (4).
In this paper, by using the concepts of -calculus operator theory and -Noor integral operator for harmonic functions , we define a new class of harmonic functions In this class, we prove a necessary and sufficient convolution condition for the functions and prove that this sufficient coefficient condition is sense preserving and univalent in the class . It is proved that this coefficient condition is necessary for the functions in its subclass . By using this necessary and sufficient coefficient condition, we obtained results based on the convexity and compactness and results on the radii of -starlikeness and -convexity of order and extreme points for the functions in the class This research work will motivate future research to work in the area of -calculus operators together with harmonic functions.
Definition 4. Let be the family of harmonic functions that satisfy the subordination condition where Inequalities (22) is equivalent to the condition We denote by a subclass of harmonic functions where for , functions and are of the form:
2. Main Results
Theorem 5. Let . Then the function if and only if where
Proof. Let be of the form (3). Then the function if and only if (22) holds or equivalently which by (21) is given by On using (20), the condition (29) may also be given by Which on using the convolution between two harmonic functions, we get where the harmonic function is given by (27).
Theorem 6. Let be of the form (3) and If where where is given by (19), then. (i)the function is locally univalent and sense-preserving as (ii)the function Equality occurs for the function
Proof. For part (i), it is clear that the theorem is true for the function Let and assume that there exist such that or Since we observe from (33) and (34) that by which the condition (32) implies the condition
in which implies as that in that is a function is locally univalent and sense-preserving in
For part (i), to prove that , we only need to show that satisfy the condition (24). Consider for and for , we can write (24) as
if the condition (32) holds. This proves the condition (24). This completes the proof of Theorem 6.
Theorem 7. Let where and are given by (25). Then if and only if the condition (32) holds that is where and are given by (33) and (34).
Proof. If part is proved in Theorem 6. To prove only if part, let Then by the class condition (22), we have from (24) that for any . where For we obtain which proves for and defined by (33) and (34) that Let be the sequence of partial sums of the series Then is a nondecreasing sequence, and by (42), it is bounded above. Thus, as it is convergent and This gives the condition (32).
Remark 8. Theorem 7 gives a necessary and sufficient condition for the functions where and are given by (25) to be -starlike and -convex of order in if we put and , respectively, in (38) and are given by
Theorem 9. The class is a convex and compact subclass of the class of functions where and are given by (25).
Proof. Let ; and let this is of the form
Then for and by Theorem 7, we get and given by (33) and (34) that
Therefore, Hence, the class is convex.
On the other hand, if we consider of the form (47), and then by Theorem 7, we get for and defined by (33) and (34).
Hence, for Similarly, we get for Therefore, class is locally uniformly bounded.
If we assume that then we conclude that and as for any Hence, from (50), we get
which proves that . Therefore, the class is closed. This proves that class is compact
Corollary 10. Let Then for Furthermore,
Remark 11. The minimum of all values of the radius for functions such that
is called the radius of -starlikeness of order and is denoted by
Now in next theorem, we obtain the radius of -starlikeness of order for functions
Theorem 12. Let and and are defined by (33) and (34). Then where defined by (19).
Proof. Let then by Theorem 7, we have where and are defined, respectively, by (33) and (34). Let be the radius of -starlikeness of order . Then if and only if from (45) that which is true if Or if
It follows that the radius in (57).
Similarly, we may find the radius of -convexity of order for functions , which is as below:
Theorem 13. Let and and are defined by (33) and (34). Then where is defined by (19).
Theorem 14. be of the form (25). Then if and only if where In particular the points and are called the extreme points of the closed convex hull of the class denoted by clco
Proof. Let be given by (63). Then from (64), it is of the form which by Theorem 7 proves that Since for this function Conversely, let and set Then on using (64), we obtain which is of the form (63). This proofs Theorem 14.
Corollary 15. Let be of the form (25). Then where and are defined, respectively, by (33) and (34). Equality in the inequalities (69) occurs for the extremal functions and given in (64) for
3. Conclusion
In this paper, we defined a new class of harmonic functions associated with newly defined -Noor integral operator for harmonic functions . In this class, we proved necessary and sufficient convolution condition for the functions We proved that, sufficient coefficient condition for the functions to be sense preserving and univalent and also this coefficient condition is necessary for subclass . By using the necessary and sufficient coefficient condition, we obtained results based on the convexity and compactness and results on the radii of -starlikeness and -convexity of order in the class . Also we investigated extreme points for the functions
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that they have no competing interests.