Abstract
In this study, the idea of -calculus is applied to define new classes of spirallike functions and . The coefficient inequalities, rotational invariance, and containment results are proved for these classes. Further, by introducing a parameter , two more subclasses are defined and their properties are investigated. Finally, we have probed into these classes by fixing finitely several coefficients.
1. Introduction
The class of functionswhich are analytic and univalent in and are normalized by , , will be denoted by .
, the class of uniformly starlike functions and , the class of uniformly convex functions, were initiated by Goodman in [1, 2]. These were further extended as and by Kanas and Wisniowska [3, 4]. The classes of uniformly convex spirallike and uniformly spirallike were introduced in [5].
Definition 1. [5] A functiongiven by (1) is iniff
Definition 2. [5] A functiongiven by (1) is iniff
Definition 3. [6] The function given by (1) is uniformly convex-spiral of orderif the image of every circular arcwith center atlying inis convex-spirallike of order.
The class of such functions is denoted by . The single variable characterization of the class is given by
Definition 4. [6] A functiondefined by (1) is iniffThe subclass of , launched by Herb Silverman [7] consists of functionsInfluenced by the studies in [8] and were defined and analyzed in [9]. Generalizing and fixing the coefficients in the above classes, new class of functions and were defined in [10].
-calculus is in the limelight of research in geometric function theory. Jackson [11, 12], the pioneer in the systematic initiation of -calculus defined -derivative aswith .
It follows that , if is differentiable in the domain. Also, . is defined by (1).where and .
Motivated by the current developments in -calculus, the -analogue of and are defined and examined.
2. Coefficient Inequalities
Definition 5. The functionis in if, , , and .
Theorem 1. Let be defined by (9). IfThen, .
Proof. From Definition 5, it is enough if we proveHowever, we haveThis is bounded by , only when
Definition 6. The function defined by (9) is inif, , and .
Theorem 2. Let be defined by (9). IfThen, .
Proof. The proof is analogous to that of Theorem 1.
Theorem 3. The class is rotationally invariant.
Proof. For , we prove that also belongs to and .
Considerby Definition 5 and whenever .The same result holds true for .
Theorem4. , whenever.
Proof. Let and .
This implies and
Definition 7. Let be the class of functions in of the form
Definition 8. Let be the class of functions in of the form
Theorem 5. The function which is defined by (20) belongs to, if and only ifThe result is sharp.
Proof. Setting in (11), we get the required result. The result is sharp forWe now state the coefficient inequality for . The proof is similar to that of Theorem 5 and hence, it is omitted.
Theorem 6. The function defined by (21) belongs toif and only if
3. Some Properties of and
We consider a few integral operators and prove preserving properties of these operators on the two classes.
3.1. Bernadi Integral Operator
With and , reduces to the Bernadi integral operator.
3.2. Alexander Operator
and for .Theorem 7. The Alexander operator preserves the classes and .
Proof. Let be in .
We prove that .
Now considering the following, gives .
As and by Theorem 5,Also, when , .
Hence,Similarly, we can prove the result for .
Theorem8. The classes and are preserved by .
Proof. Considering , , in , usingand the definition of , we getwith and .
For ,to prove . From (31), .
Using (29), we getWe can prove the result for in a similar way.
Theorem 9. The Bernadi integral operator and the integral operator preserves the classes and .
proof of this theorem is analogous to that of the above theorem.
By fixing finitely many coefficients, we define new subclasses of functions in the following section.
4. The New Class
The class denotes functions in and is of the formwhere .
reduces to , where .
The following theorem gives the coefficient estimate for the functions in this class.
Theorem 10. A function defined by (33) will be inif and only if
Proof. Assuming as for and in Theorem 1, we get (34).
By taking the functionThe sharpness of the result follows.
4.1. Closure Theorems
Theorem 11. is closed under convex linear combination.
Proof. Let and belong to and be defined aswith , , and .
Taking , , we prove.
. Consideringand aswhich shows that is in , by Theorem 10.
Hence, the class is convex.
Theorem 12. Let andwith .
Then, a function will belong to the class if and only if , where and .
Proof. Let . Then,From the above,which implies that by using Theorem 10.
To prove the converse part let us take, for and , then, we obtain .
Corollary 1. For the class , the extreme points are the functions , of Theorem 12.
Definition 9. The function , in the class is said to be(i)Starlike of order , if ;(ii)Convex of order , if for .In the following theorems, the radii results for the functions in to be starlike or convex of order are derived.
Theorem 13. , in , will be starlike of order in, where is the largest value for whichfor . The result is sharp.
Proof. To get the required result, we need to prove that , for .
Using appropriate substitutions,for , if and only ifSince , by the coefficient estimate, we can takewhere , and .
Let us now choose a positive integer , for each fixed so that is a maximum.
Hence, it follows that is starlike of order in ifWe get the value and so that is the radius of starlikeness of order for functions . The result is sharp for given by (34).
For functions in the class , we state the theorem regarding the radius of convexity of order .
Theorem 14. Let . Then, is convex of order in the disc , where is the greatest value for whichfor . The result is sharp.
5. Conclusion
The concept of q-calculus is used widely in quantum physics and fractional calculus of mathematics. The classes and were introduced and analyzed in this paper. Obtaining the coefficient inequalities, containment theorem, and rotational invariance are of paramount importance as they throw more light on these classes of functions. The results of this study have ample capacity to facilitate more research in this area and beyond.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
Geetha Balachandar performed the actualization, validation, methodology, and formal analysis. Mohammed K. A. Kaabar performed the actualization, methodology, formal analysis, validation, investigation, supervision, initial draft, and final draft. Charles Robert Kenneth performed the actualization, methodology, validation, investigation, initial draft, and formal analysis. Kins Yenoke performed the actualization, validation, methodology, and formal analysis. All authors read and approved the final manuscript.