Abstract
This article deals with the -differential subordinations for starlike functions associated with the lemniscate of Bernoulli and cardioid domain. The primary goal of this work is to find the conditions on for , where is analytic function and is subordinated by the function which is producing cardioid domain as its image domain while mapping the open unit disk. Along with this, certain sufficient conditions for -starlikeness of analytic functions are determined.
1. Introduction
Consider the class of analytic functions defined in open unit disk with normalization condition and which provides the Taylor series expansion of the form
The class consists of functions from which are univalent functions in , and the class contains the analytic functions whose codomains are bounded by the open right half plane. For more details, see [1, 2].
The concept of differential subordination plays a vital role in the study of geometric properties of analytic functions. It was first introduced by Lindelof, but Littlewood [3] did the remarkable work in this field. Many researchers contributed in the study of differential subordinations. History and the development of works in the field related to differential subordination are briefly described and included in the book by Miller and Mocanu [4]. The major development in the field of differential subordination started in 1974 by Miller et al. [5].
An analytic function is considered to be subordinated by analytic function , denoted as , if there exists another analytic function with the property that and such that Moreover, in case of univalent functions in , we can have
Recently, many mathematicians have used this concept of differential subordinations to prove many helpful results. Familiar Jack’s lemma [6] has produced several advancements for the generalization of differential subordinations and found many applications in this field. The work of Ma and Minda [7] in this field is not negligible as they studied the function which is analytic, and condition of normalization given for prescribed function is defined as and with a positive real part. With the help of the function , they introduced the following subclasses for starlike and convex functions.
These subclasses helped many researchers for further studies in the field of differential subordination. Ali et al. [8] used the concept of differential subordination to prove analytic functions to be Janowski starlike. Ali et al. [9] also evaluated several differential subordinations: and found the for . Raina and Sokol [10] used subordinations for coefficient estimation of starlike functions. Similar kinds of works have also been done by Sharma et al. [11] by using starlikeness for cardioid function, and Yunus et al. [12] studied for limacon.
Quantum calculus is the new branch of mathematics and is equally important for its applications both in physics and in mathematics as well. Jackson [13, 14] presented the functions of -derivatives and -integrals and highlighted their definitions for the first time. He also holds the credit for the systematic initiation of -calculus. Ismail et al. [15] were the pioneers to contribute in the application of -calculus in geometric function theory. The new form of the subclass of starlike functions with the involvement of -derivative was introduced by Seoudy and Aouf [16]. By choosing different image domains instead of , so many attractive subclasses of starlike functions are obtained. Mahmood et al. [17] have dealt with the class of -starlike functions by relating them with conic domains. The most recent work related to -starlikeness of functions is done by Srivastava et al. [18]. The contributions of Haq et al. [19] are remarkable. They proved differential subordinations with -analogue for cardioid and limacon domain with the involvement of Janowski function and found the sufficient conditions for -starlike functions. The version of Jack’s lemma which is the soul of our work was given by Çetinkaya and Polatoglu [20]. These recent efforts of mathematicians discussed above motivated us and provide strength to contribute in the field of differential subordinations with the involvement of its -analogue, which is the main idea of this article. The foundation of all this work in -analogue is the -derivative which is defined below.
The -derivative of a complex-valued function , defined in the domain , is given as follows:where This implies the following:provided the function is differentiable in domain The function has Maclaurin’s series representationwhere
For more details about -derivatives and recent work on it, we refer the reader to [21–25].
Definition 1. The function is said to be in the class , if
Lemma 2 (-Jack’s lemma, [20]). Consider an analytic function in with . For a maximum value of on the circle at z, where , and , then, we haveHere, is real and
By using the above lemma, we have proved our main results.
2. Main Results
Theorem 3. Assume thatand we define an analytic function on with which satisfiesIn addition, we suppose thatwhere is analytic in with Then,
Proof. Consider the functionwhich is analytic in with the condition and the functionwhere is an analytic function in with To prove the result, it would be sufficient to show that for
From (14) and (15), we deduce the following:and with this, one can have
This implies that
Now, considering the existence of a point such that
Now, we use the -Jack’s lemma which implies that there exist a number such that This, with the consideration that for , we have
The functionis clearly an even function. So, in order to find the maximum value of , we will consider the interval Thus,gives for and . Also, we can see that for , which results that Now, consider the function
So we have
Thus, is an increasing function which gives a minimum value for Then, we have
From (10), we conclude thatbut this result contradicts (11). Hence, and this leads us to the desired result.
By taking , the above result reduces to the following.
Corollary 4. Let and satisfy the subordinationThen,
Theorem 5. Assume thatand we define an analytic function on with which satisfies
In addition, we suppose thatwhere is analytic in with Then,
Proof. Consider the functionwhich is analytic in with the condition and the functionwhere is an analytic function in with Using (33) and (34), we obtain
Proving the fact that will be sufficient to prove our assertion. For this, consider
Considering the existence of a point such thatwe can make use of -Jack’s lemma which implies that there exists a number such that Now, consider that =, then for , we have
Now, one can easily see that the functionwithis clearly an even function. So, in order to find the maximum value of , we will consider the interval Now, we havefor and Also, we can see that for , thus we conclude that So we have the function
This gives
Thus, is an increasing function which gives a minimum value for . Then, we have
From (29), we conclude thatbut this result contradicts (30). Hence, which provides the required result.
By taking , the above result reduces to the following.
Corollary 6. Let and satisfy the subordinationThen,
Theorem 7. Assume thatand we define an analytic function on with which satisfies
In addition, we suppose thatwhere is analytic in with Then,
Proof. Let us define the functionwhich is analytic in with the condition and the functionwhere is an analytic function in with Using (51) and (52), we get
To prove the assertion, it would be enough to show that Therefore,which after using (9) gives
Now, we consider the functionwhere
As we see that is an even function, so at , and also we see that for . Thus, we conclude that and we get a new functionand we have
So is an increasing function, and it has its minimum value at . Then, we have
Using (47), we getbut this result contradicts (48). Hence, which proves the required result.
By taking , the above result reduces to the following.
Corollary 8. Let and satisfy the subordinationThen,
Theorem 9. Assume thatand we define an analytic function on with which satisfies
In addition, we suppose thatwhere is analytic in with Then,
Proof. Let us define the functionwhich is analytic in with the condition and the functionwhere is an analytic function in with Using (67) and (68), we obtain
To prove the result, we have to show that Therefore,
Hence, by applying (9), we obtain
Now, consider the functionwhere
The above function is clearly an even function. So in order to find its maximum value, we will consider the interval Now, we have for and . Clearly, , and hence, we have obtained the minimum value of at , and thus, we conclude that So now consider the functionwhich gives
Thus, is an increasing function. So for , it gives a minimum value. Then, we have
Using (63), we getbut this result contradicts (64). Hence which proves the required result.
By taking , the above result reduces to the following.
Corollary 10. Let and satisfy the subordinationThen,
Theorem 11. Assume thatand we define an analytic function on with which satisfies
In addition, we suppose thatwhere is analytic in with Then,
We omit the proof of this result as it can be done by using a similar technique as applied in the above results.
By taking , the above result reduces to the following.
Corollary 12. Let and satisfy the subordinationThen,
3. Conclusion
In this article, we have worked on -differential subordinations associated with lemniscate of Bernoulli and defined sufficient conditions for -starlikeness related to cardioid domain. We have also determined the conditions on to prove the starlikeness of prescribed function such asthen
We can use these results to study the sufficiency criteria of other analytic functions.
Data Availability
All data generated or analyzed during this study are included within this article.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
The authors would like to acknowledge the heads of their institutions for their supportive role and research facilities.