Abstract
In this paper, by using the concept of the symmetric -difference operator, we introduce certain classes of symmetric -starlike and symmetric -convex functions. Convolution results, coefficient estimates, and Fekete–Szegö inequalities for the analytic functions belonging to these classes are obtained.
1. Introduction
Let us represent the class of analytic (or holomorphic or regular) functions in by and suppose that is the subclass of defined as follows:
Further, let and denote the subclasses of that consist, respectively, of starlike of order and convex of order in (see [1]). For two given regular functions , we say that is subordinate to or is superordinate to , written symbolically as if there exists a Schwarz function , which (by definition) is regular in with and for all , such that
Moreover, if is univalent function in , then the following equivalence holds true (see [2, 3]):
For functions given by (1) and given bythe convolution or Hadamard product of and is defined by
Let and denote the subclasses of for which are defined by (see [4–9])
We note that and .
For , the symmetric -difference of a function is defined as follows (see [10]):and provided that is differentiable at 0. From (1) and (7), we deduce thatwhere is the -number given by
Furthermore, if and are the two functions, thenwhere are constants.
Making use of the symmetric -difference given by (7), we introduce the subclasses and of for and as follows:
We also note that(i) and (see [6, 7, 9, 11, 12]);(ii), and (see [1, 13, 14]).(iii), and (see [1, 8, 15]).(iv), and (see [16]).(v),and (see [16]).
In the present article, our aim is to investigate convolution properties, coefficient estimates, and Fekete–Szegö inequalities for the classes and . The motivation of this article is to generalize and improve previously known results.
2. Convolution Results and Coefficient Estimates
Unless otherwise mentioned, we assume throughout this investigation that
Theorem 1. If has the series form (1), then if and only ifwhere is given by
Proof. It is easy to check thatIn order to prove that equation (20) holds, we will write (12) by using the definition of the subordination as follows:where is Schwarz function; hence, we havewhich is equivalent toBy using (22) in (25), we obtainwhich proves the necessary condition (20) for Theorem 1.
Reversely, suppose that satisfies condition (20). Since it was shown in the first part of the proof that assumption (20) is equivalent to (25), we obtain thatIf we denoterelation (27) means thatThus, the simply connected domain is included in a connected component of . Therefore, using the fact that and the univalence of the function , it follows that , which implies that . This completes the proof of Theorem 1.
Putting and in Theorem 1, we get the following convolution result for .
Corollary 1. If has the series form (1), then if and only ifwhere is given by
Theorem 2. If has the series form (1), then if and only ifwhere is given by (21).
Proof. From relation (14), we haveThen, according to Theorem 1, we obtainwhere is given byand we note thatUsing relation (36) and the following identity:it is easy to check that (34) is equivalent to (32). Thus, the proof of Theorem 2 is completed.
Putting and in Theorem 2, we obtain the following result for .
Corollary 2. If has the series form (1), then if and only ifwhere is given by (31).
Theorem 3. If has the series form (1), then if and only if
Proof. From Theorem 1, we find that if and only iffor all given by (21). The left hand side of (38) can be written asThus, the proof of Theorem 3 is completed.
Putting and in Theorem 3, we obtain the following result for .
Corollary 3. If has the series form (1), then if and only if
Theorem 4. If has the series form (1), then if and only if
Proof. From Theorem 1, we find that if and only iffor all given by (21). The left hand side of (44) can be written asand this proves Theorem 4.
Putting and in Theorem 4, we obtain the following convolution result for .
Corollary 4. If has the series form (1), then if and only if
Theorem 5. If the function has the series form (1) satisfying the inequalitythen .
Proof. Hence,Thus, inequality (47) holds, and our result follows from Theorem 3.
Putting and in Theorem 5, we obtain the following result for .
Corollary 5. If the function has the series form (1) satisfying the inequalitythen .
By using arguments and analysis to those in the proof of Theorem 5, we can derive the following theorem.
Theorem 6. If the function has the series form (1) satisfying the inequalitythen .
Putting and in Theorem 6, we obtain the following result for .
Corollary 6. If the function has the series form (1) satisfying the inequalitythen .
3. Fekete–Szegö Inequalities
In this section, we obtain the Fekete–Szegö inequalities for the classes and . In order to establish our results, we need the following lemmas.
Lemma 1 (see [17]). If is a function with positive real part in and is a complex number, then
The result is sharp for the functions given by
Lemma 2 (see [17]). If is an analytic function with a positive real part in , then
When or , the equality holds if and only if is or one of its rotations. If , then the equality holds if and only if is or one of its rotations. If , the equality holds if and only ifor one of its rotations. If , the equality holds if and only if is the reciprocal of one of the functions such that equality holds in the case of .
Also, the above upper bound is sharp, and it can be improved as follows when :
Theorem 7. If defined by (1) belongs to the class , then
Proof. If , then there is a Schwarz function in such thatDefine the function bySince is a Schwarz function, we see that and . Now, by substituting (59) in (58), we haveFrom the above equation, we obtainor, equivalently,Therefore, we havewhereOur result now follows from Lemma 1. This completes the proof of Theorem 1.
Similarly, we can prove the following theorem for the class .
Theorem 8. If given by (1) belongs to the class , then
The result is sharp.
Putting and in Theorems 7 and 8, we obtain the following corollaries.
Corollary 7. If given by (1) belongs to the class , then
Corollary 8. If given by (1) belongs to the class , then
The results are sharp.
Theorem 9. Let
If given by (1) belongs to the class , then
Further, if , then
If , then
Proof. Applying Lemma 2 to (63) and (64), we can obtain our results asserted by Theorem 9.
Putting and in Theorem 9, we obtain the following result.
Corollary 9. Let
If given by (1) belongs to the class , then
Further, if , then
If , then
Similarly, we can obtain the following theorem.
Theorem 10. Let
If given by (1) belongs to the class , then
Further, if , then
If , then
Putting and in Theorem 10, we obtain the following result.
Corollary 10. Let
If given by (1) belongs to the class , then
Further, if , then
If , then
4. Conclusion
In this present investigation, we have introduced two classes and of analytic functions by using the symmetric -difference operator linked to an open unit disc . We also studied convolution results, coefficient estimates, and Fekete–Szegö inequalities for the newly defined classes. We note that our results naturally include several results that are known for those subclasses, which are listed in the introduction section.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The author would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: (22UQU4350561DSR01).