Abstract
Let be the class of functions given by which are analytic in the open unit disk For , new integral operators and are considered. The operators and satisfy and for the convolution of and In the present paper, the dominants for both operators and and subordinations for and are discussed. Also, new subclass concerning with different boundary points is defined and discussed. Moreover, some interesting problems of associated with are obtained. Furthermore, some interesting examples for our results are considered.
1. Introduction
The study of operators in geometric function theory started early. Many differential and integral operators began to be studied and introduced around twentieth century by many mathematicians like Alexander [1], Libera [2] (see also [3]), and Bernardi [4] (see also [5]), and they used integral operators early and are known in the GFT; moreover, Ruscheweyh [6] and Salagean [7] used differential operators which are also known in GFT. The Alexander integral operator [1] is an example of them, which was defined by Alexander in 1915. The importance of studying operators in geometric function theory highlights many geometric properties of families of analytic functions such as convexity, starlikeness, coefficient estimates, distortion properties, and subordination and superordination relations; also, [11–14] used the Srivastava-Attiya operator; in addition, [15–17] used integral operators, and furthermore, [18–20] used linear operators.
The new operators we introduce in this paper are generalizations of an extension of the Alexander and Libera integral operators [3].
Let be the class of functions of the form that are analytic in the open unit disk . For , we define where and
The operator for is defined by Güney and Owa [3]. Also, see [21] for related operator of generalized Libera operator in .
For the above operator for we consider the following operators: for
In view of (4) and (7), we have for
Let us consider given by (1) and which is given by
Then, the convolution (or Hadamard product) of and is defined by (see [22, 23]). This convolution shows that for
The function is said to be -valently starlike of order in if and only if for some real We denote the class of such -valently starlike of order in by
If satisfies for some real then is said to be -valently convex of order in We also denote by the class of which are -valently convex of order in
With the above definitions for and we see that
Furthermore, if satisfies for some real then is said to be -valently close-to-convex of order in Also, we write that for such functions.
For the above classes, Owa [24] has shown the following lemmas.
Lemma 1. If satisfies for some then
Lemma 2. If satisfies for some then
Lemma 3. If satisfies for some then
With the above lemmas, we have the following remark.
Remark 4. Let us consider a function given by with For such function we have If then That is, Letting we see Conversely, if satisfies the coefficient inequality (24), then satisfies which shows that Therefore, the coefficient inequality (17) in Lemma 1 is necessary and sufficient condition for of given by (20).
Using the same manner, we have that the coefficient inequality (18) in Lemma 2 is a necessary and sufficient condition for of given by (20). Also, we see that the coefficient inequality (19) in Lemma 3 is a necessary and sufficient condition for of given by (20).
In the next section, we have some interested results associated with the dominants.
2. Dominants
Let us consider a function given by with If given by (1) and given by (26) satisfy then is said to be dominated by (or dominants ). We write this dominant by
It follows from the definition for the dominant that for
We note that the function given by is the function in the class which can be written as
For the function in (31), we know that belongs to the class
Further, we consider a function defined by with This function satisfies
Therefore, and
Now, we derive the following theorems concerning with the coefficient inequalities for .
Theorem 5. If satisfies for some then
Proof. Since Therefore, We have the dominant (39). This implies the proof of theorem.
Theorem 6. If satisfies for some , then
Proof. Using (33), we obtain the dominant (44).
Theorem 7. If satisfies for some , then
Proof. It is clear using (34).
Theorem 8. For if and only if where and
Proof. If satisfies the dominant (47), then Conversely, if satisfies the dominant (48), then we have This gives That is, that This completes the proof of the theorem.
Making in Theorem 8, we have the following.
Corollary 9. For , if and only if where
Further taking in Corollary 9, we have the following.
Corollary 10. For , if and only if where , , and
Using the same manner of Theorem 8, we obtain the following.
Theorem 11. For , if and only if where and
Letting in Theorem 11, we have the following theorem.
Corollary 12. For , if and only if where
Taking in Corollary 12, we get the following corollary.
Corollary 13. For , if and only if where and
Proof. We only need to check that
Applying the function given by (34), for operator , we have the following.
Theorem 14. For , if and only if where and
Proof. We note that Therefore, if satisfies the dominant (65), then Conversely, if satisfies the dominant (66), then This implies that That is, that This completes the proof of the theorem.
Letting in Theorem 14, we see the following.
Corollary 15. For , if and only if where
Similarly, we have the following.
Theorem 16. For , if and only if where and
Proof. If satisfies the dominant (74), then Further, if satisfies the dominant (75), then we have That is, which is the same as the dominant (74).
Making in Theorem 16, we have the following.
Corollary 17. For , if and only if where
Next, we have the following.
Theorem 18. For , if and only if where
Proof. Note that Therefore, Conversely, if satisfies the dominant (82), then That is, that
Making in Theorem 18, we see the following.
Corollary 19 (see [3]). For , if and only if
Remark 20. The function given by is convex in Also, this function maps onto the strip region with
In the following section, we have some subordination relations for functions associated with the operator
3. Subordinations
Let and belong to the class . Then, is said to be subordinate to if there exists a function which is analytic in with and such that We denote this subordination by
If is univalent in , then the subordination (90) for is equivalent that and (see Miller and Mocanu [25]).
For subordinations, Miller and Mocanu [26] have given the following.
Lemma 21. Let be a positive integer, , and let be the root of the equation Also, let If is analytic in with , then satisfies
Using the above lemma, we derive the following.
Theorem 22. If satisfies then where and and are defined in Lemma 21.
Proof. We define a function by Then, is analytic in and It follows that Therefore, using Lemma 21, we have
Letting in Theorem 22, we see the following.
Corollary 23. If satisfies then where and are defined in Lemma 21.
If we take in Lemma 21, then . For such we know the following.
Corollary 24. If satisfies then where .
If we consider in Lemma 21, then we see For such we have the following.
Corollary 25. If satisfies then where and
Taking in Corollary 24, we know the following.
Corollary 26. If satisfies then where .
Remark 27. The function maps onto the domain such that Therefore, a function maps onto the domain such that
Letting in Theorem 22, we see the following.
Corollary 28. If satisfies then where and and are defined in Lemma 21.
In the next section, we get some results for subclasses of analytic functions concerning with different boundary points.
4. A Subclass concerning with Different Boundary Points
Now, we consider different boundary points with . For such boundary points , we define with and
With the above , if satisfies for some , we say that
We see that the condition (111) for the class is equivalent to
Noting that for if we consider a function such that then we have
Now we introduce the following lemma by Miller and Mocanu [25, 27] (also, due to Jack [28]).
Lemma 29. Let the function given by be analytic in with . If attains its maximum value on the circle at a point , then there exists a real number such that
Using the above lemma, we derive the following.
Theorem 30. If satisfies for some given by (110) with such that and for some real , then where and and This implies that
Proof. Define a function by Then, is analytic in and . Since we know that by (119). Suppose that there exists a point such that Then, applying Lemma 29, we write that and This shows that Since this contradicts our condition (119), we say that there is no such that . Letting for all we obtain This implies that
Taking in Theorem 30, we have the following.
Corollary 31. If satisfies for some given by (110) with such that and for some real , then where that is that
Example 1. Consider a function It follows that We consider five boundary points such that For these points we know that It follows from the above that Thus, we see that with . Using such and , we consider with That is, that With the above and , the function satisfies
Taking in Theorem 30, we have the following.
Corollary 32. If satisfies for some given by (110) with such that and for some real , then where and
Taking in Theorem 30, we have the following.
Corollary 33. If satisfies for some given by (110) with such that and for some real , then where that is,
Example 2. We consider a function given by (130). Then, satisfies We also consider five boundary points given by (132)–(136). Since we have with With the above and we consider such that Then, satisfies Using and , we have that
Next, our theorem is the following.
Theorem 34. If satisfies for some given by (110) with such that and for some real , then This means that
Proof. Define a function by Then, is analytic in , and is given by (121). Noting that We know that Consider that there exists a point such that Then, Lemma 29 gives us that and It follows that where Since this contradicts our condition (153), there is no such that This means that for all Thus, we have This completes the proof of the theorem.
Finally, we consider the coefficient problem for the class
Theorem 35. Let satisfies for ; then, where is given by (110) with and
Proof. With defined by (162), we know that
Remark 36. From Theorem 35, we know that the function with for is a member of the class
5. Conclusion
In the present paper, we defined new integral operators and of analytic functions The dominants for both operators and and subordinations for and are discussed. Also, new subclass concerning with different boundary points is defined and discussed. Moreover, some interesting problems of associated with are obtained. Furthermore, some interesting examples for our results are considered.
Data Availability
Data used to support the findings of this study are included within the article.
Disclosure
A little portion of the results in this article were presented in GFTA 2021, 15-18 October 2021, Sibiu (http://gfta2021.uab.ro/upls/Abstract.pdf).
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
The authors contributed equally to the writing of this paper. All authors approved the final version of the manuscript.