Abstract
In recent years, the theory of operators got the attention of many authors due to its applications in different fields of sciences and engineering. In this paper, making use of the Bernardi integral operator, we define a new class of starlike functions associated with the sine functions. For our new function class, extended Bernardi’s theorem is studied, and the upper bounds for the fourth Hankel determinant are determined.
1. Introduction
Let be the family of holomorphic (or analytic) functions in , and such that has the series representation:
Let be the subfamily of containing univalent functions in . Despite the fact that function theory was first proposed in 1851, it only became a viable research topic in 1916. Many academics attempted to prove or refute the conjecture , which was initially proven by de Branges in 1985, and as a result, they identified multiple subfamilies of the class that are connected to various image domains. The starlike, convex, and close-to-convex functions are among those families which are defined by
Let and be the two analytic functions in ; then, is subordinate to , denoted by if there exists a Schwarz function satisfying the conditions: such that
Let be an analytic and regular function in with , satisfying the criteria:
Then, this function is referred to as the Janowski function which is represented by . Geometrically, , and is inside the domain specified by having diameter end points:
Let be the class of functions , where are holomorphic in and meet the following requirements:
Distinct subclasses of analytic functions associated with various image domains have been introduced by many scholars. For example, Cho et al. [1] and Dziok et al. [2] discussed various properties of starlike functions related to Bell numbers and a shell-like curve connected with Fibonacci numbers, respectively. Similarly, Kumar and Ravichandran [3] and Mendiratta et al. [4] investigated subclasses of starlike functions associated with rational and exponential functions, respectively. Kanas and Raducanu [5] and Sharma et al. [6] explored some subclasses of analytic functions related to conic and cardioid domains, respectively. Raina and Sokól [7] investigated some important properties related to a certain class of starlike functions. Sokól and Stankiewicz [8] discussed radius problems of some subclasses of strongly starlike functions. Recently, Cho et al. [9] explored a family of starlike functions related to the sine function, which is defined as follows:
The th Hankel determinant for and of the functions is introduced by Noonan and Thomas [10], which is given by
The following options are provided for some special choices of and : (1)For ,is the famed Fekete-Szegő functional. (2)For ,is the second Hankel determinant.
There are relatively few findings in the literature in connection with the Hankel determinant for functions belonging to the general family . Hayman [11] established the well-known sharp inequality: where is the absolute constant. Similarly for the same class , it was obtained in [12] that
For different subclasses of the set of univalent functions, the growth of has been estimated many times. For example, Janteng [13] investigated the sharp bounds of for the classes , , and as given below:
The sharp bound of for the class of close-to-convex functions is unknown. On the other hand, Krishna and Reddy [14] calculated a precision estimate of for the Bazilevic function class. (3)For ,is the third Hankel determinant.
The calculations in (17) represent that estimating is significantly more difficult than estimating the bound of . In the first paper of Babalola [15] on , he obtained the upper bound of for the classes , , and . Later, some more contributions have been made by different authors to calculate the bounds of for different subclasses of analytic and univalent functions. Zaprawa [16] enhanced the results of Babalola [15] and demonstrated that
He also observed that the bounds are still not sharp. (4)For ,is the fourth Hankel determinant.
Since and , thus , where
In the last few years, many articles have been published to investigate the upper bounds for the second-order Hankel determinant , third-order Hankel determinant , and fourth Hankel determinant . For the functions with bounded turning, Arif et al. [17, 18] estimated the bound for the fourth- and fifth-order Hankel determinants. Khan et al. [19] also addressed this issue and derived upper bounds for the third- and fourth-order Hankel determinants for a class of functions with bounded turning that are related to sine functions. For more study about the Hankel determinant, we refer to [19–29].
In geometric function theory (GFT), especially in the category of univalent functions, integral and differential operators are extremely helpful and important. Convolution of certain analytic functions has been used to introduce certain differential and integral operators. This approach is developed to facilitate further exploration of geometric features of analytic and univalent functions. Libera and Bernardi were the ones who investigated the classes of starlike, convex, and close-to-convex functions by introducing the idea of integral operators. Recently, some researchers have shown a keen interest in this field and developed various features of the integral and differential operators. Srivastava et al. [30] investigated a new family of complex-order analytic functions by using the fractional -calculus operator. Mahmood et al. [31] looked at a group of analytic functions that were defined using -integral operators. Using the -analogue of the Ruscheweyh-type operator, Arif et al. [32] constructed a family of multivalent functions. Srivastava [33] presented a review on basic (or -) calculus operators, fractional -calculus operators, and their applications in GFT and complex analysis. This review article has been proven very helpful to investigate some new subclasses from different viewpoints and perspectives [34–40].
Inspired from the above recent developments, in this study, we investigate the inclusion of the Bernardi integral operator in the class of starlike function associated with sine function in . The Bernardi integral operator was defined by Bernardi [41], which is given by the following relation:
In the first part of the study, we extend Bernardi’s theorem to a certain class of univalent starlike functions in . Particularly, we prove that if , then . In the second part of the study, we investigate the upper bounds for the fourth-order Hankel determinant with respect to the function class associated with the sine function.
2. Main Results
In order to obtain our desired results, we first need the following lemmas.
Lemma 1. Let and be holomorphic functions in such that maps onto many sheeted starlike regions with and If , then
Proof. We know that Also, maps onto the disc . But takes values in the same disc, and therefore, Choose so that Then, Fix in . Let be the segment joining and , which lies in one sheet of the starlike image of by Let be the preimage of under Then, That is, This implies that and hence,
Lemma 2 (see [12]). Let and be regular in and map onto many sheeted starlike regions: Then,
Lemma 3. Let such that Then, is -valent starlike for in .
Proof. The proof is analogous to the one given in [41] and hence omitted.
Lemma 4. Let and for where is given by (31) in Lemma 3. Then, .
Proof. Let Then, where By Lemma 3, is -valent starlike for in : and since , From Lemma 1, we can get the conclusion:
Lemma 5 (see [42]). If , then there exists some with , such that
Lemma 6 (see [43]). Let ; then,
Lemma 7 (see [44]). Let ; then,
Now, we are in position to present our main results.
Theorem 8. If the function and is of the form (1), then
Proof. Since , according to the definition of subordination, there exists a Schwarz function with and such that
Now,
where . We define a function:
It is easy to see that and
On the other hand,
When the coefficients of are compared between the equations (51) and (48), then we get
Using Lemma 6, we can simply obtain
with and
If , then by comparing like powers of , we have
For sharpness, if we take
and thus , , and , then . This shows that the obtained second coefficient bound is sharp.
Again, by Lemma 6,
Let , with ; then, by Lemma 7, we can get
Now, suppose that
Then obviously,
Setting we can get , and hence, the maximum value of is given by
Also,
Let , with ; then, again by Lemma 7,
Assume that
Obviously, we meet the requirement:
So the function attains its maximum value at , and it is given by
Next,
Take , with ; then, according to Lemma 7, we have
Suppose that
Then obviously,
We see that , and we get the maximum value at :
Finally,
Again, taking , with , and using the result of Lemma 7, we can obtain
Let
Then obviously, . As a result, the function attains its maximum value at . Hence,
Theorem 9. If the function and is of the form (1), then we have
Proof. From (52), we can write Using Lemma 5, we get We suppose that and . Also, if we apply the triangle inequality to the above equation, then we get Assume that Obviously, we can write is increasing on . Therefore, at , the function will obtain its maximum value: Let us take It is clear that is decreasing on . So at , the function will obtain its maximum value: This complete the proof.
Theorem 10. If the function and is of the form (1), then we have
Proof. From (52), we can write From Lemma 5, we can deduce that We suppose that and . Once again, if we apply the triangle inequality to the above equation, then we get Suppose that Then, we get The above expression shows that is a decreasing function about on the closed interval . This implies that will attain its maximum value at , which is Now define Since , the function has maximum value at . That is, and this completes the proof.
Theorem 11. If the function and is of the form (1), then we have
Proof. Again from (52), we can write Using the result of Lemma 5, we can obtain Also, by Lemma 7, we have where Clearly is an increasing function about on the closed interval . This means that will attain its maximum value at , which is . Thus,
Theorem 12. If the function and is of the form (1), then we have
Proof. From (52) and (53), we have Using the result of Lemma 7, we can write where Obviously, For , we can get and consequently As , the maximum value at is Also, where attains its maximum value at , so Using the results of (109) and (111) in (106), we can get
Theorem 13. If the function and is of the form (1), then we have
Proof. From (52) and (53), we have Letting and and using the results of Lemmas 6 and 7, we get Suppose that where We see that and the maximum value of can be attained at , which is Also, If we set , then we get . Consequently, As , the maximum value at is given by Hence,
Theorem 14. If the function and is of the form (1), then we have
Proof. From (52) and (53), we have Now, using the results of Lemmas 6 and 7, we obtain where It is clear that is an increasing function, so it attains its maximum value at , which is Also, for all , we have When we set , then we get . Obviously, and it attains its maximum value at , which is given by Hence, which completes the proof of Theorem 14.
Theorem 15. If the function and is of the form (1), then we have
Proof. We know that which further implies that Using the triangle inequality, we can write By substituting the results of (41), (80), (89), (98), (104), (113), (120), and (128) in (132), we can get the desired result in (129).
3. Conclusion
In the present investigation, first, we have extended the well-known Bernardi theorem to a specific class of univalent starlike functions in the open unit disk . We have proven that if is a starlike univalent function in the unit disk and , then , where
Additionally, we have estimated the upper bounds of the fourth-order Hankel determinant for the functions class associated with the sine function systematically.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors equally contributed to this manuscript and approved the final version.