Sharp Bounds of the Coefficient Results for the Family of Bounded Turning Functions Associated with a Petal-Shaped Domain

Abstract

The goal of this article is to determine sharp inequalities of certain coefficient-related problems for the functions of bounded turning class subordinated with a petal-shaped domain. These problems include the bounds of first three coefficients, the estimate of Fekete-Szegö inequality, and the bounds of second- and third-order Hankel determinants.

1. Preliminary Concepts

Let the family of holomorphic (or analytic) functions in the region (or domain) of unit disc be described by the symbol and let be the subfamily of which is defined by

Further, the set contains all normalized univalent functions in . For two functions , we say that is subordinate to , written symbolically by , if there exists a Schwarz function with and that is analytic in such that . However, if is univalent in , then the following relation holds:

In geometric function theory, the most basic and important subfamilies of the set are the family of starlike functions and the family of convex functions which are defined as follows: with

By varying the function in (18), we get some subfamilies of the set which have significant geometric sense. For example, (i)If we take with , then the deduced familyis described by the functions of the Janowski starlike family established in [1] and later studied in different directions in [2, 3] (ii)The family with was developed in [4] by Sokól and Stankiewicz. The image of the function demonstrates that the image domain is bounded by the Bernoulli’s lemniscate right-half plan specified by (iii)By selecting , the class lead to the family which was explored in [5] while has been produced in the article [6] and later studied in [7](iv)The family with was contributed by Sharma and his coauthors [8] which contains function such that is located in the region bounded by the cardioid given by(v)The family with is studied in [9] while and were recently examined by Bano and Raza [10] and Alotaibi et al. [11], respectively(vi)If we consider , then the class was provided by Kumar and Arora [12] and is defined as a function which is in the family if (18) holds for the function , where

Clearly, the function is a multivalued function and has the branch cuts about the line segments , on the imaginary axis, and hence, it is holomorphic in In a geometric point of view, the function maps the unit disc onto a petal-shaped region

Using this idea, we now consider a subfamily of analytic functions as

If we take the function , given by (4), instead of in (9), we get the familiar class of bounded turning functions. From the statement of the Nashiro-Warschowski theorem, it follows that the functions in are univalent in . The properties of this class was studied extensively by the researchers, see [13–16].

The Hankel determinant for a function of the series form (1) was given by Pommerenke [17, 18] as

Specifically, the first-, second-, and third-order Hankel determinants, respectively, are

In literature, there are relatively few findings in relation to the Hankel determinant for the function belongs to the general family . For the function , the best established sharp inequality is , where is absolute constant, which is due to Hayman [19]. Further, for the same class , it was obtained in [20] that

The growth of has often been evaluated for different subfamilies of the set of univalent functions. For example, the sharp bound of , for the subfamilies , , and of the set , was measured by Janteng et al. [21, 22]. These bounds are

The exact bound for the collection of close-to-convex functions of such a specific determinant is still unavailable (see [23]). On the other hand, for the set of Bazilevi functions, the best estimate of was proved by Krishna and RamReddy [24]. For more work on , see References [25–29].

It is very obvious from the formulae provided in (11) that the estimate of is far more complicated compared with finding the bound of . In the first paper on , published in 2010, Babalola [30] obtained the upper bound of for the families of , , and . He obtained the following bounds:

Later on, using the same methodology, some other authors [31–35] published their work concerning for different subfamilies of analytic and univalent functions. In 2017, Zaprawa [36] improved Babalola’s [30] results by applying a new technique which is given as

He argues that such limits are indeed not the best ones. After that, in 2018, Kwon et al. [37] enhanced Zaprawa’s bound for and showed that , but it is still not the best possible. The firstly examined papers in which the authors obtained the sharp bounds of came to the reader’s hands in 2018. Such papers have been written by Kowalczyk et al. [38] and Lecko et al. [39]. These results are given as where indicates the starlike function family of order We would also like to acknowledge the research provided by Mahmood et al. [40] in which they examined the third Hankel determinant in the -analog for a subfamily of starlike functions and for more contribution of such type families, see [41, 42]. In the present article, our aim is to calculate the sharp bounds of some of the problems related to Hankel determinant for the class of bounded turning functions connected with a petal-shaped domain.

2. A Set of Lemmas

Definition 1. Letrepresent the class of all functionsthat are holomorphic inwithand has the series representation

For the proofs of our key findings, we need the following lemma. It contains the well-known formula for , see [43], the formula for due to Libera and Zlotkiewicz [44], and the formula for proved in [45].

Lemma 2. Lethas the series form ((17)). Then, for,

Lemma 3. Ifand has the series form ((17)), thenwith and for with , we have

The inequalities (21), (22), (23), and (24) in the above lemma are taken from [43, 46], [47–49], and [50], respectively.

3. Coefficient Inequalities for the Class

We begin this section by finding the absolute values of the first three initial coefficients for the function of class

Theorem 4. Ifand has the series representation ((1)), thenThese bounds are sharp.

Proof. Let Then, (9) can be written in the form of the Schwarz function as Now, if , then it may be written in terms of the Schwarz function by equivalently, From (1), we easily get By simplification and using the series expansion (28), we obtain Comparing (29) and (30),we get For , implementing (22) in (31), we obtain For , reordering (32), we get and using (21), we have For , we can rewrite (33) as Application of triangle inequality plus (23), we get By simple calculations, we obtain These outcomes are sharp. For this, we consider a function Thus, we have

Now, we discussed about the Hankel determinant problem, which is explicitly related to the Fekete-Szegö functional which is an extraordinary instance of the Hankel determinant.

Theorem 5. Ifof the form ((1)) belongs tothenThis inequality is sharp.

Proof. Employing (31) and (32), we may write By rearranging, it yields Application of (21) leads us to After the simplification, we obtain The required result is sharp and is determined by

Theorem 6. Ifhas the form ((1)) belongs to, thenThis inequality is sharp.

Proof. Using (31), (32), and (33), we have From (24), we have and also satisfy Thus, by using (24), we have Equality is achieved by using

Next, we will determine the second-order Hankel determinant for

Theorem 7. Ifbelongs to, then the second Hankel determinantThis result is the best possible.

Proof. From (31), (32), and (33), we have Using (18) and (19) to express and in terms of and noting that without loss in generality we can write , with , we obtain with the aid of the triangle inequality and replacing , , with So, It is a simple exercise to show that on , so that Putting gives Also, and so is a decreasing function. Thus, the maximum value at is The required second Hankel determinant is sharp and is obtained by

4. Third-Order Hankel Determinant

We will now determine the third-order Hankel determinant for .

Theorem 8. Ifbelongs to, then the third Hankel determinantThis result is sharp.

Proof. The third Hankel determinant can be written as After simplification of the above equation, we have Let and putting the estimations of ’s from (31), (32), (33), and (34), we get To simplify computation, let in (18), (19), and (20). Now, using the simplified form of these formulae, we obtain Substituting these expressions in (65), by simple but too long computation, Since where and Now, by using and utilizing the fact , we get where with Clearly, in the last four functions, and are nonnegative in the interval So from (70) along with in the interval we get Therefore, Here, we shall maximize over the interval For this purpose, we consider possible cases: (i)By taking we haveSince in , so, is decreasing over Thus, has its maxima at which is equal to 34560 (ii)By taking , we haveAs has its optimum point at Therefore, is an increasing function for and decreasing for Hence, has its maxima at that is approximately equal to 22764.68167 (iii)By taking , we haveAs over , so, is decreasing over Thus, has its maxima at which is equal to 34560. Now, by taking , we obtain (iv)When lies in Then, some simple computation shows that there exists real solution for these equationslies inside this region at Consequently, we obtain Thus, from all the above cases, we conclude that From (71), we can write If , then the equality is obtained from the function

5. Conclusion

For the family of bounded turning functions connected with a petal-shaped domain, we studied the problems such as the bounds of the first three coefficients, the estimate of the Fekete-Szegö inequality, and the bounds of Hankel determinants of order three. All the bounds which we investigated are sharp.

Data Availability

We have not used any data.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed equally in this research paper.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. DF-762-830-1441. Therefore, the authors gratefully acknowledge the DSR technical and financial support.