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

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.


Preliminary Concepts
Let the family of holomorphic (or analytic) functions in the region (or domain) of unit disc D = fz ∈ ℂ : jzj < 1g be described by the symbol H ðDÞ and let A be the subfamily of H ðDÞ which is defined by Further, the set S ⊂ A contains all normalized univalent functions in D. For two functions F 1 , F 2 ∈ H ðDÞ, we say that F 1 is subordinate to F 2 , written symbolically by F 1 ≺ F 2 , if there exists a Schwarz function v with vð0Þ = 0 and jvðzÞj < 1 that is analytic in D such that f ðzÞ = gðvðzÞÞ,z ∈ D. However, if F 2 is univalent in D, then the following relation holds: In geometric function theory, the most basic and important subfamilies of the set S are the family S * of starlike functions and the family C of convex functions which are defined as follows: is described by the functions of the Janowski starlike family established in [1] and later studied in different directions in [2,3] (ii) The family S * L ≡ S * ðΛðzÞÞ with ΛðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p was developed in [4] by Sokól and Stankiewicz. The image of the function ΛðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p demonstrates that the image domain is bounded by the Bernoulli's lemniscate right-half plan specified by |w 2 − 1 | <1 (iii) By selecting ΛðzÞ = 1 + sin z, the class S * ðΛðzÞÞ lead to the family S * sin which was explored in [5] while S * e ≡ S * ðe z Þ has been produced in the article [6] and later studied in [7] (iv) The family S * c ≔ S * ðΛðzÞÞ with ΛðzÞ = 1 + ð4/3Þ z + ð2/3Þz 2 was contributed by Sharma and his coauthors [8] which contains function f ∈ A such that z f ′ðzÞ/f ðzÞ is located in the region bounded by the cardioid given by (v) The family S * R ≡ S * ðΛðzÞÞ with ΛðzÞ = 1 + ðz/ð√2 + 1ÞÞðð√2 + 1 + zÞ/ð√2 + 1 − zÞÞ is studied in [9] while S * cos ≔ S * ðcos ðzÞÞ and S * cosh ≔ S * ðcosh ðzÞÞ were recently examined by Bano and Raza [10] and Alotaibi et al. [11], respectively (vi) If we consider ΛðzÞ = 1 sinh −1 z, then the class S * ρ ≔ S * ð1 + sinh −1 zÞ was provided by Kumar and Arora [12] and is defined as a function f ∈ A which is in the family S * ρ if (18) holds for the function ΛðzÞ = ρðzÞ, where Clearly, the function ρðzÞ is a multivalued function and has the branch cuts about the line segments ð−i∞,−iÞ ∪ ði, i∞Þ, on the imaginary axis, and hence, it is holomorphic in D: In a geometric point of view, the function ρðzÞ maps the unit disc D onto a petal-shaped region Ω ρ , Using this idea, we now consider a subfamily BT s of analytic functions as If we take the function ΛðzÞ, given by (4), instead ofΛðzÞ in (9), we get the familiar class R of bounded turning functions. From the statement of the Nashiro-Warschowski theorem, it follows that the functions in R are univalent in D. The properties of this class was studied extensively by the researchers, see [13][14][15][16].
The Hankel determinant HD q,n ð f Þðwith q, n ∈ ℕ = f1, 2, ⋯ g and a 1 = 1Þ for a function f ∈ S 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 f belongs to the general family S. For the function f ∈ S, the best established sharp inequality is jHD 2,n ð f Þj ≤ λ ffiffiffi n p , where λ is absolute constant, which is due to Hayman [19]. Further, for the same class S, it was obtained in [20] that The growth of jHD q,n ð f Þj has often been evaluated for different subfamilies of the set S of univalent functions. For example, the sharp bound of jHD 2,2 ð f Þj, for the subfamilies C, S * , and R of the set S, was measured by Janteng et al. [21,22]. These bounds are 2 Journal of Function Spaces 1, for f ∈ S * , 4 9 , for f ∈ R: 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 Bazilevic functions, the best estimate of jHD 2,2 ðf Þj was proved by Krishna and RamReddy [24]. For more work on jHD 2,2 ð f Þj, see References [25][26][27][28][29].
It is very obvious from the formulae provided in (11) that the estimate of jHD 3,1 ð f Þj is far more complicated compared with finding the bound of jHD 2,2 ð f Þj. In the first paper on jHD 3,1 ð f Þj, published in 2010, Babalola [30] obtained the upper bound of jHD 3,1 ðf Þj for the families of C, S * , and R. He obtained the following bounds: 16, for f ∈ S * , 0:742 ⋯ , for f ∈ R: Later on, using the same methodology, some other authors [31][32][33][34][35] published their work concerning jHD 3,1 ð f Þj 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 1, for f ∈ S * , 41 60 , forf ∈ R: He argues that such limits are indeed not the best ones. After that, in 2018, Kwon et al. [37] enhanced Zaprawa's bound for f ∈ S * and showed that jHD 3,1 ð f Þj ≤ 8/9, but it is still not the best possible. The firstly examined papers in which the authors obtained the sharp bounds of jHD 3,1 ð f Þj 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 S * ð1/2Þ indicates the starlike function family of order 1/2: We would also like to acknowledge the research provided by Mahmood et al. [40] in which they examined the third Hankel determinant in the q-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 determi-nant for the class BT s of bounded turning functions connected with a petal-shaped domain.

A Set of Lemmas
Definition 1. Let P represent the class of all functions p that are holomorphic in D with ReðpðzÞÞ > 0 and has the series representation For the proofs of our key findings, we need the following lemma. It contains the well-known formula for c 2 , see [43], the formula for c 3 due to Libera and Zlotkiewicz [44], and the formula for c 4 proved in [45].

Coefficient Inequalities for the Class BT s
We begin this section by finding the absolute values of the first three initial coefficients for the function of class BT s :

Journal of Function Spaces
Theorem 4. If f ∈ BT s and has the series representation ((1)), then These bounds are sharp.
Proof. Let f ∈ BT s : Then, (9) can be written in the form of the Schwarz function as Now, if p ∈ P , then it may be written in terms of the Schwarz function w by equivalently, From (1), we easily get f ′ z ð Þ = 1 + 2a 2 z + 3a 3 z 2 + 4a 4 z 3 + 5a 5 z 4 +⋯: By simplification and using the series expansion (28), we obtain Comparing (29) and (30),we get For a 2 , implementing (22) in (31), we obtain For a 3 , reordering (32), we get and using (21), we have For a 4 , 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 f n ′ z ð Þ = 1 + sinh −1 z n ð Þ, for n = 1, 2, 3: 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. If f of the form ((1)) belongs to BT s , then This inequality is sharp.

Journal of Function Spaces
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. If f has the form ((1)) belongs to BT s , then a 2 a 3 − a 4 j j≤ This 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 HD 2,2 ðf Þ for f ∈ BT s : Theorem 7. If f belongs to BT s , then the second Hankel determinant This result is the best possible.
Proof. From (31), (32), and (33), we have Using (18) and (19) to express c 2 and c 3 in terms of c 1 and noting that without loss in generality we can write c 1 = c, with 0 ≤ c ≤ 2, we obtain with the aid of the triangle inequality and replacing jzj ≤ 1, j xj = b, with b ≤ 1: So, It is a simple exercise to show that ϕ ′ ðc, bÞ ≥ 0 on ½0, 1, so that ϕðc, bÞ ≤ ϕðc, 1Þ: Putting b = 1 gives Also, ϕ ′ ðc, 1Þ < 0, and so ϕðc, 1Þ is a decreasing function. Thus, the maximum value at c = 0 is The required second Hankel determinant is sharp and is obtained by

Third-Order Hankel Determinant
We will now determine the third-order Hankel determinant HD 3,1 ðf Þ for f ∈ BT s .

Journal of Function Spaces
Theorem 8. If f belongs to BT s , then the third Hankel determinant This result is sharp.

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.