Some Sharp Results on Coefficient Estimate Problems for Four-Leaf-Type Bounded Turning Functions

In this study, we focused on a subclass of bounded turning functions that are linked with a four-leaf-type domain. The primary goal of this study is to explore the limits of the ﬁ rst four initial coe ﬃ cients, the Fekete-Szegö type inequality, the Zalcman inequality, the Kruskal inequality, and the estimation of the second-order Hankel determinant for functions in this class. All of the obtained ﬁ ndings have been sharp.


Introduction and Definitions
Before getting into the key findings, some prior information on function theory fundamentals is required. In this case, the symbols A and S indicate the families of normalised holomorphic and univalent functions, respectively. These families are specified in the set-builder form: where QðU d Þ stands for the set of analytic (holomorphic) functions in the disc U d = fz ∈ ℂandjzj < 1g: Thus, if g ∈ A, then, it can be stated in the series expansion form by For the given functions G 1 , G 2 ∈ QðU d Þ, the function G 1 is subordinated by G 2 (stated mathematically by G 1 ≺ G 2 ) if there exists a holomorphic function v in U d with the restrictions vð0Þ = 0 and jvðzÞj < 1 such that G 1 ðzÞ = G 2 ðvðzÞÞ: Moreover, if G 2 is univalent in U d , then Although the function theory was created in 1851, Biberbach [1] presented the coefficient hypothesis in 1916, and it made the topic a hit as a promising new research field. De-Brages [2] proved this conjecture in 1985. From 1916 to 1985, many of the world's most distinguished scholars sought to prove or disprove this claim. As a result, they investigated a number of subfamilies of the class S of univalent functions that are associated with various image domains [3][4][5]. The most fundamental and significant subclasses of the set S are the families of starlike and convex functions, represented by S * and K, respectively. Ma and Minda [6] defined the unified form of the family in 1992 as where ϕ indicates the analytic function with ϕ ′ ð0Þ > 0 and Reϕ > 0. Also, the region ϕðU d Þ is star-shaped about ϕð0Þ = 1 and is symmetric along the real axis. They examined some interesting aspects of this class. Some significant subfamilies of the collection A have recently been investigated as unique instances of the class S * ðϕÞ. In particular; (i) The class S * ½L, M ≡ S * ð1 + Lz/1 + MzÞ, −1 ≤ M < L ≤ 1, is obtained by selecting ϕðzÞ = 1 + Lz/1 + Mz and was established in [7]. Moreover, S * ðξÞ ≔ S * ½1 − 2ξ,−1 displays the well-known order ξ (0 ≤ ξ < 1) starlike function class (ii) The class S * L ≔ S * ðϕðzÞÞ with ϕðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p was designed by the researchers Sokól and Stankiewicz in [8]. Also, they showed that the image of the function ϕðzÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p is bounded by jw 2 − 1j < 1: (iii) The set S * car ≔ S * ðϕðzÞÞ with ϕðzÞ = 1 + 4/3z + 2/3 z 2 has been deduced by Sharma and his coauthors [9] in which they located the image domain of ϕðzÞ = 1 + 4/3z + 2/3z 2 , which is bounded by the below cardioid 9x 2 + 9y 2 − 18x + 5 À Á 2 − 16 9x 2 + 9y 2 − 6x + 1 À Á = 0: ð6Þ (iv) By selecting ϕðzÞ = 1 + sin z, we get the class S * ðϕ ðzÞÞ = S * sin , which was defined in [10] while S * e ≡ S * ðe z Þ was contributed by the authors [11] and, subsequently, explored some more properties of it in [12]. This class was recently generalized by Srivastava et al. [13] in which the authors determined upper bound of Hankel determinant of order three (v) The family S * cos ≔ S * ðcos ðzÞÞ and S * cosh ≔ S * ðcosh ðzÞÞ were offered, respectively, by Raza and Bano [14] and Alotaibi et al. [15]. In both the papers, the authors studied some good properties of these families (vi) By choosing ϕðzÞ = 1 + sinh −1 z, we obtain the recently studied class S * ρ ≔ S * ð1 + sinh −1 zÞ created by Al-Sawalha [16]. Barukab and his coauthors [17] studied the sharp Hankel determinant of third-order for the following class in 2021 In [18,19], Pommerenke provided the following Hankel determinant D q,n ðgÞ containing coefficients of a function g ∈ S D q,n g ð Þ ≔ a n a n+1 ⋯ a n+q−1 with q, n ∈ ℕ = f1, 2, ⋯g. By varying the parameters q and n, we get the determinants listed below: that referred as first-, second-, and third-order Hankel determinants, respectively. The Hankel determinant for functions belonging to the general family S has just a few references in the literature. The best established sharp inequality for the function g ∈ S is jD 2,n ðgÞj ≤ λ ffiffiffi n p , where λ is a constant, and it is because of Hayman [20]. Additionally, it was determined in [21] for the class S that Several mathematicians were drawn to the problem of finding the sharp bounds of Hankel determinants in a given family of functions. In this context, Janteng et al. [22,23] estimated the sharp bounds of jD 2,2 ðgÞj, for three basic subfamilies of the set S. These families are K,S * , and R (functions of a bounded turning class), and these bounds are stated as 1, for g ∈ S * , 4 9 , forg ∈ R:

Journal of Function Spaces
This determinant's exact bound for the unified collection S * ðϕÞ was determined in [24] and subsequently investigated in [25]. In [26][27][28], this problem was also solved for various families of biunivalent functions.
The formulae provided in (11) make it abundantly evident that the computation of jD 3,1 ðgÞj is much more difficult than determining the bound of jD 2,2 ðgÞj. Babalola [29] was the first mathematician who studied third-order Hankel determinant for the K,S * , and R families in 2010. Following that, several academics [30][31][32][33][34] used the same method to publish papers regarding jD 3,1 ðgÞj for specific subclasses of univalent functions. However, Zaprawa's work [35] caught the researcher's attention, in which he improved Babalola's results by utilising a revolutionary method to show that 1, for g ∈ S * , 41 60 , for g ∈ R: He also pointed out that these bounds are not sharp. In 2018, Kwon et al. [36] achieved a more acceptable finding for g ∈ S * and demonstrated that jD 3,1 ðgÞj ≤ 8/8, and this limit was further enhanced by Zaprawa and his coauthors [37] in 2021. They got jD 3,1 ðgÞj ≤ 5/9 for g ∈ S * : In recent years, Kowalczyk et al. [38] and Lecko et al. [39] got a sharp bound of third Hankel determinant given by where S * ð1/2Þ is the starlike functions family of order 1/2: In [40], the authors obtained the sharp bounds of third Hankel determinant for the subclass of S * sin , and Mahmood et al. [41] calculated the third Hankel determinant for starlike functions in q-analogue. For some new literature on sharp third-order Hankel determinant, see [42][43][44][45].
In [46], Gandhi introduced a family of bounded turning function connected with a four-leaf function defined by and characterized it with some important properties. Similar to the definition of S * 4L , we now define a new subfamily of bounded turning functions by the following set builder notation: The aim of the current manuscript is to determine the exact bounds of the coefficient inequalities, Fekete-Szegö type problem, Kruskal inequality, and Hankel determinant of order two for functions of bounded turning class linked with four-leaf domain.

A Set of Lemmas
We say a function p ∈ P if and only if it has the series expansion along with the RpðzÞ ≥ 0ðz ∈ U d Þ: Let p ∈ P be represented by (19). Then These inequalities (20), (21), and (22) are taken from [47,48]: Lemma 2. Let p ∈ P and be given by (19). Then, for x, δ, ρ ∈ U d , we have For the formula c 2 , see [48]. The formula c 3 was due to Zlotkiewicz and Libera [49] while the formula for c 4 was proved in [50].

Coefficient Inequalities for the Class BT 4L
We begin this section by finding the absolute values of the first four initial coefficients for the function BT 4L : 3 Journal of Function Spaces Theorem 4. If g ∈ BT 4L and has the series representation (3), then These bounds are best possible.
Proof. Let g ∈ BT 4L : Then, (18) can be written in the form of Schwarz function as If p ∈ P , and it may be written in terms of Schwarz function wðzÞ as Equivalently, we have where From (3), we get g′ z ð Þ = 1 + 2a 2 z + 3a 3 z 2 + 4a 4 z 3 + 5a 5 z 4 +⋯: By simplication and using the series expansion of (34), we get By comparing (35) and (36), we obtain a 5 = 1 5 For a 2 , implementing (20), in (37), we get For a 3 , (38) can be written as Using (21), we get For a 4 , we can write (39) as From (22), we have Application of triangle inequality plus (22)  After simplifying, we have a 5 j j = 1 12 Comparing the right side of (49) with we get It follows that From (26), we deduce that These bounds are best possible and can be determined by the following extremal functions: This inequality is sharp.
Proof. By using (37) and (38), we may have By rearranging, it yields Application of (21) leads us to After the simplification, we get This required result is sharp and is determined by This inequality is best possible and can be obtained by Theorem 7. If g belongs to BT 4L , and be of the form (3). Then This result is sharp.
Proof. From (37), (39), and (40), we obtain After simplifying, we have Comparing the right side of (73) with we get It follows that and From (26), we deduce that The required result is sharp and can be determined by Theorem 8. If g ∈ BT 4L , and be of the form (3). Then This inequality is best possible.
Proof. By using (38) and (40), we have After simplifying, we have Comparing the right side of (82) with we get It follows that From (26), we deduce that This inequality is best possible and can be achieved by In this section, we will give a direct proof of the inequality over the class BT 4L for the choice of n = 4,p = 1, and for n = 5,p = 1: Krushkal introduced and proved this inequality for the whole class of univalent functions in [52].

Theorem 9.
If g belongs to BT 4L , and be of the form (3). Then This result is sharp.
Proof. From (37) and (39), we obtain From (22), we have Using (22), we obtain This result is sharp and can be obtained by Theorem 10. If g belongs to BT 4L , and be of the form (3). Then This inequality is best possible.
Proof. From (37) and (40) Comparing the right side of (98) with we get It follows that From (26), we deduce that This inequality is best possible and can be achieved by Next, we will calculate the Hankel determinant of order two jD 2,2 ðgÞj for the class g ∈ BT 4L : Theorem 11. If g belongs to BT 4L , then This inequality is sharp.
Proof. The D 2,2 ðgÞ can be written as follows: Using (23) and (24) 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 jδj ≤ 1, jxj = k, where k ≤ 1 and taking c ∈ ½0, 2. So, It is not hard to observe that Ξ ′ ðc, kÞ ≥ 0 for ½0, 1, so we have Ξðc, kÞ ≤ Ξðc, 1Þ: Putting k = 1 gives It is clear that Ξ′ðc, 1Þ < 0, so Ξðc, 1Þ is a decreasing function and attains its maximum value at c = 0: Thus, we have The required second Hankel determinant is sharp and is obtained by

Conclusion
In our present investigation, we considered a subclass of bounded turning functions associated with a four-leaf-type domain. We obtained some useful results for such a class, such as the limits of the first four initial coefficients, as well as the Fekete-Szego type inequality, the Zalcman inequality, the Kruskal inequality, and the estimation of the secondorder Hankel determinant. All of the obtained results have been proven to be sharp. This work has been used to obtain higher-order Hankel determinants, such as in the investigation of the bounds of fourth-order and fifth-order Hankel determinants. These two determinants have been studied in [45,[53][54][55][56], respectively. Also, one can easily use this new methodology to obtain sharp bounds of the thirdorder Hankel determinant for other subclasses of univalent functions.

Data Availability
The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.