Sharp Bounds of Hankel Determinant on Logarithmic Coefﬁcients for Functions of Bounded Turning Associated with Petal-Shaped Domain

: The purpose of this article is to obtain the sharp estimates of the ﬁrst four initial logarithmic coefﬁcients for the class BT s of bounded turning functions associated with a petal-shaped domain. Further, we investigate the sharp estimate of Fekete-Szegö inequality, Zalcman inequality on the logarithmic coefﬁcients and the Hankel determinant H 2,1 (cid:16) F f /2 (cid:17) and H 2,2 (cid:16) F f /2 (cid:17) for the class BT s with the determinant entry of logarithmic coefﬁcients.


Introduction and Definitions
For a good sense of the terminology used throughout our primary results, some basic pertinent information from Geometric Function Theory must always be given and explained.Let us start with the letter A, which stands for the normalised analytic functions family and S for the normalised univalent functions family.These fundamental concepts are defined in the open unit disc D = {z ∈ C : |z| < 1} and are provided by the set builder in the form of where H(D) represents the family of analytic functions, and Recently, Aleman and Constantin [1] gave a beautiful interaction between univalent function theory and fluid dynamics.In fact, they demonstrated a simple method that shows how to use a univalent harmonic map to obtain explicit solutions of incompressible two-dimensional Euler equations.The logarithmic coefficients β n of f ∈ S are given by the below formula β n z n for z ∈ D.
These coefficients contribute significantly, in many estimations, to the theory of univalent functions.In 1985, de Branges [2] obtained that for n ≥ 1, and the equality holds if and only if f takes the form z/ 1 − e iθ z 2 for some θ ∈ R. Clearly, this inequality gives the famous Bieberbach-Robertson-Milin conjectures about Taylorcoefficients of f belonging to S in its most general form.For more about the proof of de Brange's result, we refer to [3][4][5].In 2005, Kayumov [6] was able to solve Brennan's conjecture for conformal mappings by considering the logarithmic coefficients.We list a few papers that have conducted significant work on the study of logarithmic coefficients [7][8][9][10][11][12][13][14].
For the given functions g 1 , g 2 ∈ A, the subordination between g 1 and g 2 (mathematically written as g 1 ≺ g 2 ), if an analytic function v appears in D with the restriction v(0) = 0 and |v(z)| < 1 in such a manner that f (z) = g(v(z)) hold.Moreover, if g 2 in D is univalent, the following connection holds: By employing the principle of subordination, Ma and Minda [15] considered a unified version of the class S * (φ) in 1992, which is stated below as where φ is a univalent function with φ (0) > 0 and φ > 0.Moreover, the region φ(D) is star-shaped about the point φ(0) = 1 and is symmetric along the real line axis.In the past few years, numerous sub-families of the collection S have been examined as particular choices of the class S * (φ).For example, (iii) By selecting φ(z) = 1 + sin z, the class S * (φ(z)) lead to the family S * sin , which was explored in [17], while S * e ≡ S * (e z ) has been produced in the article [18].(iv) The family S * cos := S * (cos(z)) and S * cosh := S * (cosh(z)) were contributed, respec- tively, by Raza and Bano [19], and Alotaibi et al. [20].In both the papers, the authors studied good properties of these families.
For given parameters q, n ∈ N = {1, 2, . ..}, the Hankel determinant H q,n ( f ) was defined by Pommerenke [21,22] for a function f ∈ S of the form Equation (1), which is given by The growth of H q,n ( f ) has been investigated for different sub-collections of univalent functions.Specifically, the absolute sharp bounds of the functional H 2,2 ( f ) = a 2 a 4 − a 2 3 were found in [23,24] for each of the sets C, S * and R, where the family R contained functions of bounded turning.This determinant has also been recently studied for two new subfamilies of bi-univalent functions in [25,26].However, the exact estimate of this determinant for the family of close-to-convex functions is still undetermined [27].Later on, many authors published their work regarding the upper bounds of the Hankel determinant for different sub-collections of univalent functions, see [28][29][30][31][32][33][34][35][36][37].
According to the definition, it is not hard to calculate that for f ∈ S, its logarithmic coefficients are given by (5) Recently, Kowalczyk and Lecko [38,39] proposed the study of the Hankel determinant H q,n F f /2 , whose elements are logarithmic coefficients of f , that is It is observed that 2 over the class S or its subclasses.Some basic calculations gives the expressions of H q,n F f /2 in the following, which we will discuss in the present paper.
In [40], Kumar and Arora introduce an interesting subclass of the starlike function, defined by Let φ(z) = 1 + sinh −1 z.It can be noted that φ(z) = 1 + ln z + √ 1 + z 2 and is convex in D. In geometry, it maps the unit disk onto a petal-shaped domain Ω ρ = {ω ∈ C : |sinh(ω − 1)| < 1} symmetric about the line ω = 1.Using this function, Barukab and his coauthors [41] considered a subclass of the bounded turning function, given by In the current article, our main goal is to calculate the sharp logarithmic coefficientrelated problems for the class BT s of bounded turning functions linked with the petalshaped domain.The sharp bounds of Fekete-Szegö inequality, Zalcman inequality of logarithmic coefficients, H 2,1 F f /2 and H 2,2 F f /2 are obtained for the class BT s .

A Set of Lemmas
Let P represent the class of all functions p that are holomorphic in D with (p(z)) > 0 and has series representation given in the form of To prove the main results, we need the following lemmas.

Coefficient Inequalities for the Class BT s
We begin this section by finding the absolute values of the first four initial logarithmic coefficients for the function of class BT s .Theorem 1.If f ∈ BT s and has the series representation (1), then These bounds are the best possible.
Proof.Let f ∈ BT s .Then, (11) can be written in the form of a Schwarz function, as If p ∈ P, and it may be written in terms of Schwarz function w(z) as From ( 1), we obtain By simplification and using the series expansion of ( 25), we obtain Comparing ( 26) and ( 27), we obtain Plugging (28) in ( 3)-( 6), we obtain For γ 1 , implementing (16), in (29), we obtain For γ 2 , we can write (30), as Using (17) we have For γ 3 , we can write (31) as From (18), we have and Application of triangle inequality plus (18) lead us to For γ 4 , we can rewrite (32) as where 19) and ( 33), we have These outcomes are best possible.For this, we consider a function where n = 1, 2, 3, 4. Thus, we have This inequality is sharp.
Proof.Employing (29), and (30), we may write Application of (17), leads us to After the simplification, we obtain The required result is sharp and is determined by using ( 3) and ( 4) and This inequality is sharp.
Proof.Using ( 29)-( 31), we have From (18), we have and Using (18), we obtain This result is the best possible and is obtained by using (3)-(5) and Let f ∈ BT s be of the form Equation (1).Then This inequality is the best possible.
Proof.From ( 30) and (32), we obtain After simplifying we have Comparing the right side of (34) with where therefore by Equations ( 19) and ( 35), we have This required inequality is sharp and is determined by using Equations ( 4) and ( 6) and

Hankel Determinant with Logarithmic
Coefficients for the Class BT s Theorem 5.If f belongs to BT s , then The inequality is sharp.
Proof.From ( 29)-( 31),we have Using ( 13) and ( 14) 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 It is a simple exercise to show that Since ∂φ(c,1) ∂c < 0, so φ(c, 1) is a decreasing function, and obtains its maximum value at The required Hankel determinant is sharp and is obtained by using ( 3)-(5) and If f belongs to BT s , and has the form Equation (1).
This result is the best possible.
Proof.The H 2,2 F f /2 can be written as Putting ( 30)- (32), with c 1 = c we obtain Let w = 4 − c 2 in ( 13)- (15).Now using the simplified form of these lemmas, we obtain 432c 4 c 2 = 216c 6 + 216c 4 wx, Putting the above expressions in (36), we obtain, where ρ, x, τ ∈ D, and Now, by using |x| = x, |τ| = y and utilizing the fact |ρ| ≤ 1, we obtain where Now, we have to maximize S(c, x, y) in the closed cuboid For this, we have to discuss the maximum values of S(c, x, y) in the interior of Θ, in the interior of its six faces and on its twelve edges.
Let g(x) = 4(15−x) 21−x .As g (x) < 0 in (0, 1), it can be observed that g(x) is decreasing over (0, 1).Hence c 2 > 14 5 .It is not difficult to be verified that the inequality Equation ( 38) can not hold true in this situation for x ∈ 2 5 , 1 .Thus, there is no critical point of S(c, x, y) exist in (0, 2) × 2 5 , 1 × (0, 1).Suppose that there is a critical point ( c, x, y) of S existing in the interior of cuboid Θ, clearly, it must satisfy that x < 2 5 .From the above discussion, it can be also known that c 2 ≥ 292 103 and y ∈ (0, 1).Presently, we will prove that S( c, x, y) < 276480.For (c, x, y) ∈ 292 103 , 2 × 0, 2 5 × (0, 1), by invoking x < 2 5 and 1 − x 2 < 1; it is not hard to observe that Therefore, we have Obviously, it can be observed that and 103 , 2 , we obtain that ∂ 2 Γ 2 ∂y 2 ≤ 0 for y ∈ (0, 1) and thus it follows that Therefore, we have It is easy to be calculated that Υ 2 (c) attains its maximum value 74510.302at c ≈ 1.683731.Thus, we have Hence S( c, x, y) < 276480.This implies that S is less than 276480 at all the critical points in the interior of Θ.Therefore, S has no optimal solution in the interior of Θ.
2. Interior of all the six faces of cuboid Θ : (i) On the face c = 0, S(c, x, y) takes the form Then, Thus L 1 (x, y) has no critical point in the interval (0, 1) × (0, 1).
(iii) On the face x = 0, S(c, x, y) reduces to Differentiating L 2 (c, y) partially with respect to y Putting ∂L 2 ∂y = 0, we obtain For the given range of y, y 1 should belong to (0, 1),, which is possible only if c > c 0 , c 0 ≈ 1.76094199.Moreover, the derivative of L 2 (c, y), partially with respect to c, is By substituting the value of y in (40), plugging ∂L 2 ∂c = 0 and simplifying, we obtain A calculation gives the solution of (41) in the interval (0, 1) that is c ≈ 1.3851278.Thus, L 2 (c, y) has no optimal point in the interval (0, 2) × (0, 1).
(v) On the face y = 0, S(c, x, y) becomes Presently, differentiating partially with respect to c, then, with respect to x and simplifying, we have and A numerical computation demonstrates that the solution does not exist for the system of Equations ( 42) and ( 43) in (0, 2) × (0, 1).Hence L 4 (c, x) has no optimal solution in the interval (0, 2) × (0, 1).
(vi) On the face y = 1, S(c, x, y) yields Partial derivative of L 5 (c, x) with respect to c and then with respect to x, we have and As in the above case, we conclude the same result for the face y = 0, that is the system of Equations ( 44) and ( 45) has no solution in (0, 2) × (0, 1).
If f ∈ BT s , then the sharp bound for this Hankel determinant is determined by using Equations ( 4)-( 6) and

Conclusions
Due to the great importance of logarithmic coefficients, Kowalczyk and Lecko [38,39] proposed the topic of studying the Hankel determinant with the entry of logarithmic coefficients.In the current article, we considered a subclass of bounded turning functions denoted as BT s .This family of univalent functions was connected with a petal-shaped domain with f (z) subordinated to 1 + sinh −1 z.We gave an estimate for some initial
with the aid of the triangle inequality and replacing |τ| ≤ 1, |x| = b, where b ≤ 1 and taking