Sharp Coefﬁcient Problems of Functions with Bounded Turnings Subordinated by Sigmoid Function

: This study deals with analytic functions with bounded turnings, deﬁned in the disk O d = { z : | z | < 1 } . These functions are subordinated by sigmoid function 2 1 + e − z and their class is denoted by BT Sg . Sharp coefﬁcient inequalities, including the third Hankel determinant for class BT Sg , were investigated here. The same was also included for the logarithmic coefﬁcients related to functions of the class BT Sg .


Introduction and Definitions
Let the class of analytic functions in disk O d = {z ∈ C : |z| < 1} be denoted by the notation H(D) and suppose that A is the sub-family of H(D) defined as follows.
A := f ∈ H(D) : f (z) = ∞ ∑ k=1 a k z k , with a 1 = 1 . ( Moreover, all univalent functions from class A are composed in a class, named as S. For two functions g 1 , g 2 ∈ H(D), the function g 1 is said to be subordinated by g 2 , mathematically denoted as g 1 ≺ g 2 , if a regular function υ defined in O d exists with the property that υ(0) = 0 and |υ(z)| < 1 such that f (z) = g(υ(z)). Moreover, if g 2 is univalent in O d , then the relation g 1 ≺ g 2 with g 1 (0) = g 2 (0) implies that g 1 (O d ) ⊂ g 2 (O d ). For details, see [1][2][3] and the references therein. Although the univalent function theory was initiated in 1851, the coefficient conjecture, proposed by Bieberbach [4] in 1916 and laterally proved by de-Branges [5] in 1985, turned the theory into one of the emerging areas of potential research. During the era between 1916 and 1985, several researchers attempted to prove or disprove this conjecture, which resulted in the formation of many subclasses of the class S that are based on the geometry of image domains. The most studied and fundamental subclasses of S are S * and K, which contain starlike and convex functions, respectively. Ma and Minda [6], in 1992, introduced the following general form of the class S * : axis. With the variation in function φ, the class S * (φ) generates several sub-families of S * , which include some of the ones listed below.
(ii). The family S * (φ) with φ(z) = √ 1 + z was developed by Sokól and Stankiewicz in [8], which maps the symmetric disk O d onto the region bounded by |w 2 − 1| < 1. (iii). By choosing the function φ(z) = 1 + sinh −1 z, we obtain S * ρ := S * 1 + sinh −1 z , which was recently introduced by Kumar and Arora [9]. In 2021, Barukab et al. [10] found the sharp upper bound of the third Hankel determinant for functions of the following class: Later, in 2022, Shi et al. [11] determined the sharp second-order Hankel determinant for the above class, but with logarithmic coefficients. (iv). By choosing φ(z) = 1 + 4 5 z + 1 5 z 4 , the class S * (φ) reduces to class S * 3l , which was studied by Gandhi et al. [12]. In 2022, Arif et al. [13] determined the sharp third-order Hankel determinant for functions of class S * 3l . Later, Shi et al. [14] determined the sharp second Hankel determinant of the same class with logarithmic coefficients.
(v). Raza and Bano [15] and Alotaibi et al. [16] contributed the families S * cos := S * (cos(z)) and S * cosh := S * (cosh(z)), respectively. These researchers investigated some geometric characteristics of the functions of these families. (vi). We obtain the family S * sin by choosing φ(z) = 1 + sin z, which was established in [17]. In this paper, the authors determined radii problems for the defined class S * sin . For functions f ∈ S of the series form as given in (1), the Hankel determinant H q,n ( f ) (with q, n ∈ N = {1, 2, . . .} and a 1 = 1) was given by Pommerenke [18,19] and is defined as follows: With certain variations in q and n, we have the following second and third-order Hankel determinants, respectively: For the functions f ∈ S, the best established sharp inequality is |H 2,n ( f )| ≤ λ √ n, where λ is an absolute constant and this inequality is due to Hayman [20]. Furthermore, the following results for the class S can be accessed from [21].
where R denotes the class of functions with bounded turnings which is defined as To date, a number of researchers have contributed to the work on Hankel determinants and have achieved remarkable milestones. Some of the recent developments can be accessed from [24][25][26][27][28][29][30][31][32][33][34] and the references therein. The logarithmic function associated to function f ∈ S is defined as If f ∈ S assumes the series of the form given in (1), then (3) gives the following relations: Recently, in [35,36], Kowalczyk and Lecko introduced the following qth-order Hankel determinant H q,n F f /2 containing the logarithmic coefficients of f .
From above, one can easily deduce that We now define the class of functions with bounded turnings and associated with the sigmoid function φ(z) = 2 1+e −z with φ(0) = 1 and φ(z) > 0 as follows: We intended to find the sharp bound of |H 3,1 ( f )| and a 3 a 5 − a 2 4 for the class BT Sg . In addition, we investigated the sharp bounds of H 2,1 F f /2 and H 2,2 F f /2 for the class BT Sg .

A Set of Lemmas
Definition 1. A function p ∈ P if and only if, it has the series expansion along with the p(z) ≥ 0 for z ∈ O d .

Lemma 1.
If the function p ∈ P has the series representation as given in (12), then and |c n | ≤ 2 for n ≥ 1.

Lemma 2 ([39]
). If the function p ∈ P has the series representation as given in (12) Lemma 3. If the function p ∈ P has the series representation as given in (12), then, for x, δ, ρ ∈ O d = {z ∈ C : |z| ≤ 1}, we have Here, the readers can refer to the formula for c 2 given in [37]. The formula for c 3 is due to Libera and Zlotkiewicz [40], and the formula for c 4 is proved in [41].

Theorem 1.
If the function f ∈ BT Sg assumes the series representation as given in (1), then The inequality is sharp and sharpness can be achieved from Proof. Let f ∈ BT Sg . Then, If p ∈ P, then This implies that Using (21) in (20), we obtain From the series defined in (1), it follows that By comparing (22) and (23), one may have Substituting (24)- (27) in (2) and setting c 1 = c, we obtain where L = 8,847,360. Now, assuming c 1 = c and m = 4 − c 2 1 in (16)-(18), we obtain the following: By inserting the above expressions into (28), one may have Since m = 4 − c 2 , it follows that By replacing |δ| with y and |x| with x, if we apply the statement |ρ| ≤ 1, it follows that where with 1]. For this purpose, we need to find max (c, x, y) in the interior of ∆, in the interior of all of its six faces, and on the twelve edges of cuboid ∆.
Taking y = 0, we obtain A computation indicates that the solution for the system of equations From the computation, we conclude that the solution for the system of equations (III). Finally, we look forward to the maximum of (c, x, y) at the twelve edges of ∆. By substituting y = 0 and x = 0, it yields p 6 (c) = (c, 0, 0) = 15c 6 − 3456c 4 + 13824c 2 .
For the critical point, the equation ∂p 9 ∂c = 0 gives c 0 = 1.110119, at which, the maximum value of p 9 (c) is attained. Thus, one may conclude that By setting y = 1 and c = 0, we obtain By simple computation, we see that p 12 (x) obtains its maximum value at 0, so we have Hence, from the above situations, we conclude that By using Equation (29), it follows that Thus, the required result is accomplished.
By inserting the above expressions into (36), one may have Since m = 4 − c 2 , it follows that By replacing |δ| by y and |x| by x, if we apply the relation |ρ| ≤ 1, it follows that Now, we have to maximize (c, x, y) in the closed cuboid ∆ : [0, 2] × [0, 1] × [0, 1]. For this purpose, we have to find the maximum of (c, x, y) in the interior of ∆, in the interior of all its six faces, and on the twelve edges of ∆. (i). Initially, we will look for the maxima of (c, x, y) in the interior of ∆. Let (c, x, y) ∈ (0, 2) × (0, 1) × (0, 1). Then, on differentiating (38) partially with respect to parameter y, one may obtain Taking ∂ ∂y = 0, we obtain For y 0 to belong to (0, 1), we must have and Now, in order to find the solutions that meet both inequalities (39) and (40), we see that c 2 > 4, and an easy calculation shows that (39) does not hold for all x ∈ (0, 1). This implies that we found no optimal point of in (0, 2) × (0, 1) × (0, 1).
(ii). Next , we look forward to the maximum of (c, x, y) in the interior of all six faces of ∆. If we choose c = 0, then we obtain which shows that there does not exist any point of extrema for (0, x, y) in (0, 1) × (0, 1). Taking c = 2, we obtain (2, x, y) = 320.
We note that (42) is free of y. It follows that For the critical point, ∂j 9 ∂c = 0 gives c 0 = 1.310480, at which, the maximum value is attained for j 9 (c). Thus, j 9 (c) ≤ 2980.76554.
By choosing y = 0 and c = 0, the edge (c, x, y) yields For the critical point, ∂j 10 ∂x = 0 gives x 0 = 0.577350, at which, the maximum value is achieved for j 10 (x). Thus, By taking y = 1 and c = 0, we obtain By simple computation, we see that j 11 (x) obtains its maximum value at 0, so we have (0, x, 1) ≤ 46080.
The required inequality is accomplished.
Setting y = 0, we have A computation shows that a solution for a system of the following equations in (0, 2) × (0, 1) does not exist. Substituting y = 1, we find that In this situation, we came to the same conclusion as for t 4 ; that is, the system has no solution in (0, 2) × (0, 1). (iii). Finally, we now intend to find the maximum value of W (c, x, y) at the twelve edges of ∆. For this, we proceed as follows. By choosing y = 0 and x = 0, it yields that t 6 (c) = W (c, 0, 0) = 345c 6 − 55296c 4 + 221184c 2 .
By simple computation, we see that t 7 (c) obtains its maximum value at 0, so we have W (c, 0, 1) ≤ 4423680.
By simple computation, we see that t 10 (x) obtains its maximum value at 0, so we have t 10 (x) ≤ 4423680.
By choosing y = 0 and c = 0, we have For the critical point, ∂t 11 ∂x = 0 gives x 0 ≈ 0.6221710, at which, the maximum value is achieved for t 11 (x). Therefore, we have which completes the proof.

Conclusions
We have obtained the sharp bounds of Hankel determinants of order three for the class of functions with bounding turning that are associated with the sigmoid function. All of the bounds that we found here were sharp. Moreover, we investigated the sharp bounds of logarithmic coefficients linked with the functions of bounded turnings. This also includes the third-order Hankel determinant for these logarithmic coefficients. This work will help in finding the fourth-order Hankel determinants for the same types of analytic functions that have been considered in this study.