Analytic Functions Related to a Balloon-Shaped Domain

: One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article deﬁnes a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X deﬁned in the new domain, including the sharp estimates for the coefﬁcients a 2 , a 3 , and a 4 , and for three second-order and third-order Hankel determinants, H 2,1 X , H 2,2 X , and H 3,1 X . The optimality of each obtained estimate is given as well.


Introduction
Let A be the class of all analytic functions X defined in the open unit disc U = {z ∈ C : |z| < 1} with X (0) = 0 and X (0) = 1.Thus, each analytic function in A has the following Taylor series representation Let S be the subclass of all analytic functions in A that are univalent in U.
An analytic function X is said to be subordinate to an analytic function g in U, denoted as X ≺ g, if there exists a Schwarz function ξ that is analytic in U with ξ(0) = 0 and |ξ(z)| < 1, such that X (z) = g(ξ(z)).In particular (see [1]), if g is univalent in U, then X ≺ g if and only if X (0) = g(0) and X (U) ⊂ g(U).
Using the concept of subordination, many subclasses have been defined and studied, such as S * , C, K and R of starlike, convex, close to convex, and functions with bounded turnings, respectively.See [2][3][4][5][6] for the new results about more subclasses.
For two analytic functions X and ζ in A with the series representation of X given in (1) and b t z t the convolution (Hadamard product) X * ζ is defined by Shanmugam [7] generalized the idea of Padmanabhan et al. [8] and introduced the general form of function class S * h (ϕ) as follows where h is a fixed function in A and ϕ is a convex univalent function on U with ϕ(0) = 1 and Re(ϕ(z)) > 0. Ma and Minda [9] defined a more general form of function class S * (ϕ) by applying for some restrictions h(z) = z 1−z (and hence X * h = X ) with ϕ(0) = 1 and ϕ (0) > 0. The generic form of Ma and Minda-type class of starlike functions is defined as In recent years, many authors have established important subfamilies of analytic functions by varying ϕ(z) in S * (ϕ), and they proved significant geometric properties of those subfamilies.For details, see [10][11][12][13][14].
We discuss the following two classes that have some interesting geometric properties.
(i) For ϕ 1 (z) = √ 1 + z, the class S * (ϕ) becomes S * L , which was introduced by Sokol and Stankiewicz [15], and it contains those functions X ∈ A such that zX (z) X (z) lies in the region bounded by the right half of the lemniscate of Bernoulli defined by 1+e −z , the class S * (ϕ) becomes S * sig , which was defined and investigated by Geol et al. [16].Geometrically, a function X ∈ S * sig if and only if By taking inspiration from all of the previous works mentioned, we introduce the following new class of analytic functions by using the quotient of ϕ 1 (z) = √ 1 + z and ϕ 2 (z) = 2 1+e −z .
Definition 1.Let X ∈ A, given in (1).We say X ∈ R sl if it satisfies the following condition Geometrically, each X ∈ R sl maps the open unit disc into a balloon-shaped domain, which is symmetric about the real axis, as shown in the following Figure 1.For X ∈ A and n, k ≥ 0, Pommerenke [17]  . ( Recently, finding the sharp upper bounds of the Hankel determinants H k,n (X ) for certain n and k for various subfamilies of analytic functions has been identified as an interesting and important problem.Many researchers have observed sharp upper bounds of Hankel determinants for many subfamilies of analytic functions.In particular, the upper bounds of second and third-order Hankel determinants have been estimated in [18][19][20][21][22][23] for several subclasses of normalized analytic function.
Hayman [24] was the first to give the sharp inequality for X ∈ S, and subsequently proved that |H 2,n (X )| ≤ λ √ n, where λ > 0. This inequality is further explained in [25] and showed that |H 2,2 (X )| ≤ λ, where 1 ≤ λ ≤ 11  3 .Janteng et al. [26] determined the sharp bounds of H 2,2 (X ) for the subfamilies of K, S * , and R. Babalola [27] studied a third-order Hankel determinant for the subclasses of S * and C, while Zaprawa [28] amended Babalola's results and gave the following estimates, which it is believed may not be the best possible results.
Kwon et al. [29] improved this determinant for starlike functions as |H 3,1 (X )| ≤ 8 9 .Zaprawa et al. [30] extended his work by estimating |H 3,1 (X )| ≤ 5 9 for X ∈ S * .Arif et al. [31] calculated the sharpness of the bounds of the coefficients and H 3,1 (X ) for a subfamily of starlike functions related to sigmoid functions; see [32] for the modified sigmoid functions.Orhan et al. [33] estimated the sharp Hankel determinants for a subfamily of analytic functions associated with the lemniscate of Bernoulli.Moreover, Shi et al. [34,35] estimated the sharpness of Hankel determinants for the functions with bounded turning associated with a petal-shaped domain and inverse functions, respectively.
Moreover, the estimation of various bounds can be considered for many classes of functions; for example, see [36][37][38].
It is natural to ask what the upper bounds for the analytic functions in the newly defined class R sl related to the coefficients of the Taylor series representation (1) and Hankel determinants are.
The aim and novelty of this article are the sharp upper bounds of the modulus of the coefficients a 2 , a 3 , and a 4 and the second-order and third-order Hankel determinants, H 2,1 X , H 2,2 X , and H 3,1 X , for the analytic functions in the new class R sl .

A Set of Lemmas
Let P represent the class of analytic functions p, such that p(0) = 1, Re(p(z)) > 0 for z ∈ U, which has the following Taylor series form, The subsequent Lemmas 1-4 will help to demonstrate our main findings, where c t , c t+k , and c t+2k for t, k ∈ N are coefficients of the Taylor series (6).

Main Results
Theorem 1.Let X ∈ R sl .Then, the following inequalities for the coefficients in (1) are true.
The sharpness of these inequalities can be obtained using the function In particular, if n = 1, 2, 3, and 4, then we have Proof.As X ∈ R ls , from (4), we obtain Then, (1) gives Let p ∈ P be written by

This implies that
It follows from ( 21) and ( 22) that Using Lemma 1, ( 23) and ( 24) imply By (25), Using Lemma 3, we obtain From (26), we have By applying Lemma 4, Theorem 2. Let X ∈ R ls .Then, the sharp upper bound for the following second-order Hankel determinant is given by The function (17) gives the sharpness of the inequality (27).
Proof.Applying to the identities ( 23) and ( 24), Using Lemma 1, we obtain It is easy to verify that the function (17) gives the sharpness of the inequality (27).
Theorem 3. Let X ∈ R ls .Then, the sharp upper bound for the following second-order Hankel determinant is given by The function (17) gives the sharpness of the inequality (28).
Proof.By the identities ( 23)-( 25), Now, using Lemma 2, we have Using the triangular inequality by taking Then ∂F ∂b which shows that F(b, c) is an increasing function for all b ∈ [0, 1] and c ∈ [0, 2].Thus, the maximum value occurs at b = 1.Consequently, this shows that G(c) is a decreasing function for all c ∈ [0, 2], and the maximum value occurs at c = 0.By referring to (29), we can deduce the required inequality, It is also easy to verify that the function (17) provides the sharpness of the inequality (28).
Theorem 4. Let X ∈ R ls .Then, we have the sharp upper bound for the following third-order Hankel determinant.
The sharpness of this inequality can occur according to the function given in (18).
Similarly, we obtain It follows that Differentiating with regard to "y", we have Then, for all c ∈ 372 157 , 2 and y ∈ (0, 1), we have Thus, we obtain It can be seen that ζ (c) = 0, for any c ∈ 372 157 , 2 .Also, ζ(c) is a decreasing function and its maximum value occurs at c ≈ 1.53928554, which is 37,437.

II. On the six faces of the cuboid
Next, we proceed to examine the maximum value of the function G(c, q, y) on all six faces of the cuboid .
Solving for c within the range (0, 2), we find that c ≈ 1.4228.This indicates that there is no optimal solution for G(c, 0, y).

III. On the twelve edges of the cuboid
Finally, we need to find the maximum values of G(c, q, y) along the twelve edges.(i) On q = 0 and y = 0: G(c, 0, 0) becomes then ∂h 6 ∂c = 0 gives the critical point c ≈ 1.4343, where the maximum value is obtained as follows.

Conclusions
In the present article, we defined a class of analytic functions by considering the ratio of two well-known functions.We investigated the sharp upper bounds of the modulus of coefficients a 2 , a 3 , and a 4 ; and the sharp upper bounds for the modulus of three secondorder and third-order Hankel determinants, H 2,1 X , H 2,2 X , and H 3,1 X , for the normalized analytic functions X belonging to the newly defined class.These findings contribute to the existing body of knowledge and provide valuable insights for further research in the field.This work provides a direction to define more interesting generalized domains and to extend to new subclasses of starlike and convex functions by using quantum calculus.