A Study of Third Hankel Determinant Problem for Certain Subfamilies of Analytic Functions Involving Cardioid Domain

In the present article, we consider certain subfamilies of analytic functions connected with the cardioid domain in the region of the unit disk. The purpose of this article is to investigate the estimates of the third Hankel determinant for these families. Further, the same bounds have been investigated for two-fold and three-fold symmetric functions.


Introduction and Definitions
Let A be the family of all functions that are holomorphic (or analytic) in the open unit disc ∆ = {z ∈ C : |z| < 1} and having the following Taylor-Maclaurin series form: Further, let S represent a subfamily of A, which contains functions that are univalent in ∆. The familiar coefficient conjecture for the function f ∈ S of the form (1) was first presented by Bieberbach [1] in 1916 and proven by de-Branges [2] in 1985. In between the years 1916 and 1985, many researchers tried to prove or disprove this conjecture. Consequently, they defined several subfamilies of S connected with different image domains. Among these, the families S * , C, and K of starlike functions, convex functions, and close-to-convex functions, respectively, are the most fundamental subfamilies of S and have a nice geometric interpretation. These families are defined as: we can find a function w, called the Schwarz function, that is holomorphic in ∆ with w (0) = 0 and |w(z)| < 1 such that h 1 (z) = h 2 (w (z)) (z ∈ ∆) . In the case of the univalency of h 2 in ∆, then the following relation holds: In [3], Padmanabhan and Parvatham in 1985 defined a unified family of starlike and convex functions using familiar convolution with the function z/ (1 − z) a , for a ∈ R. Later on, Shanmugam [4] generalized this idea by introducing the family: where " * " stands for the familiar convolution, φ is a convex, and h is a fixed function in A. Furthermore, if we replace h in S * h (φ) by z/ (1 − z) and z/ (1 − z) 2 , we obtain the families S * (φ) and C (φ) respectively. In 1992, Ma and Minda [5] reduced the restriction to a weaker supposition that φ is a function, with Re φ(z) > 0 in ∆, whose image domain is symmetric about the real axis and starlike with respect to φ(0) = 1 with φ (0) > 0 and discussed some properties including distortion, growth, and covering theorems. The family S * (φ) generalizes various subfamilies of the family A, for example; 1+Bz is the family of Janowski starlike functions; see [6]. Further, if A = 1 − 2α and B = −1 with 0 ≤ α < 1, then we get the family S * (α) of starlike functions of order α. (ii). The family S * L := S * ( √ 1 + z) was introduced by Sokól and Stankiewicz [7], consisting of functions f ∈ A such that z f (z)/ f (z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w 2 − 1| < 1. (iii). For φ(z) = 1 + sin z, the family S * (φ) leads to the family S * sin , introduced in [8]. (iv). When we take φ(z) = e z , then we have S * e := S * (e z ) [9]. (v). The family S * R := S * (φ(z)) with φ(z) = 1 + z k k+z k−z , k = √ 2 + 1 was studied in [10]. (vi). By setting φ(z) = 1 + 4 3 z + 2 3 z 2 , the family S * (φ) reduces to S * car , introduced by Sharma and his coauthors [11], consisting of functions f ∈ A such that z f (z)/ f (z) lies in the region bounded by the cardioid given by: and also by the Alexandar-type relation, the authors in [11] defined the family C car by: see also [12,13]. For more special cases of the family S * (φ), see [14,15]. We now consider the following family connected with the cardioid domain: For given parameters q, n ∈ N = {1, 2, . . .}, the Hankel determinant H q,n ( f ) was defined by Pommerenke [16,17] for a function f ∈ S of the form (1) given by: The growth of H q,n ( f ) has been investigated for different subfamilies 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 [18,19] for each of the families C, S * and R, where the family R contains functions of bounded turning. However, the exact estimate of this determinant for the family of close-to-convex functions is still undetermined [20]. Recently, Srivastava and his coauthors [21] found the estimate of the second Hankel determinant for bi-univalent functions involving the symmetric q-derivative operator, while in [22], the authors studied Hankel and Toeplitz determinants for subfamilies of q-starlike functions connected with the conic domain. For more literature, see [23][24][25][26][27][28][29][30].
The Hankel determinant of third order is given as: The estimation of the determinant |H 3,1 ( f )| is very hard as compared to deriving the bound of |H 2,2 ( f )|. The very first paper on H 3,1 ( f ) was given in 2010 by Babalola [31], in which he obtained the upper bound of H 3,1 ( f ) for the families of S * , C, and R. Later on, many authors published their work regarding |H 3,1 ( f )| for different subfamilies of univalent functions; see [32][33][34][35][36]. In 2017, Zaprawa [37] improved the results of Babalola as under: 540 , for f ∈ C, 41 60 , for f ∈ R.
. and claimed that these bounds are still not the best possible. Further, for the sharpness, he examined the subfamilies of S * , C, and R consisting of functions with m-fold symmetry and obtained the sharp bounds. Moreover, in 2018, Kwon et al. [38] improved the bound of Zaprawa for f ∈ S * and proved that |H 3,1 ( f )| ≤ 8/9, but it is not yet the best possible. The authors in [39][40][41] contributed in a similar direction by generalizing different families of univalent functions with respect to symmetric points. In 2018, Kowalczyk et al. [42] and Lecko et al. [43] obtained the sharp inequalities: for the recognizable families K and S * (1/2), respectively, where the symbol S * (1/2) stands for the family of starlike functions of order 1/2. Furthermore, we would like to cite the work done by Mahmood et al. [44] in which they studied the third Hankel determinant for a subfamily of starlike functions in the q-analogue. Additionally, Zhang et al. [45] studied this determinant for the family S * e and obtained the bound |H 3,1 ( f )| ≤ 0.565. In the present article, our aim is to investigate the estimate of |H 3,1 ( f )| for the subfamilies S * car , C car , and R car of analytic functions connected with the cardioid domain. Moreover, we also study this problem for families of m-fold symmetric functions connected with the cardioid domain.

A LEMMA
Let P denote the family of all functions p that are analytic in ∆ with (p(z)) > 0 and having the following series representation: Lemma 1. If p ∈ P and it has the form (6), then: |c m c n − c k c l | ≤ 4 for m + n = k + l, where the inequalities (7), (10), (11), and (9) are taken from [46]. Proof. Let f ∈ S * car . Then, in the form of the Schwarz function, we have:

Bound of |H
Furthermore, we easily get: and from series expansion of w with simple calculations, we can write: By comparing (12) and (13), we get: Applying (7) in (14) and (15), we have: Now, reshuffling (16), we get: If we insert |c 1 | = x ∈ [0, 2], then we have: The above function has its maximum value at x = 2. Therefore: Equalities are obtained if we take: Theorem 2. If f ∈ S * car and it has the series form (1) , then: Proof. From (5), the third Hankel determinant can be written as: Now, rearranging, it yields: Applying the triangle inequality: besides, (7), (10), (11) and (8) lead us to: If we insert |c 1 | = x ∈ [0, 2], then we have: Then, the function Φ (x) is increasing. Therefore, we get its maximum value by putting x = 2, Thus, the proof follows.
From the function given by (18), we conclude the following conjecture.
Conjecture 3.1. Let f ∈ S * car and in the form (1). Then, the sharp bound is: These bounds are the best possible.

Proof.
Let the function f ∈ C car . Then, by the Alexandar-type relation, we say that z f ∈ S * car , and hence, using the coefficient bounds of the family S * car , which was proven in the last Theorem, we get the needed bounds.
Theorem 4. Let f have the form (1) and belong to C car . Then: Proof. From (5), the third Hankel determinant can be obtained as: Utilizing the definition of the family C car , we easily have: After reordering, it yields: Using the triangle inequality, we get: The application of (7) and (11) These results are the best possible.
Proof. Let f ∈ R car . Then, we can write (3), in the form of the Schwarz function, as: . Since: by comparing (19) and (13), we may get: Using (7) in (20), we get: Applying (11) in (21) and (22), we obtain: Thus, the proof is completed. Equalities in each coefficient |a 2 | , |a 3 |, and |a 4 | are obtained respectively by taking: Theorem 6. Let f ∈ R car and be given in the form (1). Then: Proof. From (5), the third Hankel determinant can be written as: Utilizing (20)-(23), we have: Implementing the triangle inequality, we have: Thus, the proof of this result is completed. By S (m) , we define the family of m-fold univalent functions having the following Taylor series form: