Third Hankel Determinant for the Logarithmic Coefﬁcients of Starlike Functions Associated with Sine Function

: The logarithmic functions have been used in a verity of areas of mathematics and other sciences. As far as we know, no one has used the coefficients of logarithmic functions to determine the bounds for the third Hankel determinant. In our present investigation, we first study some well-known classes of starlike functions and then determine the third Hankel determinant bound for the logarithmic coefficients of certain subclasses of starlike functions that also involve the sine functions. We also obtain a number of coefficient estimates. Some of our results are shown to be sharp.


Introduction
We denote by A the class of analytic (holomorphic) functions f defined in the open unit disk U = {z : z ∈ C and |z| < 1}, which satisfy the following normalization conditions f (0) = 0 and f (0) = 1.
Thus, each f ∈ A has the following series form: Moreover, we denote by S the subclass of A of functions which are univalent in U. For two functions h 1 , h 2 ∈ A, we say that the function h 1 is subordinate to the function h 2 (written as h 1 ≺ h 2 ) if there exists an analytic function w with the property |w(z)| ≤ |z| and w(0) = 0 such that h 1 (z) = h 2 (w(z)) (z ∈ U).
Moreover, if h 2 ∈ S, then the above conditions can be written as: In 1992, Ma and Minda [1] introduced the class S * (Φ) as follows: where the function Φ is assumed to be analytic with positive real part on U such that Φ(U) is axially symmetric and starlike with respect to Moreover, they investigated a number of useful geometric properties such as growth, distortion and covering results. By putting specifically, then we can see that the functions class S * (Φ) is similar to that of the wellknown class of starlike functions. For the various choices of the function Φ, we have the following function classes: then we obtain the class S * sin = S * (1 + sin z), of starlike functions whose image under an open unit disk is eight-shaped (see [2]).

2.
For the choice we obtain the class whose image is bounded by a nephroid-shaped region (see [3]). 3.
If we put then the function class leads to the class the class of starlike functions associated with the lemniscate of Bernoulli (see [4]).

4.
Moreover, if we take we obtain the class S * car = S * 1 + which is the class of starlike functions whose image under open unit is a cardioid shape and was introduced by Sharma et al. [5].

5.
Furthermore, if we pick Φ(z) = e z we obtain the class S * exp = S * (e z ), which was introduced and studied by Mendiratta et al. [6]. 6.
If we put Φ(z) = √ 1 + z + z, then we have the class of starlike functions associated with the crescent-shaped region as discussed in [7].
The generalizations of the class S * were studied by many authors. Indeed, they replaced Φ in (2) with Fibonacci numbers, Bell numbers, shell-like curves, conic domains and a modified sigmoid function [8][9][10][11], and they have defined some other generalized subclasses of the class of starlike functions.
It was Pommerenke [12,13] who studied the Hankel determinant H q,n ( f ) for a function f ∈ A written as in (1). The Hankel determinant H q,n ( f ) is given as follows: For different values of q and n, the Hankel determinants for various orders are derived. For example, when n = 1 and q = 2, the above-defined determinant becomes as follows: We note that the nth coefficient of a function class S is well known to be bounded by n, and the coefficient limits give information about the function's geometric characteristics. The famous problem solved by Fekete-Szegö [14] is to determine the greatest value of the coefficient functional |a 3 − σa 2 2 | over the class S for each σ ∈ [0, 1], which was demonstrated using the Loewner technique. For a detailed study about this well-known functional, see [15][16][17]. Furthermore, if we take q = n = 2, then we have the second Hankel determinant In recent years, many authors have studied and investigated the upper bound of |H 2,2 ( f )| for different subclasses of analytic functions. A few of them are Noonan and Thomas [18], Hayman [19], Ohran et al. [20] and Shi et al. [21]. Furthermore, the bounds for the third Hankel determinant were first investigated by Babalola [22]. Some recent and interesting works on this topic maybe found in [23][24][25][26].
In [2], Cho et al. defined and studied a class of starlike functions associated with the sine function, defined as follows: The logarithmic coefficients of f ∈ S, denoted by γ n = γ n ( f ), are defined by the following series expansion: Logarithmic coefficients have recently attracted considerable interest. For instance, Milin's conjecture highly depends on logarithmic coefficients (see [27]; see also ([28], page 155)). Ali et al. [29] investigated the logarithmic coefficients of some close-to-convex functions, while the third logarithmic coefficient in some subclasses of close-to-convex functions was studied by Cho et al. [30]. Moreover, logarithmic coefficients of univalent functions can be found in [31]. Very recently, Kowalczyk and Lecko [32] have studied the Hankel matrices whose entries are logarithmic coefficients of univalent functions and have given sharp bounds for the second Hankel determinant of logarithmic coefficients of convex and starlike functions. For some other related works, see [33][34][35]. For a function f given by (1), the logarithmic coefficients are as follows: Based on all of the above ideas, we propose the study of the Hankel determinant, whose entries are logarithmic coefficients of f ∈ S, that is The main aim of this paper is to find upper bounds for H 3,1 ( f ) for the class of starlike functions associated with the sine functions.

A Set of Lemmas
We denote by P the class of analytic functions p which are normalized by and have the following form: To prove our main results, we need the following lemmas.
where I, X and V are real numbers.

Main Results
Theorem 1. If f ∈ S * sin and it has the form given in (1), then The following functions are examples for the sharpness of the above first four inequalities respectively.
Proof. Let f ∈ S * sin and then, by the definitions of subordinations, there exists a Schwartz function w(z) with the properties that w(0) = 0 and w(z) < 1, Define the function It is clear that p(z) ∈ P. This implies that and 1 + sin(w(z)) = 1 + 1 2 Comparing (31) and (32), we achieve Now, from (5) to (9) and (33) to (37), we obtain Applying (14) to (38), we get From (39) and using (18), we have Clearly, H(c 1 ) is a decreasing function and its maximum is attained at c 1 = 0, hence Applying Lemma 3 on Equation (40), we get Moreover, using Lemma 3 on (41), we get Rearranging (42), we obtain By making use of (14) and (15), along with the triangular inequality, we can easily obtain the desired result.
To prove the sharpness of (21) to (24), observe that It follows that these inequalities are sharp.
Theorem 2. If f ∈ S * sin and it has the form given in (1), then The function f 2 given in (27) is an example of sharpness for this result.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
No data were used to support this study.