On the upper bounds of Hankel determinants for some subclasses of univalent functions associated with sine functions

ABSTRACT


Introduction
Let A be the family of functions of the form which are analytic in the open unit disk E = {z ∈ C : |z| < 1} and let S denote the subclass of A consisting of all univalent functions in E. With a view to recalling the principal of subordination between analytic functions, let f (z) and g (z) be analytic functions in E. Then we say that the function f (z) is subordinate to g (z) in E, if there exits a Schwarz function w (z) , analytic in E with w (0) = 0 and |w (z)| < 1, (z ∈ E) such that f (z) = g (w (z)) .We denote this subordination by If g is a univalent function in E, then f (z) ≺ g (z) ⇔ f (0) = g (0) and f (E) ⊂ g (E) .
The famous coefficient conjucture Beiberbach conjucture for the functions f ∈ S of the form (1) was first presented by Beiberbach [1] in 1916 and proven by de-Branges [2] in 1985.
In between the years 1916 and 1985, many mathematicians worked to prove Beiberbach's conjucture.Consequently, they defined several subclasses 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 subclasses of S and have a nice geometric interpretation.These families are defined as follows: We recall here which are connected with trigonometric functions and are defined as folows: The class S * sin of analytic function defined in (5) was introduced by Cho et al. [3].In the 1960s Pommerenke [4], [5] defined the Hankel determinant H q,n (f )for a given f of the form (1) f as follows where q, n ∈ N = {1, 2, 3, ...} .In particular, The studies on Hankel determinants are concentrated on estimating H 2,2 (f ) and H 3,1 (f ) for different subclasses of S.The absolute sharp bounds of the functional H 2,2 (f ) were found in [6], [7] for each of the families S * ,C, and R, where the familyR contains functions of bounded turning.In [7] [8], 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 subclasses of univalent functions; see [9][10][11][12][13][14][15][16].In 2017, Zaprawa [17] improved the results of Babalola.In 2018, Kowalczyk et al. [18] and Lecko et al. [19] 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 .Arif M. et al. [20] obtained the upper bound of |H 3,1 (f )|for the subclasses S * sin ,C sin and R sin in 2019.In 2019, Shi et al. [21] investigated the estimate of |H 3,1 (f )|for the subclasses S * car ,C car and R car in of analytic functions connected with the cardioid domain.In 2019, Zaprawa [22] studied the Hankel Determinant for Univalent Functions Related to the Exponential Function.
, ...} and λ ≥ 0. We let D n λ denote [24] the operator defined by We observe that D n λ is a linear operator and for Now, we define a subclass of analytic functions as follows: and f (z) is defined by (1).We define the classes of S * (λ,n) and C (λ,n) in the following way ) and In this present article, our aim is to investigate the estimate of |H 2,2 (f )| and |H 3,1 (f )| for the subclasses S * (λ,n) and C (λ,n) of analytic functions related with sine function.

Auxiliary lemmas
Let P denote the family of all functions p which are analytic in E with Re p (z) > 0 and has the following series representation Here p (z) is called the Caratheodory function [26].28], [29]) Let the function p (z) ∈ P be given by ( 11), then for some x, |x| ≤ 1, and for some complex value η, |η| ≤ 1.

Main results
Proof.From the definition of the class S * (λ,n) , we have where w is analytic in E with w (0) = 0 and After some simple calculations, we obtain From ( 20), ( 21) and ( 22), it follows that Applying Lemma 1 in ( 23) and ( 24), we obtain If the expression (25) is calculated by taking J = − 1 24 , K = 1 4 , L = 1 in the inequality (19), we obtain If the expression (26) is calculated using the Lemma 1 and ( 16) inequality, we have Proof.To obtain the (29) inequality, we will use the expression (18).If equations ( 23) and ( 24) are used,we can write Here, considering that and of the form (1), then Here, it is defined as ν = 18 Proof.From (23), ( 24) and ( 25), it follows that If the expression ( 31) is calculated by taking in the inequality (19),we obtain Now, Let's Look at the sign of With a simple calculation, we write Thus, since λ ≥ 0 must be, we get Proof.From ( 23), ( 24) and ( 25), it follows that If the triangle inequality is applied to this last equation, we obtain according to inequality, we can write Thus , we obtain Here, it is defined as ν = 18 17 Proof.If the absolute value of both sides of the expression is taken and the triangle inequality is applied, we can write In this last inequality, if the upper bound expressions discussed in Theorem 1, Theorem 2, Theorem 3, and Theorem 4, are written instead of and necessary operations are done, we obtain Here, it is defined as ν = 18 Proof.From the definition of the class C (λ,n) , we have where w is analytic in E with w (0) = 0 and |w (z)| < 1, z ∈ E.

U n c o r r e c t e d P r o o f
On the upper bounds of Hankel determinants for some subclasses of univalent functions . . .
Proof.To obtain the inequality (43) we will use the expression (18).If equations ( 36) and ( 37) are used,we can write Here , 3(1+2λ)  Proof.The proof of this theorem is similar to the one in Theorem 5. □

3. 1 .Theorem 1 .
The upper bound of the modules of Hankel's determinants for the coefficients of functions belonging to class S * (λ,n) If the function f (z) ∈ S * (λ,n) and of the form (1), then

Theorem 5 .
If the function f (z) ∈ S * (λ,n) and of the form (1), then U n c o r r e c t e d P r o o f