Coe ﬃ cient functionals for a class of bounded turning functions related to modiﬁed sigmoid function

: The main objective of the present article is to deﬁne the class of bounded turning functions associated with modiﬁed sigmoid function. Also we investigate and determine sharp results for the estimates of four initial coe ﬃ cients, Fekete-Szeg¨o functional, the second-order Hankel determinant, Zalcman conjucture and Krushkal inequality. Furthermore, we evaluate bounds of the third and fourth-order Hankel determinants for the class and for the 2-fold and 3-fold symmetric functions. and


Introduction
Let A represent the collections of analytic functions defined in open unit disc D = {z ∈ C : |z| < 1} whose normalization is of the form f (z) = z + ∞ n=2 a n z n (z ∈ D). (1.1) Let S denote the subclass of A comprising of functions of the form (1.1) which are also univalent in D.
Let P represent the class of all functions p that are analytic in D with (p(z)) > 0 and has the series representation p(z) = 1 + ∞ n=1 c n z n (z ∈ D). (1.2) Also we note that lately many subclasses of starlike functions are introduced see [7,9,12] by choosing some particular functions such as functions associated with Bell numbers, shell-like curve connected with Fibonacci numbers, functions connected with conic domains and rational functions instead of ϕ in (1.3). Pommerenke [24,25] introduced the Hankel determinant H q,n ( f ) for function f ∈ S of the form (1.1), where the parameters q, n ∈ N = {1, 2, 3, · · · } as follows: (1.4) The Hankel determinants for different orders are obtained for different values of q and n. When q = 2 and n = 1, the determinant is Note that H 2,1 ( f ) = a 3 − a 2 2 , is the classical Fekete-Szegö functional. For various subclasses of A, the best possible value of the upper bound for H 2,1 ( f ) was investigated by different authors (see [13][14][15] for details). Furthermore, when q = 2 and n = 2, the second Hankel determinant is The upper bound of H 2,2 ( f ) has been studied by several authors in the last few decades. For instance, the readers may refer to the works of Hayman [11], the Noonan and Thomas [22], Ohran et al. [23] and Shi et al. [34]. Moreover, Babalola [3] studied the Hankel determinant H 3,1 ( f ) for some subclasses of analytic functions. For some recent works on third order Hankel determinant we may refer the interested reader to such more recent works as (for example) [28,32,38]. The bound of the fourth Hankel determinant for a class of analytic functions with bounded turning associated with cardoid domain was approximated by Srivastava et al. in [37]. It should be remarked that a wide variety of applications of Hankel systems arise in linear filtering theory, discrete inverse scattering, and discretization of certain integral equations arising in mathematical physics [40]. Evaluating these Hankel determinants for various new subclasses has been an attracting area lately. One such field of interest is the Quantum Calculus (q-calculus), which is a generalization of classical calculus by replacing the limit by a parameter q. For the basics and preliminaries, the readers are advised to see the works and expositions in [31,35,37]. It is important to mention here the work on a q-differential operator by Srivastava et al. [33], in which they determined the upper bound of second Hankel determinant for a subclass of bi-univalent functions in q-analogue. Recently, the upper bound estimate for q-analogue of a subclass of starlike functions in connection with exponential function were evaluated in [36].
Recently, a class of starlike functions associated with Modified sigmoid function was defined by Goel and Kumar [10], i.e, Motivated by all the works mentioned above and [4], in this article we introduce and investigate the class R S G , which is defined as follows: We also establish some sharp results such as coefficient bounds, Fekete-Szegö inequality, second-order determinant, Zalcman conjecture and Krushkal inequality for functions belonging to the class R S G . Moreover, we estimate bounds of the third and forth-order Hankel determinants for this class R S G and for the 2-fold and 3-fold symmetric functions.

A set of lemmas
For the proofs of our main findings, we need the following lemmas.
Lemma 2. If p ∈ P and has the series of the form (1.2), then We note that the inequalities (2.3), (2.4) and (2.6) in the above can be found in [2,25] and (2.5) is given by [13].

Lemma 3. [2]
If p ∈ P and has the series of the form (1.2), then where J, K and L are real numbers.

Lemma 4.
[27] Let m, n, l and r satisfy the inequalities 0 < m < 1, 0 < r < 1 and If p ∈ P and has power series (1.2), then 3. Bounds of H 3,1 ( f ) for the class R S G Theorem 1. Let f ∈ R S G and be of the form (1.1). Then The first four inequalities are sharp for the functions defined below respectively Proof. Let f ∈ R S G . Then, (1.5) can be put in the form of Schwarz function w (z) as Also, if p ∈ P, then it may be written in terms of the Schwarz function w as or equivalently, By a simplification and using the series expansion (3.9), we have Comparing (3.10) and (3.11), we get For a 2 , putting (2.4) in (3.12), we have For a 3 , simplifying (3.13), we get and applying (2.3), we have For a 4 , using (3.14), we obtain By applying Lemma 3 to (3.18), we get For a 5 , applying Lemma 4 to (3.15), we get For a 6 , re-arranging (3.16) and applying the triangle inequality, we get By applying (2.3) and (2.4) to the above, we get For a 7 , re-arranging (3.17) and applying the triangle inequality, we get Also by using (2.3) and (2.4) to the above, we obtain |a 7 | ≤ 381 377 282 240 .
Next, we consider the Fekete-Szegö problem and the Hankel determinants for the class R S G .
Theorem 2. If f of the form (1.1) belongs to R S G , then The result is sharp for the function f 2 defined by (3.7) for |ζ| ≤ 8/3 and the functiom f 1 defined by (3.7) for |ζ| ≥ 8/3.
Proof. Using (3.12) and (3.13), we can write By rearranging we have Applying (2.5) we get Then with simple calculations, we obtain For the sharpness consider the function For the case |ζ| ≥ 8 3 consider which gives If we put ζ = 1, then the above result becomes: Corollary 1. If f of the form (1.1) belongs to R S G , then The result is sharp for the function f 3 defined by (3.7).
Theorem 4. If f of the form (1.1) belongs to R S G , then The result is sharp for the function f 2 defined by (3.7).

Using (2.3) and (2.4), we get the required result.
Theorem 6. If f ∈ A belongs to R S G , then The result is sharp for function f 4 defined by (3.7).
Proof. From (3.12)-(3.15), we have By applying of Lemma 4, we get the desired result.
We will now determine the bound of the third Hankel determinant H 3,1 ( f ) for f ∈ R S G .

Bounds of |H 4,1 ( f )| for the 2-fold and 3-fold symmetric functions
A function f is said to be m-fold symmetric if the following condition holds true for ε = exp 2πi m , The set of all m-fold symmetric functions belonging to the familiar class S of univalent functions is denoted by S (m) , represented by the following series expansion An analytic function f of the form (5.1) belongs to the class R (m) S G if and only if where p (z) belong to the class P (m) which is defined as follows: If a function f belongs to S (2) , then its series representation is and H 4,1 ( f ) = a 3 a 5 a 7 − a 3 3 a 7 + a 2 3 a 2 5 − a 3 5 . .

Zalcman functional
One of the main conjectures in Geometric function theory, suggested by Lawrence Zalcman in 1960, is that the coefficients of class S satisfy the inequality, Only the well-known Koebe function k(z) = z (1−z) 2 and its rotations have equality in the above form. For the popular Fekete-Szego inequality, when n = 2, the equality holds. Many researchers have researched Zalcman functional in the literature [5,8,19].
Theorem 12. Let f ∈ A belong to R S G . Then The result is sharp for the function f 4 defined by (3.7).
Proof. We use the Eqs (3.13) and (3.15) to get the Zalcman functional, and then we get Using Lemma 4, we can get the necessary result for the last expression.

Krushkal inequality for the class R S G
In this section we will give a direct proof of the inequality a p n − a p(n−1) 2 ≤ 2 p(n−1) − n p over the class R S G for the choice of n = 4, p = 1, and for n = 5, p = 1. Krushkal introduced and proved this inequality for the whole class of univalent functions in [17].
Theorem 13. Let f ∈ A belong to R S G . Then The result is sharp for the function f 3 defined by (3.7).
Proof. From Eqs (3.12) and (3.14), we get By applying (2.6) to the above,we get the required result.
The result is sharp for the function f 4 defined by (3.7).
Proof. From Eqs (3.12) and (3.14), we get By using Lemma 4, we can get the necessary result for the last expression.

Conclusions
In the present study, we have defined the class of bounded turning functions associated with modified sigmoid function. Also we have determined the sharp results for some coefficient functionals which play a very important role in the study of the geometric function theory. Furthermore, we have evaluated bounds of the third and fourth-order Hankel determinants for the 2-fold and 3-fold symmetric functions.
Recently, the usages of the quantum (or q-) calculus happens to provide another popular direction for researches in geometric function theory of complex analysis. This is evidenced by the recentlypublished survey-cum-expository review article by Srivastava [31]. Therefore the quantum (or q-) extensions of the results, which we have presented in this paper, are worthy of investigation.