Applications of Modified Sigmoid Functions to a Class of Starlike Functions

The main focus of this investigation is the applications of modified sigmoid functions. Due to its various uses in physics, engineering, and computer science, we discuss several geometric properties like necessary and sufficient conditions in the form of convolutions for functions to be in the special class SSG earlier introduced by Goel and Kumar and obtaining third-order Hankel determinant for this class using modified sigmoid functions. Also, the third-order Hankel determinant for 2and 3-fold symmetric functions of this class is evaluated.


Introduction
In this section, we present the related material for better understanding of the concepts discussed later in this article. We start with the notation of A, the class of functions f which are analytic in U = fz ∈ ℂ : jzj < 1g and its series representation is f z ð Þ = z + 〠 ∞ n=2 a n z n , z ∈ U: Further, a subclass of class A which is denoted by S contains all univalent functions in U: Bieberbach conjectured in 1916 that |a n | ≤n, n = 2, 3, ⋯. De Branges proved this in 1985; see [1]. During this period, a lot of coefficient results were established for some subfamilies of S. Some of these classes are the class S * , known as the class of starlike functions, the class K, known as class of convex functions, and R of bounded turning functions. These are defined as K ψ ð Þ = f ∈ S : Now, recall the subordination definition; we say that an analytic function f 1 ðzÞ is subordinate to f 2 ðzÞ in U and is symbolically written as f 1 ðzÞ ≺ f 2 ðzÞ if there occurs a Schwarz function uðzÞ with properties that juðzÞj ≤ 1 and uð0Þ = 1 such that f 1 ðzÞ = f 2 ðuðzÞÞ. Moreover, if f 2 ðzÞ is in the class S, then we have the following equivalency, due to [2,3], For two functions f 1 ðzÞ = z + ∑ ∞ n=2 a n,1 z n and f 2 ðzÞ = z + ∑ ∞ n=2 a n,2 z n in U, then the convolution or Hadamard product is defined by a n,1 a n,2 z n : By varying the right-hand side of subordinated inequality in (2), several familiar classes can be obtained such as the following: (1) For ψ = ð1 + AzÞ/ð1 + BzÞ , we get the class S * ðA, BÞ; see [4] for details (2) While for different values of A and B the class S * ðαÞ = S * ð1 − 2α,−1Þ is obtained and investigated in [5] (3) For ψ = 1 + ð2/π 2 Þðlog ðð1 + ffiffi ffi z p Þ/ð1 − ffiffi ffi z p ÞÞÞ 2 , the class was defined and studied in [6] (4) For ψ = ffiffiffiffiffiffiffiffiffi ffi 1 + z p , the class is denoted by S * L ; details can be seen in [7,8], and for further study, see [9] (5) For ψ = cosh ðzÞ, the class is denoted by S * cosh ; see [10] (6) For ψ = 1 + sin ðzÞ, the class is denoted by S * sin ; see [11] for details, and for further investigation, see [12] (7) While for ψ = z + ffiffiffiffiffiffiffiffiffiffiffi ffi 1 + z 2 p , the class is denoted by S * l ; see [13] (8) For ψ = e z , the class denoted by S * e was defined and studied in [14,15] (9) Similarly, if ψ = 1 + ð4/3Þz + ð2/3Þz 2 , then such a class is denoted by S * C and was introduced in [16], and for further study, the reader is referred to [17] Also, several other subclasses of starlike functions were introduced recently in [18][19][20][21][22] by choosing some particular function for ψ such functions are associated with Bell numbers, shell-like curve connected with Fibonacci numbers, and functions connected with the conic domains.
In this paper, we investigate starlike functions associated with a kind of special functions known as modified sigmoid function ψðzÞ = 2/ð1 + e −z Þ. In mathematics, the theory of special functions is the most important for scientists and engineers who are concerned with actual mathematical calculations. To be specific, it has applications in problems of physics, engineering, and computer science. The activation function is an example of special function. These functions act as a squashing function which is the output of a neuron in a neural network between certain values (usually 0 and 1 and -1 and 1). There are three types of functions, namely, piecewise linear function, threshold function, and sigmoid function. In the hardware implementation of neural network, the most important and popular activation function is the sigmoid function. The sigmoid function is often used with gradient descendent type learning algorithm. Due to differentiability of the sigmoid function, it is useful in weightlearning algorithm. The sigmoid function increases the size of the hypothesis space that the network can represent. Some of its advantages are the following: (1) It gives real numbers between 0 and 1 (2) It maps a very large output domain to a small range of outputs (3) It never loses information because it is a one-to-one function (4) It increases monotonically For more details, see [23]. The class S * SG defined by Goel and Kumar in [24] is defined as For a parameter q, with n ∈ ℕ = f1, 2, 3,⋯g, Pommerenke [25,26] defined Hankel determinant H q,n ð f Þ for functions f ∈ S of the form (1) as follows: The growth of H q,n ð f Þ has been evaluated for different subcollections of univalent functions. Exceptionally, for each of the sets K, S * , and R, the sharp bound of the determinant H 2,2 ð f Þ = ja 2 a 4 − a 2 3 j was found by Jangteng et al. [7,27], while for the family of close-to-convex functions the sharp estimate is still unknown (see [28]). On the other hand, for the set of Bazilevic functions, the best estimate of jH 2,2 ð f Þj was proved by Krishna et al. [29]. For more work on H 2,2 ð f Þ, see [30][31][32][33][34].
The determinant is known as the third-order Hankel determinant, and the estimation of this determinant jH 3,1 ðf Þj is the focus of various researchers of this field. In 2010, the first article on H 3,1 ð f Þ was published by Babalola [35], in which he obtained the upper bound of jH 3,1 ð f Þj for the classes of S * , K, and R. Later on, a few mathematicians extended this work for various subcollections of holomorphic and univalent 2 Journal of Function Spaces functions; see [36][37][38][39][40][41]. In 2017, Zaprawa [42] improved their work by proving 60 , for f ∈ R: And he asserted that these inequalities are not sharp as well.
Additionally, for the sharpness, he investigated the subfamilies of S * , C, and R comprising functions with m-fold symmetry and acquired the sharp bounds. Recently, in 2018, Kowalczyk et al. [43] and Lecko et al. [44] got the sharp inequalities which are for the classes K and S * ð1/2Þ, respectively, where the symbol S * ð1/2Þ indicates the family of starlike functions of order 1/2. Additionally, in 2018, the authors [45] got an improved bound jH 3,1 ðf Þj ≤ 8/9 for f ∈ S * , which is yet not the best possible. In this article, our main purpose is to study necessary and sufficient conditions for functions to be in the class S * SG in the form of convolutions results, coefficient inequality, and important third-order Hankel determinant for this class in (7) and also for its 2-and 3-fold symmetric functions:

A Set of Lemmas
Let P be the family of functions pðzÞ that are holomorphic in D with RepðzÞ > 0 and its series form is as follows: Lemma 1. If pðzÞ ∈ P and it is of the form (12), then c n+2k − δc n c 2 and for ξ ∈ ℂ.
Theorem 5. Let f ðzÞ ∈ A be of the form (1), then the necessary and sufficient condition for function f ðzÞ that belongs to class S * SG is Proof. In the light of Theorem 4, we show that S * SG if and only if Þ a n z n−1 Hence, the proof is completed.
Theorem 6. Let f ∈ A be of the form (1) and satisfies Proof. To show f ∈ S * SG , we have to show that (28) is satisfied. Consider Þ a n z n−1 Proof. Since f ∈ S * SG , then there exists an analytic function wðzÞ, jwðzÞj ≤ 1 and wð0Þ = 0, such that Denote Obviously, the function kðzÞ ∈ P and wðzÞ = ðkðzÞ − 1Þ/ ðkðzÞ + 1Þ. This gives Journal of Function Spaces On equating coefficients of (36) and (37), we get Now from (38) and (39), we have Now, using (18), we get the required result. If we put λ = 1, the above result becomes as follows.
Corollary 8. Let f ðzÞ ∈ S * SG be of the form (1) then The result is best possible for function Theorem 9. Let f ðzÞ ∈ S * SG be of the form (1), then a 2 a 3 − a 4 j j≤ The result is best possible for function defined as Applying Lemma 3, we get the required result.
For the third Hankel determinant, we need the following result. Lemma 11. [24]. Let f ðzÞ ∈ S * SG be of the form (1). Then, These results are sharp for function defined as , for a n n = 2, 3, 4, 5 ð Þ : Theorem 12. Let f ðzÞ ∈ S * SG be of the form (1). Then, Proof. Since by applying triangle inequality, we obtain Now, using Corollary 8, Theorems 9 and 10, and Lemma 11, we get the required result.
By S ðmÞ , we mean the set of m-fold symmetric univalent functions having the following series form The subclass S * ðmÞ SG is a set of m-fold symmetric starlike functions associated with modified sigmoid function. More precisely, an analytic function f of the form (57) belongs to class S * ðmÞ where the set P ðmÞ is defined by Theorem 13. If f ∈ S * ð2Þ SG be of the form (57), then Proof. Since f ∈ S * ð2Þ SG ; therefore, there exists a function p ∈ P ð2Þ such that Using the series form (57) and (59), when m = 2 in the above relation, we have Now, Therefore, Using (13) and (14) along with triangle inequality, we get Theorem 14. If f ∈ S * ð3Þ SG be of the form (57), then Proof. Since f ∈ S * ð3Þ SG ; therefore, there exists a function p ∈ P ð3Þ such that Using the series form (57) and (59), when m = 3 in the above relation, we have Now, Therefore, Using (13), we get The result is best possible for function defined as follows: Data Availability The data used in this article are artificial and hypothetical, and anyone can use these data before prior permission by just citing this article.