Applications of Modified Sigmoid Functions to a Class of Starlike Functions

Khan, Muhammad Ghaffar; Ahmad, Bakhtiar; Murugusundaramoorthy, Gangadharan; Chinram, Ronnason; Mashwani, Wali Khan

doi:https://doi.org/10.1155/2020/8844814

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Research Article | Open Access

Volume 2020 | Article ID 8844814 | https://doi.org/10.1155/2020/8844814

Applications of Modified Sigmoid Functions to a Class of Starlike Functions

Muhammad Ghaffar Khan,¹Bakhtiar Ahmad,²Gangadharan Murugusundaramoorthy,³Ronnason Chinram,⁴and Wali Khan Mashwani¹

Academic Editor: Acu Ana Maria

Received17 Sept 2020

Revised11 Oct 2020

Accepted01 Nov 2020

Published19 Nov 2020

Abstract

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 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 2- and 3-fold symmetric functions of this class is evaluated.

1. 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 , the class of functions which are analytic in and its series representation is

Further, a subclass of class which is denoted by contains all univalent functions in Bieberbach conjectured in 1916 that , . De Branges proved this in 1985; see [1]. During this period, a lot of coefficient results were established for some subfamilies of . Some of these classes are the class , known as the class of starlike functions, the class , known as class of convex functions, and of bounded turning functions. These are defined as

Now, recall the subordination definition; we say that an analytic function is subordinate to in and is symbolically written as if there occurs a Schwarz function with properties that and such that . Moreover, if is in the class , then we have the following equivalency, due to [2, 3],

For two functions and in , then the convolution or Hadamard product is defined by

By varying the right-hand side of subordinated inequality in (2), several familiar classes can be obtained such as the following: (1)For , we get the class ; see [4] for details(2)While for different values of and the class is obtained and investigated in [5](3)For , the class was defined and studied in [6](4)For , the class is denoted by ; details can be seen in [7, 8], and for further study, see [9](5)For , the class is denoted by ; see [10](6)For , the class is denoted by ; see [11] for details, and for further investigation, see [12](7)While for , the class is denoted by ; see [13](8)For , the class denoted by was defined and studied in [14, 15](9)Similarly, if , then such a class is denoted by 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 [18–22] by choosing some particular function for 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 . 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 weight-learning 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 defined by Goel and Kumar in [24] is defined as

For a parameter , with , Pommerenke [25, 26] defined Hankel determinant for functions of the form (1) as follows:

The growth of has been evaluated for different subcollections of univalent functions. Exceptionally, for each of the sets , , and , the sharp bound of the determinant 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 Bazilevi functions, the best estimate of was proved by Krishna et al. [29]. For more work on , see [30–34].

The determinant is known as the third-order Hankel determinant, and the estimation of this determinant is the focus of various researchers of this field. In 2010, the first article on was published by Babalola [35], in which he obtained the upper bound of for the classes of , , and . Later on, a few mathematicians extended this work for various subcollections of holomorphic and univalent functions; see [36–41]. In 2017, Zaprawa [42] improved their work by proving And he asserted that these inequalities are not sharp as well. Additionally, for the sharpness, he investigated the subfamilies of , , and comprising functions with -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 and , respectively, where the symbol indicates the family of starlike functions of order . Additionally, in 2018, the authors [45] got an improved bound for , 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 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

2. A Set of Lemmas

Let be the family of functions that are holomorphic in with and its series form is as follows:

Lemma 1. If and it is of the form (12), then and for .

For the results in (13), (14), (15), (16), and (17), see [46]. Also, see [47] for (18).

Lemma 2. [48]. If and is represented by (12), then

Lemma 3. Let have representation of the form (12), then

Proof. Consider the left-hand side of (20) and then rearranging the terms, we have where we have used (13) and (14).

3. Convolution Results for Class

Theorem 4. Let be the form (1), then , if and only if for all and also for

Proof. Since is analytic in domain , so , for all , that is for , which is equivalent to (22) for . In this case, the proof is completed. Now, from definition (7), there occurs a Schwarz function , such that and , such that Equivalently,

which implies that

We know that By simple computation, equation (25) becomes where is given above.

Conversely, suppose equation (22) holds true for , it implies that , for all . Let be analytic in , with . Also, suppose that , . It is clear from (24) that . Hence, the simply connected domain is contained in connected component of . The univalence of , together with the fact , shows that and implies that .

Theorem 5. Let be of the form (1), then the necessary and sufficient condition for function that belongs to class is

Proof. In the light of Theorem 4, we show that if and only if Hence, the proof is completed.

Theorem 6. Let be of the form (1) and satisfies then .

Proof. To show , we have to show that (28) is satisfied. Consider so by Theorem 5, .

4. Upper Bound for Set

Theorem 7. Let and is of the form (1), then

Proof. Since , then there exists an analytic function , and , such that

Denote

Obviously, the function and . This gives while

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 , the above result becomes as follows.

Corollary 8. Let be of the form (1) then

The result is best possible for function

Theorem 9. Let be of the form (1), then

The result is best possible for function defined as Applying Lemma 3, we get the required result.

Proof. By using (38), (39), and (40), we get Applying Lemma 3, we get the required result.

Theorem 10. Let be of the form, (1) then

Proof. With the help of (38), (39), and (40), we get Now, rearranging the terms Using (13), (14), and (15), we get the required result.

For the third Hankel determinant, we need the following result.

Lemma 11. [24]. Let be of the form (1). Then,

These results are sharp for function defined as

Theorem 12. Let 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.

5. Bounds of for 2-Fold and 3-Fold Symmetric Functions

Let , if a rotation of domain about the origin through an angle carries itself on the domain is called -fold symmetric. It is very much clear to see that an analytic function is -fold symmetric in , if

By , we mean the set of -fold symmetric univalent functions having the following series form

The subclass is a set of -fold symmetric starlike functions associated with modified sigmoid function. More precisely, an analytic function of the form (57) belongs to class if and only if where the set is defined by

Theorem 13. If be of the form (57), then

Proof. Since ; therefore, there exists a function such that

Using the series form (57) and (59), when in the above relation, we have

Now,

Therefore, Using (13) and (14) along with triangle inequality, we get

Theorem 14. If be of the form (57), then

Proof. Since ; therefore, there exists a function such that

Using the series form (57) and (59), when 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.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors jointly worked on the results, and they read and approved the final manuscript.

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Copyright © 2020 Muhammad Ghaffar Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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