# Applications of Modified Sigmoid Functions to a Class of Starlike Functions

Authors: Muhammad Ghaffar Khan; Bakhtiar Ahmad; Gangadharan Murugusundaramoorthy; Ronnason Chinram; Wali Khan Mashwani
Journal: Journal of Function Spaces
 (2020)

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## 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 classSSG∗ 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.


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## Body

## 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 ofA, the class of functions f which are analytic in U=z∈ℂ:z<1 and its series representation is
(1)fz=z+∑n=2∞anzn,z∈U.Further, a subclass of classA which is denoted by S contains all univalent functions in U. Bieberbach conjectured in 1916 that ∣an∣≤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
(2)S∗ψ=f∈S:zf′zfz≺ψ=1+z1−z,z∈U,(3)Kψ=f∈S:zf′z′f′z≺ψ=1+z1−z,z∈U,(4)Rψ=f∈S:f′z≺ψ=1+z1−z,z∈U.Now, recall the subordination definition; we say that an analytic functionf1z is subordinate to f2z in U and is symbolically written as f1z≺f2z if there occurs a Schwarz function uz with properties that uz≤1 and u0=1 such that f1z=f2uz. Moreover, if f2z is in the class S, then we have the following equivalency, due to [2, 3],
(5)f10=f20,f1U⊆f2U.For two functionsf1z=z+∑n=2∞an,1zn and f2z=z+∑n=2∞an,2zn in U, then the convolution or Hadamard product is defined by
(6)f1∗f2z=z+∑n=2∞an,1an,2zn.By varying the right-hand side of subordinated inequality in (2), several familiar classes can be obtained such as the following:
(1)
Forψ=1+Az/1+Bz, we get the class S∗A,B; see [4] for details(2)
While for different values ofA and B the class S∗α=S∗1−2α,−1 is obtained and investigated in [5](3)
Forψ=1+2/π2log1+z/1−z2, the class was defined and studied in [6](4)
Forψ=1+z, the class is denoted by SL∗; details can be seen in [7, 8], and for further study, see [9](5)
Forψ=coshz, the class is denoted by Scosh∗; see [10](6)
Forψ=1+sinz, the class is denoted by Ssin∗; see [11] for details, and for further investigation, see [12](7)
While forψ=z+1+z2, the class is denoted by Sl∗; see [13](8)
Forψ=ez, the class denoted by Se∗ was defined and studied in [14, 15](9)
Similarly, ifψ=1+4/3z+2/3z2, then such a class is denoted by SC∗ 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 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 functionψz=2/1+e−z. 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 monotonicallyFor more details, see [23].The classSSG∗ defined by Goel and Kumar in [24] is defined as
(7)zf′zfz≺21+e−z,z∈U.For a parameterq, with n∈ℕ=1,2,3,⋯, Pommerenke [25, 26] defined Hankel determinant Hq,nf for functions f∈S of the form (1) as follows:
(8)Hq,nf=anan+1⋯an+q−1an+1an+2⋯an+q⋮⋮⋯⋮an+q−1an+q⋯an+2q−2.The growth ofHq,nf 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 H2,2f=a2a4−a32 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 H2,2f was proved by Krishna et al. [29]. For more work on H2,2f, see [30–34].The determinant(9)H3,1f=1a2a3a2a3a4a3a4a5is known as the third-order Hankel determinant, and the estimation of this determinant H3,1f is the focus of various researchers of this field. In 2010, the first article on H3,1f was published by Babalola [35], in which he obtained the upper bound of H3,1f for the classes of S∗, K, and R. 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
(10)H3,1f≤1,forf∈S∗,49540,forf∈K,4160,forf∈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
(11)H3,1f≤4/135,H3,1f≤1/9,for the classes K and S∗1/2, respectively, where the symbol S∗1/2 indicates the family of starlike functions of order 1/2. Additionally, in 2018, the authors [45] got an improved bound H3,1f≤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 SSG∗ 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

LetP be the family of functions pz that are holomorphic in D with Repz>0 and its series form is as follows:
(12)pz=1+∑n=1∞cnznz∈D.Lemma 1.
Ifpz∈P and it is of the form (12), then
(13)cn≤2forn≥1,(14)cn+k−δcnck≤2for0≤δ≤1,(15)cncm−clck≤4forn+m=l+k,(16)cn+2k−δcnck2≤21+2δforδ∈ℝ,(17)c2−c122≤2−c122,and for ξ∈ℂ.
(18)c2−ξc12≤2max1;2ξ−1.For the results in (13), (14), (15), (16), and (17), see [46]. Also, see [47] for (18).Lemma 2.
[48]. If pz∈P and is represented by (12), then
(19)c2−νc12≤−4v+2v≤0,20≤v≤1,4v−2v≥1.Lemma 3.
Letp∈P have representation of the form (12), then
(20)αc13−βc1c2+γc3≤2α+2β−2α+2α−β+γ.Proof.
Consider the left-hand side of (20) and then rearranging the terms, we have
(21)αc13−βc1c2+γc3=αc13−2c1c2+c3−β−2αc1c2−c3+α−β+γc3≤αc13−2c1c2+c3+β−2αc1c2−c3+α−β+γc3≤2α+2β−2α+2α−β+γ,where we have used (13) and (14).


## 3. Convolution Results for ClassSSG∗

Theorem 4.
Letfz∈A be the form (1), then fz∈SSG∗, if and only if
(22)1zfz∗z−αz21−z2≠0,for all α=αξ=2/1−e−ξ and also for α=1.Proof.
Sincefz∈SSG∗ is analytic in domain U, so fz≠0, for all z∈U∗=U0, that is 1/zfz≠0 for z∈U, which is equivalent to (22) for α=1. In this case, the proof is completed. Now, from definition (7), there occurs a Schwarz function uz, such that uz<1 and u0=0, such that
(23)zf′zfz=21+e−uz.Equivalently,
(24)zf′zfz≠21+e−ξ,ξ=1,which implies that(25)zf′z−fz21+e−ξ≠0.We know that(26)zf′z=fz∗z1−z2,fz=fz∗z1−z.By simple computation, equation (25) becomes
(27)1zfz∗z−αz21−z2≠0,where α is given above.Conversely, suppose equation (22) holds true for α=1, it implies that 1/zfz≠0, for all z∈U. Let Φz=zf′z/fz be analytic in U, with Φ0=1. Also, suppose that Ψz=2/1+e−z, z∈U. It is clear from (24) that Ψ∂U∩ΦU=ϕ. Hence, the simply connected domain ΦU is contained in connected component of Ψ∂U. The univalence of "Ψ", together with the fact Φ0=Ψ0=1, shows that Φ≺Ψ and implies that fz∈SSG∗.Theorem 5.
Letfz∈A be of the form (1), then the necessary and sufficient condition for function fz that belongs to class SSG∗ is
(28)1−∑n=2∞n1−e−ξ−21−e−ξanzn−1≠0.Proof.
In the light of Theorem4, we show that SSG∗ if and only if
(29)0≠1zfz∗z−αz21−z2=1zzf′z−αzf′z−fz=1−∑n=2∞α−1n−αanzn−1=1−∑n=2∞n1−e−ξ−21−e−ξanzn−1.Hence, the proof is completed.Theorem 6.
Letf∈A be of the form (1) and satisfies
(30)∑n=2∞n1−e−ξ−21−e−ξan<1,then f∈SSG∗.Proof.
To showf∈SSG∗, we have to show that (28) is satisfied. Consider
(31)1−∑n=2∞α−1n−αanzn−1>1−∑n=2∞α−1n−αanzn−1=1−∑n=2∞α−1n−αanzk−1>1−∑n=2∞α−1n−αan=1−∑n=2∞n1−e−ξ−21−e−ξan>0,so by Theorem 5, fz∈SSG∗.


## 4. Upper BoundH3,1f for Set SSG∗

Theorem 7.
Letf∈SSG∗ and is of the form (1), then
(32)a3−λa22≤14max1,2λ−12.Proof.
Sincef∈SSG∗, then there exists an analytic function wz, wz≤1 and w0=0, such that
(33)zf′zfz=21+e−wz.Denote(34)Ψwz=21+e−wz,kz=1+c1z+c2z2+⋯=1+wz1−wz.Obviously, the functionkz∈P and wz=kz−1/kz+1. This gives
(35)wz=kz−1kz+1=c1z+c2z2+c3z3+⋯2+c1z+c2z2+c3z3+⋯,(36)21+e−wz=1+14c1z+14c2−18c12z2+11192c13−14c2c1+14c3z3+14c12c2−12c3c1−14c22+12c4z4+⋯,while
(37)zf′zfz=1+a2z+2a3−a22z2+a23−3a2a3+3a4z3+−a24+4a22a3−4a2a4−2a32+4a5z4+⋯.On equating coefficients of (36) and (37), we get
(38)a2=14c1,(39)a3=18c2−132c12,(40)a4=71152c13−596c2c1+112c3,(41)a5=−116171152c14−724c12c2+23c3c1+38c22−c4.Now from (38) and (39), we have
(42)a3−λa22=18c2−2λ+14c12.Now, using (18), we get the required result.If we putλ=1, the above result becomes as follows.Corollary 8.
Letfz∈SSG∗ be of the form (1) then
(43)a3−a22≤14.The result is best possible for function(44)fz=zexp∫0zet2−1tet2+1dt=z+14z3+⋯.Theorem 9.
Letfz∈SSG∗ be of the form (1), then
(45)a2a3−a4≤16.The result is best possible for function defined as(46)fnz=zexp∫0zet3−1tet3+1dt=z+16z4+⋯.Applying Lemma 3, we get the required result.Proof.
By using (38), (39), and (40), we get
(47)a2a3−a4=172c13−112c2c1+112c3.Applying Lemma 3, we get the required result.Theorem 10.
Letfz∈SSG∗ be of the form, (1) then
(48)a2a4−a32≤55576.Proof.
With the help of (38), (39), and (40), we get
(49)a2a4−a32=148c3c1−79216c14−1192c12c2−164c22.Now, rearranging the terms
(50)a2a4−a32=c1192c3−c1c2−c1c3−c2264−79216c14.Using (13), (14), and (15), we get the required result.For the third Hankel determinant, we need the following result.Lemma 11.
[24]. Let fz∈SSG∗ be of the form (1). Then,
(51)a2≤12,a3≤14,a4≤16,a5≤18.These results are sharp for function defined as(52)fnz=zexp∫0zetn−1−1tetn−1+1dt,forann=2,3,4,5.Theorem 12.
Letfz∈SSG∗ be of the form (1). Then,
(53)H3,1f≤1912304≃0.0829.Proof.
Since(54)H3,1f=a3a2a4−a32−a4a4−a2a3+a5a3−a22,by applying triangle inequality, we obtain
(55)H3,1f≤a3a2a4−a32+a4a4−a2a3+a5a3−a22.Now, using Corollary8, Theorems 9 and 10, and Lemma 11, we get the required result.


## 5. Bounds ofH3,1f for 2-Fold and 3-Fold Symmetric Functions

Letm∈ℕ=1,2,3,⋯, if a rotation of domain D about the origin through an angle 2π/m carries itself on the domain D is called m-fold symmetric. It is very much clear to see that an analytic function f is m-fold symmetric in D, if
(56)fe2πmz=e2πmfz,z∈D.BySm, we mean the set of m-fold symmetric univalent functions having the following series form
(57)fz=z+∑k=2∞amk+1zmk+1,z∈D.The subclassSSG∗m 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 SSG∗m if and only if
(58)zf′zfz=21+e−pz−1/pz+1,p∈Pm,where the set Pm is defined by
(59)Pm=p∈P:pz=1+∑k=1∞cmkzmk,z∈D.Theorem 13.
Iff∈SSG∗2 be of the form (57), then
(60)H3,1f≤132.Proof.
Sincef∈SSG∗2; therefore, there exists a function p∈P2 such that
(61)zf′zfz=21+e−pz−1/pz+1.Using the series form (57) and (59), when m=2 in the above relation, we have
(62)a3=18c2,a5=116c4−3128c22.Now,(63)H3,1f=a3a5−a32.Therefore,(64)H3,1f=c2128c4−38c22.Using (13) and (14) along with triangle inequality, we get
(65)H3,1f≤132.Theorem 14.
Iff∈SSG∗3 be of the form (57), then
(66)H3,1f≤136.Proof.
Sincef∈SSG∗3; therefore, there exists a function p∈P3 such that
(67)zf′zfz=21+e−pz−1/pz+1.Using the series form (57) and (59), when m=3 in the above relation, we have
(68)a4=112c3.Now,(69)H3,1f=−a42.Therefore,(70)H3,1f=1144c32.Using (13), we get
(71)H3,1f≤136.The result is best possible for function defined as follows:(72)f4z=zexp∫0zet3−1tet3+1dt=z+16z4+⋯.



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