An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function

Lei Shi 1 , Hari Mohan Srivastava 2,3 , Muhammad Arif 4,* and Shehzad Hussain 4 and Hassan Khan 4 1 School of Mathematics and Statistics, Anyang Normal University, Anyan 455002, Henan, China; shimath@163.com 2 Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada; harimsri@math.uvic.ca 3 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 4 Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan; shehzad873822@gmail.com (S.H.); hassanmath@awkum.edu.pk (H.K.) * Correspondence: marifmaths@awkum.edu.pk


Introduction and Definitions
Let the collection of functions f that are holomorphic in ∆ = {z ∈ C : |z| < 1} and normalized by conditions f (0) = f (0) − 1 = 0 be denoted by the symbol A. Equivalently; if f ∈ A, then the Taylor-Maclaurin series representation has the form: Further, let we name by the notation S the most basic sub-collection of the set A that are univalent in ∆.The familiar coefficient conjecture for the function f ∈ S of the form (1) was first presented by Bieberbach [1] in 1916 and proved by de-Branges [2] in 1985.In 1916-1985, many mathematicians struggled to prove or disprove this conjecture and as result they defined several subfamilies of the set S of univalent functions connected with different image domains.Now we mention some of them, that is; let the notations S * , C and K, shows the families of starlike, convex and close-to-convex functions respectively and are defined as: where the symbol " ≺ " denotes the familiar subordinations between analytic functions and is define as; the function h 1 is subordinate to a function h 2 , symbolically written as h 1 ≺ h 2 or h 1 (z) ≺ h 2 (z), if we can find a function w, which is holomorphic in ∆ with w (0 In case of univalency of h 1 in ∆, then the following relation holds: In [3], Padmanabhan and Parvatham in 1985 defined a unified families of starlike and convex functions using familiar convolution with the function z/ (1 − z) a , for all a ∈ R. Later on, Shanmugam [4] generalized the idea of paper [3] and introduced the set where " * " stands for the familiar convolution, φ is a convex and h is a fixed function in A. We obtain the families S * (φ) and C (φ) when taking z/ (1 − z) and z/ (1 − z) 2 instead of h in S * h (φ) respectively.In 1992, Ma and Minda [5] reduced the restriction to a weaker supposition that φ is a function, with Reφ > 0 in ∆, whose image domain is symmetric about the real axis and starlike with respect to φ(0) = 1 with φ (0) > 0 and discussed some properties.The set S * (φ) generalizes various subfamilies of the set A, for example: 1+Bz is the set of Janowski starlike functions, see [6].Further, if A = 1 − 2α and B = −1 with 0 ≤ α < 1, then we get the set S * (α) of starlike functions of order α.

The class S *
L := S * ( √ 1 + z) was introduced by Sokól and Stankiewicz [7], consisting of functions f ∈ A such that z f (z)/ f (z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w 2 − 1| < 1. 3. For φ(z) = 1 + sin z, the class S * (φ) lead to the class S * sin , introduced in [8]. 4. The family S * e := S * (e z ) was introduced by Mediratta et al. [9] given as: S * e = f ∈ S : or, equivalently They investigated some interesting properties and also links these classes to the familiar subfamilies of the set S. In [9], the authors choose the function f (z) = z + 1 4 z 2 (Figure 1) and then sketch the following figure of the function class S * e by using the form (3) as: Similarly, by using Alexandar type relation in [9], we have; From the above discussion, we conclude that the families S * e and C e considered in this paper are symmetric about the real axis.
For given parameters q, n ∈ N = {1, 2, . ..}, the Hankel determinant H q,n ( f ) was defined by Pommerenke [10,11] for a function f ∈ S of the form (1) as follows: ( The concept of Hankel determinant is very useful in the theory of singularities [12] and in the study of power series with integral coefficients.For deep insight, the reader is invited to read [13][14][15].Specifically, the absolute sharp bound of the functional H 2,2 ( f ) = a 2 a 4 − a 2  3 for each of the sets S * and C were proved by Janteng et al. [16,17] while the exact estimate of this determinant for the family of close-to-convex functions is still unknown (see, [18]).On the other side for the set of Bazilevič functions, the sharp estimate of |H 2,2 ( f )| was given by Krishna et al. [19].Recently, Srivastava and his coauthors [20] found the estimate of second Hankel determinant for bi-univalent functions involving symmetric q-derivative operator while in [21], the authors discussed Hankel and Toeplitz determinants for subfamilies of q-starlike functions connected with a general form of conic domain.For more literature see [22][23][24][25][26][27][28][29].The determinant with entries from (1) is known as Hankel determinant of order three and the estimation of this determinant |H 3,1 ( f )| is very hard as compared to derive the bound of |H 2,2 ( f )|.The very first paper on H 3,1 ( f ) visible in 2010 by Babalola [30] in which he got the upper bound of H 3,1 ( f ) for the families of S * and C. Later on, many authors published their work regarding |H 3,1 ( f )| for different sub-collections of univalent functions, see [8,[31][32][33][34][35][36].In 2017, Zaprawa [37] upgraded the results of Babalola [30] by giving and claimed that these bounds are still not best possible.Further for the sharpness, he examined the subfamilies of S * and C consisting of functions with m-fold symmetry and obtained the sharp bounds.Moreover this determinant was further improved by Kwon et al. [38] and proved |H 3,1 ( f )| ≤ 8/9 for f ∈ S * , yet not best possible.The authors in [39][40][41] contributed in similar direction by generalizing different classes of univalent functions with respect to symmetric points.In 2018, Kowalczyk et al. [42] and Lecko et al. [43] got the sharp inequalities for the recognizable sets K and S * (1/2) respectively, where the symbol S * (1/2) indicates to the family of starlike functions of order 1/2.Also we would like to cite the work done by Mahmood et al. [44] in which they studied third Hankel determinant for a subset of starlike functions in q-analogue.Additionally Zhang et al. [45] studied this determinant for the set S * e and obtained the bound In the present article, our aim is to investigate the estimate of |H 3,1 ( f )| for both the above defined classes S * e and C e .Moreover, we also study this problem for m-fold symmetric starlike and convex functions associated with exponential function.

A Set of Lemmas
Let P denote the family of all functions p that are analytic in D with (p(z)) > 0 and has the following series representation Lemma 1.If p ∈ P and has the form , then and for complex number λ, we have For the inequalities (7), ( 11), ( 8), ( 10), ( 9) see [46] and (12) is given in [47].Proof.Let f ∈ S * e .Then we can write (2), in terms of Schwarz function as

Improved Bound of |H
If h ∈ P, then it can be written in form of Schwarz function as From above, we can get and from the series expansion of w along with some calculations, we have After some computations and rearranging, it yields Comparing (13) and (14), we have From (5), the Third Hankel determinant can be written as Using (15), ( 16), ( 17) and (18), we get After rearranging, it yields Using triangle inequality along with (7), ( 11), ( 8) and (9), provide us , we obtain a function of variable x.Therefore, we can write The above function attains its maximum value at x = 0.64036035, which is Thus, the proof is completed.
The first three inequalities are sharp.
Proof.If f ∈ C e , then we can write (4), in form of Schwarz function as From (1), we can write By comparing (23) and ( 14), we get Implementing (7), in (24) and (25), we have Reshuffling (26), we have Application of triangle inequality and ( 7) and (11) leads us to By using triangle inequality along with (7), and (8), we get Equalities are obtain if we take where where γ is a complex number.
Proof.From ( 24) and (25), we get By reshuffling it, provides Application of (12), leads us to Substituting γ = 1, we obtain the following inequality.
The above function gets its maximum at x = 0.7024858, Therefore, we have Thus the proof is completed.By S (m) , we define the set of m-fold univalent functions having the following Taylor series form The where the set P (m) is defined by Here we prove some theorems related to 2-fold and 3-fold symmetric functions. .Then, there exists a function p ∈ P (2) , such that Using the series form (33) and (36), when m = 2 in the above relation, we can get Now, Utilizing (37) and (38), we get By rearranging, it yields Using triangle inequality long with (8) and (7), gives us Hence, the proof is done.
This result is sharp for the function Proof.As, f ∈ S * (3) e , therefore there exists a function p ∈ P (3) , such that Utilizing the series form (33) and (36), when m = 3 in the above relation, we can obtain Then, Utilizing (7) and triangle inequality, we have .
Thus the proof is ended.
e and has the form given in (33) .
Proof.As, f ∈ C (2) e , then there exists a function p ∈ P (2) , such that Utilizing the series form (33) and (36), when m = 2 in the above relation, we can obtain Application of (7), ( 8) and triangle inequality, leads us to Thus, the required result is completed.
e and has the form given in (33), then This result is sharp for the function f (z) = Proof.Let, f ∈ C (3) e .Then there exists a function p ∈ P (3) , such that 1 + z f (z) f (z) = exp p (z) − 1 p (z) + 1 .
Utilizing the series form (33) and (36), when m = 3 in the above relation, we can obtain Then, Implementing (7) and triangle inequality, we have Hence, the proof is done.

Conclusions
In this article, we studied Hankel determinant H 3,1 ( f ) for the families S * e and C e whose image domain are symmetric about the real axis.Furthermore, we improve the bound of third Hankel determinant for the family S * e .These bounds are also discussed for 2-fold symmetric and 3-fold symmetric functions.

4 .Theorem 2 .
Bound of |H 3,1 ( f )| for the Set C e Let f has the form (1) and belongs to C e .Then
sub-families S * (m) e and C (m) e of S (m) are the sets of m-fold symmetric starlike and convex functions respectively associated with exponential functions.More intuitively, an analytic function f of the form (33), belongs to the families S * (m) e and C (m) e , if and only if