Third Hankel Determinant Problem for Certain Subclasses of Analytic Functions Associated with Nephroid Domain

In this research article we consider two well known subclasses of starlike and bounded turning functions associated with nephroid domain. Our aims to find third Hankel determinant for these classes.


Introduction and Definitions
Let A be the collections of all normalized analytic functions defined in the unit disc D = {z ∈ C : |z| < 1} and of the form f (z) = z + ∞ n=2 a n z n , (z ∈ D). (1.1) where "≺" represent subordination. Two analytic functions f and g, a function f is subordinate to g symbolically f ≺ g if there exist an analytic function w (z) with limitation w (0) = 0 and |w (z)| < |z| such that f (z) = g (w (z)) . If g ∈ S, then equivalence conditions f (0) = g (0) and f (D) ⊂ g (D) .
Let P denote class of all analytic functions p such that Re (p (z)) > 0, and of the form By changing the function right hand side of subordinations in (1.2), we obtain some subclasses of the class S which have interesting geometric properties, see [3,4,5,6,7,8,9,10,11,12,13]. From among these subfamilies we recall here the families that are associated with trigonometric function as follows; Recently, authors in [18], introduced the class S * N e which are associated with nephriod domain.
The Hankel determinant H q,n (f ) where parameters q, n ∈ N = {1, 2, 3, · · · } for function f ∈ S of the form (1.1) was introduced by Pommerenke [14,15] as; (1.6) The growth of H q,n (f ) has been evaluated for different subcollections of univalent functions. Exceptionally, the sharp bound of the determinant H 2,2 (f ) = a 2 a 4 − a 2 3 for class S * , C and R were found by Janteng et al. [16,17] while for the family of close-to-convex functions the sharp estimation is still unknown (see, [19]). On the other hand, for the set of Bazilevič functions, the best estimate of |H 2,2 (f )| was proved by Krishna et al. [20]. For more work on H 2,2 (f ) , see [21,22,23,24,25].
is known as third order Hankel determinant and the estimation of this determinant |H 3,1 (f )| is so hard. In 2010, the first article on H 3,1 (f ) by Babalola [26], in which he obtained the upper bound of |H 3,1 (f )| for the groups of S * , K and R. Later on, a few creators distributed their work regarding |H 3,1 (f )| for various subcollections of holomorphic and univalent functions, see [34,35,36,37,38], which served as a base for the research in this field. Recently various authors explored some interesting classes for the said property of Hankel determinant.
Srivastava et al. [27] discussed this result for a class of Bi-valent functions defined by q-derivative and gave various interesting properties of it. Then he along with coauthors in [28] investigated the class of close to convex functions associated with lemniscate of Bernouli and evaluated its Hankel determinant. Continuing the same trend he in [29] incorporated the research on Toeplitz forms and Hankel determinant for some q-starlike functions associated with a generalized domain. Many other domains were also investigated for its Hankel determinant like a class of starlike functions associated with k-Fibonacci numbers. Whose third Hankel was evaluated by Shafiq et al. [30]. Further related work on the subject the reader is referred to [31,32,33]. Motivated from above discussed work on the topic we investigate |H 3,1 (f )| for classes of functions defined in the relations (1.4) and (1.5).

Sets of Lemma
The following lemmas are important as they help in our main results.
Proof. Since using (3.5) and (3.6), we get applying Lemma 2, we get the required results.
Theorem 3. Let f (z) ∈ S * N e be of the form (1.1). Then for ξ ∈ C, we have Proof. Since using (3.5) and (3.6), we get applying Lemma 2.6, we get the required results.
If we put ξ = 1, the above result become as: Theorem 4. Let f (z) ∈ S * N e be of the form (1.1). Then This results is sharp.

Application of Lemma 3 to (4.4), lead us to
Rearranging the (4.5), we have applying (2.1) and (2.2), we get Theorem 9. Let f (z) ∈ R N e be of the form (1.1). Then for ξ ∈ C, we have Proof. Since using (4.2) and (4.3), we get