On τ -Pseudo- ν -Convex κ -Fold Symmetric Bi-Univalent Function Family

: The object of this article is to explore a τ -pseudo- ν -convex κ -fold symmetric bi-univalent function family satisfying subordinations condition generalizing certain previously examined families. We originate the initial Taylor–Maclaurin coefﬁcient estimates of functions in the deﬁned family. The classical Fekete–Szegö inequalities for functions in the deﬁned τ -pseudo- ν -convex family is also estimated. Furthermore, we present some of the special cases of the main results. Relevant connections with those in several earlier works are also pointed out. Our study in this paper is also motivated by the symmetry nature of κ -fold symmetric bi-univalent functions in the deﬁned class.

In [5], the Koebe one-quarter theorem ensures that s(D) contains a disc of radius 1/4 for every function s ∈ S. Hence, every function s ∈ S admits an inverse g = s −1 defined by s −1 (s(ς)) = ς, s(s −1 (κ)) = κ, |κ| < r 0 (s), r 0 (s) ≥ 1/4 , where ς, κ ∈ D and a compuatation shows that g has the expansion of the form A member s of A is called bi-univalent in D If both s and s −1 are univalent in D. Let σ be the family of bi-univalent functions in D given by (1). The systematic study of the family σ has its origin in a paper authored by Lewin [6], where coefficient-related investigations for elements of the family σ are examined. Lewin was the first to investigate the family σ and it was proved that |d 2 | < 1.51 for members of the family σ. Few years later, the estimation for |d 2 | was further investigated by Brannan and Clunie [7] and they proved that |d 2 | < √ 2, if s ∈ σ. In 1984, Tan [8] found initial coefficient estimates of functions in the family σ. Brannan and Taha in [9], investigated bi-starlike and bi-convex functions, which are analogous to the concepts of starlike and convex functions. An investigation by Srivastava et al. [10] resurfaced the interest in the study of family σ and it opened the space for many thinkings in the topics of discussion of the paper. The trend in the last decade was to investigate coefficient-related non-sharp bounds for members of certain subfamilies of σ as it can be seen in papers [11][12][13][14][15].
An holomorphic function s in D is said to be κ-fold symmetric if s(e 2πi κ ς) = e 2πi κ s(ς). For each s ∈ S, the function s given by s(ς) = ( f (ς κ )) 1/κ , κ ∈ N, is univalent and maps D into a region with κ-fold symmetry. The class of κ-fold symmetric univalent functions in D is symbolized by S κ . A function s ∈ S κ has the form given by Clearly S 1 = S. Following the concept of S κ , κ ∈ N, Srivastava et al. [16] examined the family σ κ of κ-fold symmetric bi-univalent functions. They found some interesting results, such as the series for s −1 = g, s ∈ σ κ , which is as follows: We obtain (2) from (4) on taking κ = 1. Note that σ 1 = σ. Some examples in the class σ κ are Inverse functions of them are as follows: The momentum on investigations of functions in certain subfamilies of σ κ was gained in recent years due to the paper [16] and it has led to a large number of papers on subfamilies of σ κ [17][18][19][20]. Inspired by these works, many researchers have investigated several interesting subfamilies of σ κ and found non-sharp estimates on initial coefficients and the Fekete-Szegö functional problem |d 2κ+1 − δd 2 κ+1 |, δ ∈ [21] of functions belonging to these subfamilies (see, for example [22][23][24][25]) and this continued to appear in [26,27], showing the developments in the subject area of this paper.
In [30], Mishra and Soren have illustrated that is a univalent starlike function of order ξ. Therefore g(ς) is bi-starlike function of order ξ. On similar lines of [30], one can show that s(ς) = ς + d 2 2 ς 2 is bi-convex function of order ξ. Hence, the function .
We obtain the results stated below, if we set τ = 1 in Theorem 1.
We obtain the results stated below, if we set ν = 1 in Theorem 1.
We obtain the results stated below, if we set τ = 1 in Corollary 8.

Conclusions
In the current study, a κ-fold bi-univalent function family M τ σ κ (ν, η, ϕ) is introduced and the original results about the upper bounds of |d κ+1 | and |d 2κ+1 | are estimated for functions belonging to this family. Furthermore, the estimate of Fekete-Szegö problem |d 2κ+1 − δd 2 κ+1 |, δ ∈ , for functions in M τ σ κ (ν, η, ϕ) is also examined. Various subfamilies of M τ σ κ (ν, η, ϕ) are also discussed. The problem to determine bound on |d κk+1 |, (k ∈ N − {1.2}) for the classes that have been examined in this paper remain open. Since the only investigation on the defined family was related to coefficient bounds, it could inspire many researchers for further investigations related to different other aspects associated with (i) q-derivative operator [35], (ii) integrodifferential operator [36], (iii) Hohlov operator linked with legendary polynomials [37] and so on.