A Comprehensive Family of Biunivalent Functions Defined by k-Fibonacci Numbers

g−1 ω ð Þ = f ω ð Þ = ω − d2ω + 2d2 − d3 ω3 − 5d2 − 5d2d3 + d4 ω4+⋯, ð2Þ such that z = g−1ðgðzÞÞ and ω = gðg−1ðωÞÞ, jωj < r0ðgÞ, r0ðgÞ ≥ 1/4, z, ω ∈D. A function g of A is called biunivalent (or bi-Schlicht) in D if both g and g−1 are univalent (or Schlicht) in D. Let Σ stands for the set of biunivalent (or bi-Schlicht) functions having the form (1). Historically, investigations of the family Σ begun five decades ago by Lewin [2] and Brannan and Clunie [3]. Later, Tan [4] found some coefficient estimates for biunivalent functions. In 1986 [5], Brannan and Taha introduced certain well-known subfamilies of Σ in D. Many interesting results related to initial bounds for some special families of Σ have appeared in [6–8]. In 2007, the concept of k-Fibonacci number sequence fFk,jg∞j=0, k ∈R+ was examined by Falcón and Plaza [9] and is given by


Introduction and Notations
Let ℂ be the set of all complex numbers and the disc fz ∈ ℂ : jzj < 1g be symbolized by D. Let ℕ = ℕ 0 \ f0g ≔ f1, 2, 3, ⋯g and ℝ be the collection of all real numbers. We denote the set of all normalized regular functions in D that have the series of the form by A and the symbol S stands for set of all functions of A that are univalent (or Schlicht) in D. As per the popular Koebe theorem (see [1]), every function g ∈ S has an inverse function given by such that z = g −1 ðgðzÞÞ and ω = gðg −1 ðωÞÞ, jωj < r 0 ðgÞ, r 0 ðgÞ ≥ 1/4, z, ω ∈ D.
A function g of A is called biunivalent (or bi-Schlicht) in D if both g and g −1 are univalent (or Schlicht) in D. Let Σ stands for the set of biunivalent (or bi-Schlicht) functions having the form (1). Historically, investigations of the family Σ begun five decades ago by Lewin [2] and Brannan and Clunie [3]. Later, Tan [4] found some coefficient estimates for biunivalent functions. In 1986 [5], Brannan and Taha introduced certain well-known subfamilies of Σ in D. Many interesting results related to initial bounds for some special families of Σ have appeared in [6][7][8].
In 2007, the concept of k-Fibonacci number sequence fF k,j g ∞ j=0 , k ∈ ℝ + was examined by Falcón and Plaza [9] and is given by where j ∈ ℕ and F 1,j = F j is the well-known Fibonacci number sequence. Özgür and Sokól in 2015 [10] proved that if then,p where t k is as in (4) and z ∈ D. Further, ifp k ðzÞ = 1 + Σp k,j z j , then, we havep Fibonacci polynomials, Pell-Lucas polynomials, Gegenbauer polynomials, Chebyshev polynomials, Horadam polynomials, Fermat-Lucas polynomials, and generalizations of them are potentially important in many branches such as architecture, physics, combinatorics, number theory, statistics, and engineering. Additional information is associated with these polynomials one can go through [11][12][13]. More details about the very popular functional of Fekete-Szegö for biunivalent functions based on k-Fibonacci numbers can be found in [14][15][16][17][18][19][20].
The recent research trends are the outcomes of the study of functions in Σ based on any one of the above-mentioned polynomials, which can be seen in the recent papers [21][22][23][24][25][26][27][28]. Generally, interest was shown to estimate the first two coefficient bounds and the functional of Fekete-Szegö for some subfamilies of Σ.
For functions g and f regular in D, g is said to subordinate f , if there is a Schwarz function ψ in D, such that ψð0Þ = 0, jψðzÞj < 1, and gðzÞ = f ðψðzÞÞ, z ∈ D. This subordination is indicated as g ≺ f . In particular, if f ∈ S, then gðzÞ ≺ f ðzÞ ⇐ gð0Þ = f ð0Þ and gðDÞ ⊂ f ðDÞ.
Inspired by the recent articles and the new trends on functions in Σ, we present a comprehensive family of Σ defined by using k-Fibonacci numbers as given by (3) with F k,j as in (4).
In the next section, we derive the estimates for jd 2 j,jd 3 j and obtain the Fekete-Szegö [33] inequalities for functions in the class SRS τ Σ ðγ, μ,p k Þ.

Coefficient Bounds and Fekete-Szegö Functional
In this section, we offer to get the upper bounds on initial coefficients and find the functional of Fekete-Szegö for functions ∈SRS τ Σ ðγ, μ,p k Þ.

Journal of Function Spaces
Similarly, it follows that where v is a regular function such that |vðωÞ | <1 in D such that pðωÞ =p k ðvðωÞÞ and the function lðωÞ is in the class P, where By virtue of (14), (15), (18), and (19), we obtain From (21) and (23), we get and also, If we add (26) and (24), then we obtain Substituting the value of ðu 2 1 + v 2 1 Þ from (26) in (27), we get which gets (10), on using Lemma 5. On using (25) in the subtraction of (24) from (26), we arrive at Then, in view of Lemma 5 and equation (28), (29) reduces to (11).
From (28) and (29), for δ ∈ ℝ, we can easily compute that where In view of (4), we find that which enable us to conclude (12) with J as in (13). Theorem 6 is proved. In Section 3, few interesting consequences and relevant observations of the main result are mentioned.