Fekete-Szegö problem for Bi-Bazilevič functions related to Shell-like curves

Abstract: In the present investigation, we define a subclass of bi-univalent functions related to shelllike curves connected with Fibonacci numbers to find the estimates of second, third Taylor-Maclaurin coefficients and Fekete-Szegö inequalities. Further, certain special cases are also discussed.


Introduction
Let A denote the class of functions of the form f (z) = z + ∞ n=2 a n z n (1.1) which are analytic in the open unit disk D = {z : z ∈ C and |z| < 1}. Further, by S we shall denote the class of all functions in A which are univalent in D.
For analytic functions f and g in D, f is said to be subordinate to g if there exists an analytic function w such that w(0) = 0, |w(z)| < 1 and f (z) = g(w(z)), z ∈ D.
This subordination will be denoted here by f ≺ g, z ∈ D or, conventionally, by f (z) ≺ g(z), z ∈ D.
Let P denote the class of functions of the form p(z) = 1 + p 1 z + p 2 z 2 + p 3 z 3 + · · · , z ∈ D (1.2) which are analytic with {p(z)} > 0. Here p(z) is called as Caratheodory functions [1]. It is well known that the following correspondence between the class P and the class of Schwarz functions w exists: p ∈ P if and only if p(z) = 1 + w(z) / 1 − w(z). Let P(β), 0 ≤ β < 1, denote the class of analytic functions p in D with p(0) = 1 and {p(z)} > β.
Recently, Sokół [2] and Dziok et al. [3] studied the classes SL( p) and KSL( p) of shell-like functions and convex shell-like functions which are characterized by [4,5] and the function p is not univalent in D, but it is univalent in the disc |z| < (3 − √ 5) / 2 ≈ 0.38. For example, p(0) = p (−1 / 2τ) = 1 and p (e ∓ arccos (1/4)) = 1 / √ 5 and it may also be noticed that 1 / |τ| = |τ| / 1 − |τ| which shows that the number |τ| divides [0, 1] such that it fulfills the golden section. The image of the unit circle |z| = 1 under p is a curve described by the equation given by 10x − which is translated and revolved trisectrix of Maclaurin. The curve p re it is a closed curve without any loops for 0 < r ≤ r 0 = (3 − √ 5) / 2 ≈ 0.38. For r 0 < r < 1, it has a loop and for r = 1, it has a vertical asymptote. Since τ satisfies the equation τ 2 = 1 + τ, this expression can be used to obtain higher powers τ n as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of τ and 1. The resulting recurrence relationships yield Fibonacci numbers u n as τ n = u n τ + u n−1 .
Also, Raina and Sokół [5] proved that This shows that the relevant connection of p with the sequence of Fibonacci numbers u n , such that Hence We note that the function p belongs to the class P(β) with β = √ 5 10 ≈ 0.2236 [5].
Recently, the initial coefficient estimates are found for functions in the class of bi-univalent functions defined through certain polynomials like the Faber polynomial, the Lucas polynomial, the Chebyshev polynomial, the Gegenbauer polynomial and the Meixner-Pollaczek polynomial. Motivated in this line, in the present work, we introduce the following new subclass of bi-univalent function, as follows: δ ≥ 0, if the following conditions are satisfied: By suitably specializing the values of µ, λ and δ, the class BSL µ, δ, λ Σ ( p) reduces to various new subclasses, we illustrate the following subclasses: 2. For λ = 1 and δ = 0, we get the class BSL µ, 0, 1 3. For µ = 1, we get the class BSL 1, δ, λ 4. For λ = µ = 1, we get the class BSL 1, δ, 1 p). A function f ∈ Σ of the form (1.1) is said to be in F SL Σ (δ, p), if the following conditions 5. For µ = 1 and δ = 0, we obtain the class BSL 1, 0, λ p). A function f ∈ Σ of the form (1.1) is said to be in BSL Σ (λ, p(z)), if the following conditions In order to prove our results for the functions in the class BSL µ, δ, λ Σ ( p), we need the following lemma. where p(z) = 1 + p 1 z + p 2 z 2 + · · · (z ∈ D).
In this investigation, we find the estimates for the coefficients |a 2 | and |a 3 | for functions in the class BSL µ, δ, λ Σ ( p) and its special cases. Also, Fekete-Szegö inequality for functions in this subclass.

Coefficient estimates and Fekete-Szegö inequality
In the following theorem, we discuss coefficient estimates and Fekete-Szegö inequality for functions in the class f ∈ BSL µ, δ, λ Σ ( p).

Corollaries and consequences
In this section, we give coefficient estimates and Fekete-Szegö inequalities for the subclasses of BSL µ, δ, λ Σ ( p).