Bounds for a new subclass of bi-univalent functions subordinate to the Fibonacci numbers

Şahsene Altınkaya

doi:10.3906/mat-1910-41

Abstract

In this investigation, by using a relation of subordination, we define a new subclass of analytic bi-univalent functions associated with the Fibonacci numbers. Moreover, we survey the bounds of the coefficients for functions in this class.

Highlights

Introduction and backgroundLet C be the complex plane and let U = {z : z ∈ C and |z| < 1}, the open unit disc
We recall that the analytic function f is said to be subordinate to the analytic function g, if there exists a Schwarz function
Bounds for the first few coefficients |a2|, |a3| of various subclasses of bi-univalent functions have been obtained by a number of sequels to [15] including [1, 9, 10, 16]

Summary

Introduction and background

By A we represent the class of functions analytic in U , satisfying the condition f (0) = f ′(0) − 1 = 0. The Carathéodory class, consisting of the functions p analytic in U satisfying p(0) = 1 and R p(z) > 0 , is usually denoted by P. Bounds for the first few coefficients |a2| , |a3| of various subclasses of bi-univalent functions have been obtained by a number of sequels to [15] including (among others) [1, 9, 10, 16]. In the literature, there are only a few works (by making use of the Faber polynomial expansions) determining the general coefficient bounds |an| for bi-univalent functions ([2, 3, 8, 14]). A function f ∈ Σ is said to be in the class () WΣ p (z, w ∈ U)

Observe that p

If we set then we arrive at

This enables us to conclude that