A Subclass of Bi-Univalent Functions Based on the Faber Polynomial Expansions and the Fibonacci Numbers

Abstract

In this investigation, by using the Komatu integral operator, we introduce the new class of bi-univalent functions based on the rule of subordination. Moreover, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient of the bi-univalent function class.

Coefficient Inequalities for Bi-univalent Functions Defined by Subordination and A Sequence of Fibonacci Numbers

“…Son yıllarda yapılan çalışmalarda bu problem tekrar çalışılmaya başlanmış [25];  sınıfı ve Taylor-Maclaurin serilerinin ilk iki katsayısı olan ve ün modülleri için üst sınırları bulma çalışmaları popüler hale gelmiştir [2,3,4,5,6,15,21]. Literatürde in modülü için üst sınırların elde edildiği çalışmalar olmasına rağmen [8,24,26] \{ } için | | genel katsayı tahmini hala açık bir problemdir.…”

A comprehensive class of bi-univalent functions subordinate to shell-like curves

“…Recently, in their pioneering work on the subject of bi-univalent functions, Srivastava et al [23] actually revived the study of the coefficient problems involving bi-univalent functions. Various subclasses of the bi-univalent function class Σ were introduced and non-sharp estimates on the first two coefficients |a 2 | and |a 3 | in the Taylor-Maclaurin series expansion (1) were found in several recent investigations (see, for example, [1,2,3,4,5,7,11,12,13,14,15,18,19,21,22,24,25] and references therein). The afore-cited papers on the subject were actually motivated by the pioneering work of Srivastava et al [23].…”

Coefficients assessment for certain subclasses of bi-univalent functions related with quasi-subordination

“…This univalent function, therefore, has an inverse one f −1 which satisfies f −1 (f (z)) = z, (z ∈ D) and f (f −1 (w)) = w, |w| < r 0 (f ), r 0 (f ) 1 4 . In fact the inverse function f −1 is given by (1.2) g(w) = f −1 (w) = w − a 2 w 2 + (2a 2 2 − a 3 )w 3 − (5a 3 2 − 5a 2 a 3 + a 4 )w 4 + • • • . A function f ∈ A is said to be bi-univalent in D if both f and f −1 are univalent in D. Let Σ denote the class of bi-univalent functions defined in the unit disc D. Ma-Minda [11] introduction the following classes using subordination:…”