# Inclusion properties for bi-univalent functions of complex order defined by combining of Faber polynomial expansions and Fibonacci numbers

## TL;DR

This paper introduces a new class of bi-univalent functions using fractional derivatives, Faber polynomial expansions, and Fibonacci numbers to establish bounds on their coefficients.

## Contribution

It combines fractional derivatives, Faber polynomials, and Fibonacci numbers to define and analyze a novel class of bi-univalent functions with coefficient bounds.

## Key findings

- Derived bounds for coefficients of the new bi-univalent function class
- Established properties of functions using Faber polynomial expansions and Fibonacci numbers
- Introduced a new function class using Tremblay fractional derivative operator

## Abstract

In this present investigation, we introduce the new class R of bi-univalent functions defined by using the Tremblay fractional derivative operator. Additionally, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient an of the bi-univalent function class.