Connection between Legendre Polynomials and classes of Bi-Bazilevic

Functions Defined by Borel Distribution and Ruscheweyh Operator

Waleed Al-Rawashdeh1,∗

1Department of Mathematics, Zarqa University, 2000 Zarqa, 13110, Jordan

Email: walrawashdeh@zu.edu.jo

Abstract

This paper introduces two new classes of bi-Bazilevic and bi-univalent functions that are defined using Borel

distribution and Ruscheweyh operator, which also associated with Legendre polynomials and modified Sig-

moid function within the open unit disk D. This paper explores the characteristics and behaviors of these

functions, we find estimates for the modulus of the initial Taylor series coefficients a2 and a3 for functions

within our newly defined classes and some of their various subclasses. Moreover, this paper explores the clas-

sical Fekete-Szeg¨o functional problem concerning functions f that are classified within our specific classes.

Additionally, we obtain the classical Fekete-Szeg¨o inequalities of functions belonging to these classes and

some of their various subclasses.

Keywords: Bi-Univalent Functions; Bi-Bazilevic; Borel Distribution; Ruscheweyh operator; Modified Sig-

moid Function; Legendre Polynomials; Coefficient estimates; Fekete-Szeg¨o functional problem; Convolution;

Hadamard Product