Applications of Orthogonal Polynomials to Subclasses of Bi-Univalent Functions

Abstract

Orthogonal polynomials have been studied extensively by legender in 1784. They are representatively related with typically real functions, which is played an important role in the geometric function theory, and and its role of estimating coefficient bounds. This chapter associates certain bi-univalent functions with certain orthogonal polynomials such as Gegnbauer polynomials, and Horadam polymials, and then explores some properties of the subclasses in hand. This Chapter is concerned with the connection between Orthogonal polynomials and bi-univalent functions. Our purpose is to inroduce certain classes of bi unvalent functions by mean of Gegenbauer polynomials and Hordam polynomials. Bounds for the initial coefficients of |a_{2}| and |a_{3}|, and results related to Fekete–Szegö functional are obtained.