Coefficient bounds for regular and bi-univalent functions linked with Gegenbauer polynomials

S. R. Swamy,S. Yalçın

doi:10.15393/j3.art.2022.10351

Abstract

Coefficient bounds for regular and bi-univalent functions linked with Gegenbauer polynomials

Highlights

IntroductionBrannan and Taha [3] presented and investigated certain subsets of Σ similar to convex and starlike functions of order σ (0 σ < 1) in D
Let the set of complex numbers be denoted by C, the set of normalized regular functions in D = {z ∈ C : |z| < 1} that have the power series of the form g(z) = z + d2z2 + d3z3 + . . . = z + ∑︁ djzj, (1)
The estimates on |d2|, |d3| and the famous inequality of Fekete-Szego were determined for bi-univalent functions linked with certain polynomials like (p, q)-Lucas polynomials, second kind Chebyshev polynomials, Horadam polynomials and Gegenbauer polynomials

Summary

Introduction

Brannan and Taha [3] presented and investigated certain subsets of Σ similar to convex and starlike functions of order σ (0 σ < 1) in D. Some interesting results concerning initial bounds for certain special sets of Σ have appeared: [7] and [10]. The estimates on |d2|, |d3| and the famous inequality of Fekete-Szego were determined for bi-univalent functions linked with certain polynomials like (p, q)-Lucas polynomials, second kind Chebyshev polynomials, Horadam polynomials and Gegenbauer polynomials. It is well-known that these polynomials and other special polynomials play a potentially important role in the approximation theory, statistical, physical, mathematical, and engineering sciences

Objectives

Findings

Conclusion