Coefficient Estimates of New Families of Analytic Functions Associated with q-Hermite Polynomials

Abstract

In this paper, we introduce two new subclasses of bi-univalent functions using the q-Hermite polynomials. Furthermore, we establish the bounds of the initial coefficients υ2, υ3, and υ4 of the Taylor–Maclaurin series and that of the Fekete–Szegö functional associated with the new classes, and we give the many consequences of our findings.

Faber Polynomial Coefficient Estimates for Bi-Close-to-Convex Functions Defined by the q-Fractional Derivative

“…It is a clearly presented fact that the transition from our q-results to the corresponding (p, q)-results is a rather trivial exercise because the additional forced-in parameter p is obviously redundant (see, for details, ( [5], p. 340) and ( [54], Section 5, pp. 1511-1512); see also [59][60][61][62]).…”

Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions

“…Furthermore, many authors have introduced and studied numerous subclasses of the bi-univalent function family Σ, analogously to the work by Srivastava et al [5]. However, several recent papers have only provided non-sharp estimates on the initial coefficients |a 2 | and |a 3 | in the Taylor Maclaurin expansion (1) (see, for example, [6][7][8][9][10][11][12][13]). The general coefficient bounds |a n | for n ∈ N with n ≧ 3 for functions f ∈ Σ have not been fully addressed for many subfamilies of Σ (see for example [14]).…”

On a Certain Class of Bi-Univalent Functions in Connection with Gegenbauer Polynomials

“…Explorations in geometric function theory show that regular functions have kin connections with orthogonal polynomials. Actually, some purposeful investigations have been carried out on regular functions in connection with the Legendre polynomials [12], Laguerre polynomials [35], Chebyshev polynomials [7], Jacobi polynomials [10], Horadam polynomials [32], Hermite polynomials [1], and Gegenbauer polynomials [26]. For more instances, see [2,3,5,9,[22][23][24][25]29].…”

“…From (1.4), we note that (1) if α = 1, then we will obtain the well-known Chebyshev polynomials and (2) if α = 1 2 , then we will obtain the well-known Legendre polynomials. For more details on Gegenbauer polynomials see [2,22,26].…”