Initial bounds for certain classes of bi-univalent functions defined by Horadam polynomials

Abstract

Our present investigation is motivated essentially by the fact that, in Geometric Function Theory, one can find many interesting and fruitful usages of a wide variety of special functions and special polynomials. The main purpose of this article is to make use of the Horadam polynomials h n (x) and the generating function Π(x, z), in order to introduce three new subclasses of the bi-univalent function class Σ. For functions belonging to the defined classes, we then derive coefficient inequalities and the FeketeSzegö inequalities. Some interesting observations of the results presented here are also discussed. We also provide relevant connections of our results with those considered in earlier investigations.

Bi-univalent Function Subclasses Subordinate to Horadam Polynomials

“…|1 − δ| ≤ Q |1−δ| |(6(1−γ)+(τ −γ)(1−4γ)+2τ 2 )(bκ) 2 −4(1+τ −γ) 2 (pbκ 2 +qa)| ; |1 − δ| ≥ Q 3 ,whereQ 3 = 1 3(2 + τ − γ) 6(1 − γ) + (τ − γ)(1 − 4γ) + 2τ 2 − 4(1 + τ − γ) 2 pbκ 2 + qa b 2Remark Allowing γ = τ = 1 in Corollary 3.3, we obtain a result of Magesh et al[17, Corollary 2.3], which is also stated as Corollary 1 in Orhan et al[19].…”

“…Remark 2.1. i) Taking τ = 1 in Theorem 2.1, we get a result of Orhan et al [19,Theorem 1]. Further by letting ν = 1, we obtain Corollary 2.3 of Magesh et al [17], which is also stated as Corollary 1 in Orhan et al [19]. Also, we get Corollary 2 of Orhan et al [19], if we let ν = 0 instead of ν = 1. ii) Taking ν = 1 in Theorem 2.1, we get Corollary 2.4 of Swamy and Sailaja [28].…”

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

“…In a similar context, Srivastava et al [29] explored analytic and bi-univalent functions in connection with the Horadam polynomials. This research was followed by further investigations conducted by Al-Amoush [30], Wanas and Alina [31], Abirami et al [32], and other researchers (see, for example, [17,[33][34][35]).…”