Certain classes of bi-univalent functions related to Shell-like curves connected with Fibonacci numbers

A Comprehensive Family of Biunivalent Functions Defined byk-Fibonacci Numbers

“…(i) By taking μ = 0 and k = 1 in the above corollary, we obtain Theorem 2.3 of Magesh et al [31] (ii) By allowing μ = 0 and τ = 1 in the above corollary, we get two results Güney et al ([18], Corollary 10 and Corollary 23)…”

“…Remark 3. We note that (i) (i) When τ = 1, the family K 1 Σ ðμ, pk Þ was introduced by Frasin et al [30] (ii) The family L 1 Σ ð0, pk Þ ≡ S * Σ ðp k ÞÞ was mentioned by Güney et al [18], when μ = 0 and τ = 1 (iii) For μ = 0 and k = 1, the class L τ Σ ð0, p1 Þ ≡ S Σ ðp 1 Þ was investigated by Magesh et al [31] We now state the following lemma, which we will be using in the proof of our theorem.…”

A comprehensive class of bi-univalent functions subordinate to shell-like curves

“…Recently, in their pioneering work on the subject of bi-univalent functions, Srivastava et al [23] actually revived the study of the coefficient problems involving bi-univalent functions. Various subclasses of the bi-univalent function class Σ were introduced and non-sharp estimates on the first two coefficients |a 2 | and |a 3 | in the Taylor-Maclaurin series expansion (1) were found in several recent investigations (see, for example, [1,2,3,4,5,7,11,12,13,14,15,18,19,21,22,24,25] and references therein). The afore-cited papers on the subject were actually motivated by the pioneering work of Srivastava et al [23].…”

“…(see [17]). For more details one could refer recent works in this line from [1,3,4,5,11,13,14,18,19] and references therein.…”

Initial estimates for certain subclasses of bi-univalent functions with $κ-$Fibonacci numbers

“…For κ = 1, the classes SL(p) and KSL(p) were introduced and studied by Sokó l [28] and Dziok et al [9] respectively (see also [10,19,27]) and the references therein. Also, one can refer [4,5,14,18] and references therein for other subclasses of bi-univalent functions which are related with shell-like curves connected with Fibonacci numbers.…”