Maclaurin Coefficient Estimates for New Subclasses of Bi- univalent Functions Connected with a q-Analogue of Bessel Function

In this paper, we introduce new subclasses of the function class Σ of bi-univalent functions connected with a q-analogue of Bessel function and defined in the open unit disc. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients a2 and a3 for functions in these new subclasses.


Introduction, Definitions, and Preliminaries
The theory of q-calculus plays an important role in many areas of mathematical, physical, and engineering sciences. Jackson (see [1,2]) was the first to have some applications of the q-calculus and introduced the q-analogue of the classical derivative and integral operators (see also [3]). Let A denote the class of analytic functions of the form and S be the subclass of A which are univalent functions in Δ.
If g ∈ A is given by then, the Hadamard (or convolution) product of f and g is defined by If f and F are analytic functions in Δ, we say that f is subordinate toF, written f ≺ F, if there exists a Schwarz functionw, which is analytic in Δ, with wð0Þ = 0, and jwðzÞj < 1 for all z ∈ Δ, such that f ðzÞ = FðwðzÞÞ, z ∈ Δ. Furthermore, if the function F is univalent in Δ, then we have the following equivalence (see [4,5]): The Bessel function of the first kind of orderν is defined by the infinite series (see [6]) where Γ stands for the Gamma function. Recently, Szász and Kupán [7] investigated the univalence of the normalized Bessel function of the first kind k ν : Δ ⟶ ℂ defined by (see also [8][9][10]) For 0 < q < 1, the q-derivative operator for k ν is defined by Using the definition formula (8), we will define the next two products: (i) For any nonnegative integer k, the q-shifted factorial is given by (ii) For any positive number r, the q-generalized Pochhammer symbol is defined by For ν > 0, λ > −1, and 0 < q < 1, El-Deeb and Bulboacă [11] define the function I λ ν,q : Δ ⟶ ℂ by A simple computation shows that where the function M q,λ+1 is given by Using the definition of q-derivative along with the idea of convolutions, El-Deeb and Bulboacă [11] introduce the lin- where Remark 1. From the definition relation (14), we can easily verify that the next relations hold for all f ∈ : The Koebe one quarter theorem (see [12]) proves that the image of Δ under every univalent function f ∈ S contains a disk of radius 1/4: Therefore, every function f ∈ S has an inverse f −1 satisfying where A function f ∈ A is said to be bi-univalent in Δ if both f ðzÞ and f −1 ðzÞ are univalent in Δ. Let Σ denote the class of bi-univalent functions in Δ given by (1). For a brief history and interesting examples in the class Σ, see [13]. Brannan and Taha [14] (see also [15][16][17]) introduced certain subclasses of the bi-univalent functions class Σ similar to the familiar subclasses S * ðαÞ and KðαÞ of starlike and convex functions of order αð0 ≤ α < 1Þ, respectively (see [13]). Thus, following Brannan and Taha [14], a function f ∈ A is 2 Abstract and Applied Analysis said to be in the class S * Σ ðαÞ of strongly bi-starlike functions of order αð0 < α ≤ 1Þ if each of the following conditions is satisfied: where the function g is given by and g is the extension of f −1 to Δ: The classes S * Σ ðαÞ and K Σ ðαÞ of bi-starlike functions of order α and biconvex functions of order αð0 < α ≤ 1Þ, corresponding to the function classes S * ðαÞ and KðαÞ, were also introduced analogously. For each of the function classes S * Σ ðαÞ and K Σ ðαÞ, they found nonsharp estimates on the first two Taylor-Maclaurin coefficients ja 2 j and ja 3 j (for details, see [14,17]).
The objective of the present paper is to introduce new subclasses of the function class Σ and find estimates on the coefficients ja 2 j and ja 3 j for functions in these new subclasses of the function class Σ employing the techniques used earlier by Srivastava et al. [18]. Now, we define the subclasses of functions Q q Σ ðα, γ, λ, νÞ, P q Σ ðβ, γ, λ, νÞ, and Q Σ ðh, p, γ, ν, q, λÞ as follows: Definition 2. Let f ðzÞ be given by (1), then f ðzÞ is said to be in the class Q q Σ ðα, γ, λ, νÞ if the following conditions are satisfied: where the function g is given by (2).
Definition 3. Let f ðzÞ be given by (1), then f ðzÞ is said to be in the class P q Σ ðβ, γ, λ, νÞ if the following conditions are satisfied: where the function g is given by (2).
To prove our results, we need the following lemma.
Lemma 4 (see [19], Lemma 3). If G ∈ P then jc k j ≤ 2 for each k, where P is the family of all functions G analytic in Δ for which RðGðzÞÞ > 0, GðzÞ = 1 + c 1 z + c 2 z 2 + ⋯ ⋯ for z ∈ Δ: Definition 5. Let f ðzÞ be given by (1), hðzÞ and pðwÞ in P have the forms then f ðzÞ is said to be in the class Q Σ ðh, p, γ, ν, q, λÞ if the following conditions are satisfied: where the function g is given by (2).
Proof. It follows from (21) and (22) that where pðzÞ and qðwÞ in P and have the forms Now, equating the coefficients in (34) and (35), we get − 1 + γ ð Þψ 2 a 2 = q 1 α, ð40Þ From (38) and (40), we get Applying Lemma 4 for the coefficients p 2 and q 2 , we immediately have This gives the bound on ja 2 j as asserted in (32). Next, in order to find the bound on ja 3 j, by subtracting (41) from (39), we get