Subclasses of Bi-Univalent Functions Associated with q−Con uent Hypergeometric Distribution Based Upon the Horadam Polynomials

In this paper, we introduce new subclasses of analytic and bi-univalent functions connected with a qcon uent hypergeometric distribution by using the Horadam polynomials. Furthermore, we nd estimates on the rst two Taylor-Maclaurin coe cients |a2| and |a3| for functions in these subclasses and obtain Fekete-Szeg® problem for these subclasses. Mathematics Subject Classi cation (2010): 30C50; 30C45; 11B65; 47B38


Introduction
In [23] Srivastava presented and motivated about brief expository overview of the classical q-analysis versus the so-called (p, q)-analysis with an obviously redundant additional parameter p. We also briey consider several other families of such extensivelyand widely-investigated linear convolution operators as (for example) the DziokSrivastava, SrivastavaWright and SrivastavaAttiya linear convolution operators, together with their extended and generalized versions. The theory of (p, q)-analysis has important role in many areas of mathematics and physics. Our usages here of the q-calculus and the fractional qcalculus in geometric function theory of complex analysis are believed to encourage and motivate signicant further developments on these and other related topics (see Srivastava and Karlsson [24, pp. 350351], Srivastava [21,22]). Let A denote the subclass of functions of the form and, let the function h ∈ A is given by The Hadamard (or convolution) product of f and h is dened by Denition 1.1. For f, g ∈ A, we say that f is subordinate to g, written f (z) ≺ g(z), if there exists a Schwarz function w, which is analytic in ∆, with w(0) = 0 and |w(z)| < 1 for all z ∈ ∆, such that f (z) = g(w(z)), z ∈ ∆. Furthermore, if the function g is univalent in ∆, then we have the following equivalence (see [4,16]): The conuent hypergeometric function of the rst kind is given by the power series where (b) k is the Pochhammer symbol dened in terms of the Gamma function by is convergent for all nite values of z (see [20]). It can be written otherwise is convergent for b, c, m > 0. Very recently, Porwal and Kumar [19] introduced the conuent hypergeometric distribution (CHD) whose probability mass function is Porwal [18] introduced a series I(b; c; m; z) whose coecients are probabilities of conuent hypergeometric distribution and dened a linear operator Ω(b; c; m)f : A → A as follows Srivastava [23] made use of various operators of q-calculus and fractional q-calculus and recalling the denition and notations. The q-shifted factorial is dened for λ, q ∈ C and n ∈ N 0 = N ∪ {0} as follows By using the q-gamma function Γ q (z), we get where (see [8]) Also, we note that and, the q-gamma function Γ q (z) is known where [k] q denotes the basic q-number dened as follows Using the denition formula (4) we have the next two products: (i) For any non negative integer k, the q-shifted factorial is given by (ii) For any positive number r, the q-generalized Pochhammer symbol is dened by It is known in terms of the classical (Euler's) gamma function Γ (z), that Also, we observe that For 0 < q < 1, the q-derivative operator [13] (see also [1,12]) for I(b; c; m; z) is dened by For λ > −1 and 0 < q < 1, we dened the linear operator I λ,q (b; c; m)f : A → A by where the function N q,λ+1 is given by A simple computation shows that where From the denition relation (6), we can easily verify that the next relations hold for all f ∈ A: Remark 1.2. Putting b = c in the operator I λ,q (b; c; m), we obtain the q-analogue of Poisson operator I λ,m q dened by El-Deeb et al. [7] as follows Remark 1.3. The Horadam polynomials h n (x) are dened by the following recurrence relation (see [10]) with for some real constants α, β, ρ and σ. The generating function of the Horadam polynomials h n (x) is given as follows (see [11]) Remark 1.4. By selecting the particular values of α, β, ρ and σ, the Horadam polynomial h n (x) reduces to several known polynomials.
These polynomials, the families of orthogonal polynomials and other special polynomials, as well as their extensions and generalizations, are potentially important in a variety of disciplines in many branches of science, especially in the mathematical, statistical and physical sciences. For more information associated with these polynomials (see [9,10,14,15]).
The Koebe one-quarter theorem (see [5]) proves that the image of ∆ under every univalent function f ∈ A contains a disk of radius 1 4 . Therefore, every function f ∈ A has an inverse f −1 that satises and Let Σ denote the class of bi-univalent functions in ∆ given by (1). Note that the following functions f 1 (z) = z 1 − z , , with their corresponding inverses g 1 (w) = w 1 + w , g 2 (w) = e 2w − 1 e 2w + 1 , g 3 (w) = e w − 1 e w , respectively, are elements of Σ (see [6,7,25]). For a brief history and interesting examples in the class Σ see, for example [2]. Brannan and Taha [3] (see also [25]) introduced certain subclasses of the bi-univalent functions class Σ similar to the familiar subclasses S * (δ) and K (δ) of starlike and convex functions of order δ (0 ≤ δ < 1), a function f ∈ A is said to be in the class S * Σ (δ) of strongly bi-starlike functions of order δ (0 < δ ≤ 1), if each of the following conditions is satised: where the function g is the analytic extension of f −1 to ∆, and is given by The classes S * Σ (α) and K Σ (α) of bi-starlike functions of order α and bi-convex functions of order α (0 < α ≤ 1), corresponding to the function classes S * (α) and K (α), were also introduced analogously. For each of the function classes S * Σ (α) and K Σ (α), they found non-sharp estimates on the rst two Taylor-Maclaurin coecients |a 2 | and |a 3 | (for details, see [3] and [25]).
The object of the present paper is to introduce new subclasses of the function class Σ involving the q−conuent Hypergeometric function connected with Horadam polynomials h n (x) that generalize the previous dened classes, and nd estimates on the coecients |a 2 |, and |a 3 | for functions in these new subclasses of the function class Σ. Denition 1.5. Let 0 ≤ γ ≤ 1, η ∈ C * = C\ {0} , b, c, m > 0, λ > −1, 0 < q < 1 and x ∈ R, then f ∈ Σ is said to be in the class K λ,q Σ (η, γ, b, c, m, x) if the following conditions are satised: and 1 + 1 η where α is real constant and the function g is the analytic extension of f −1 to ∆, and is given by (13).
Remark 3.8. We mention that all the above estimations for the coecients |a 2 |, |a 3 |, and Fekete-Szeg® problem for the function class K λ,q Σ (η, γ, b, c, m, x) are not sharp. To nd the sharp upper bounds for the above functionals remains an interesting open problem, as well as those for |a n |, n ≥ 4.