Subclass of analytic functions defined by q-derivative operator associated with Pascal distribution series

Abstract: The purpose of the present paper is to find the necessary and sufficient condition and inclusion relation for Pascal distribution series to be in the subclass TCq(λ, α) of analytic functions defined by q-derivative operator. Further, we consider an integral operator related to Pascal distribution series, and several corollaries and consequences of the main results are also considered.


Introduction and definitions
Let A denote the class of the normalized functions of the form This class was introduced by Dixit and Pal [13]. The theory of q-calculus operators are used in describing and solving various problems in applied science such as ordinary fractional calculus, optimal control, q-difference and q-integral equations, as well as geometric function theory of complex analysis. The application of q-calculus was initiated by Jackson [23]. Recently, many researchers studied q-calculus such as Srivastava et al. [52], Muhammad and Darus [31], Kanas and Rȃducanu [28], Aldweby and Darus [2][3][4] and Muhammad and Sokol [30]. For details on q-calculus one can refer [1, 5-7, 9, 20, 23, 25, 38, 39, 43, 44, 46, 48-51] and also the reference cited therein.
For 0 < q < 1 the Jackson's q-derivative of a function f ∈ A is, by definition, given as follows [23] and is sometimes called the basic number n. If q → 1−, [n] q → n. For a function h(z) = z n , we obtain where h is the ordinary derivative.
Using the above defined q-calculus, several subclasses belonging to the class A have already been investigated in geometric function theory. Ismail et al. [26] were the first who used the q-derivative operator D q to study the q-calculus analogous of the class S * of starlike functions in U (see Definition 1.1 below). However, a firm footing of the q-calculus in the context of geometric function theory was presented mainly and basic (or q-) hypergeometric functions were first used in geometric function theory in a book chapter by Srivastava (see, for details, ( [45], p.347 et seq.); see also [46]).
We now introduce a new subclass of analytic functions defined by q-derivative operator D q .
A variable X is said to be Pascal distribution if it takes the values 0, 1, 2, 3, . . . with probabilities . . , respectively, where s and m are called the parameters, and thus Very recently, El-Deeb et al. [15] (see also, [10,34]) introduced a power series whose coefficients are probabilities of Pascal distribution, that is where m ≥ 1, 0 ≤ s ≤ 1, and we note that, by ratio test the radius of convergence of above series is infinity. We also define the series Let consider the linear operator I m s : A → A defined by the convolution or Hadamard product where m ≥ 1 and 0 ≤ s ≤ 1.
Motivated by several earlier results on connections between various subclasses of analytic and univalent functions, using hypergeometric functions (see for example, [8,11,21,29,42,47]), generalized Bessel functions (see for example, [18,22,33,36]), Struve functions (see for example, [12,24]), Poisson distribution series (see for example, [14,16,19,32,35,37]) and Pascal distribution series (see for example, [10,15,17,34]), in this paper we determine the necessary and sufficient condition for Φ m s to be in the class T C q (λ, α). Furthermore, we give sufficient condition for I m s (R τ (A, B)) ⊂ T C q (λ, α) and finally, we give necessary and sufficient condition for the function f such that its image by the integral operator G m t dt belongs to the class T C q (λ, α). To establish our main results, we need the following Lemmas.
The result is sharp.

Necessary and sufficient condition for
For convenience throughout in the sequel, we use the following identities that hold for m ≥ 1 and 0 ≤ s < 1: By simple calculations we derive the following relations: Unless otherwise mentioned, we shall assume in this paper that 0 ≤ α < 1 and 0 ≤ λ ≤ 1, 0 < q < 1 and 0 ≤ s < 1.
3. Sufficient condition for I m s (R τ (A, B)) ⊂ T C q (λ, α) Making use of Lemma 1.5, we will study the action of the Pascal distribution series on the class T C q (λ, α).
is satisfied then I m s f ∈ T C q (λ, α).

Integral operator
Theorem 4.1. Let m ≥ 1 and q → 1 − .If the integral operator G m s is given by then G m s ∈ T C q (λ, α) if and only Proof. According to (1.7) it follows that Using Lemma 1.4, the function G m q (z) belongs to T C q (λ, α) if and only if Now, By a similar proof like those of Theorem 3.1 we get that G m s f ∈ T C q (λ, α) if and only if (4.2) holds.

Corollaries and consequences
Corollary 5.1. Let m ≥ 1 and q → 1 − .Then Φ m s ∈ T C q (0, α), if and only if is satisfied then I m s f ∈ T C q (0, α).

Conclusions
In this paper, we find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in a subclass of analytic functions defined by q-derivative operator. Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas (see, for example, [ [45], pp.350-351] and [ [44], p.328]). Moreover, in this recently-published surveycum-expository review article by Srivastava [44], the so-called (p, q)-calculus was exposed to be a rather trivial and inconsequential variation of the classical q-calculus, the additional parameter p being redundant (see, for details, [ [44], p.340]). This observation by Srivastava [44] will indeed apply also to any attempt to produce the rather straightforward (p, q)-variations of the results which we have presented in this paper.