Subclasses of analytic functions associated with Pascal distribution series

In the present paper we determine necessary and sufficient conditions for the Pascal distribution series to be in the subclasses S(k, λ) and C(k, λ) of analytic functions. Further, we consider an integral operator related to Pascal distribution series. Some interesting special cases of our main results are also considered.

A function f ∈ A is said to be in the class R τ (A, B),τ ∈ C\{0}, This class was introduced by Dixit and Pal [3]. A variable X is said to be Pascal distribution if it takes the values 0, 1, 2, 3, . . . with probabilities . . , respectively, where q and m are called the parameters, and thus Very recently, El-Deeb et al. [5] (see also, [14,1]) introduced a power series whose coefficients are probabilities of Pascal distribution, that is where m ≥ 1, 0 ≤ q ≤ 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 q : A → A defined by the convolution or Hadamard product where m ≥ 1 and 0 ≤ q ≤ 1. Motivated by several earlier results on connections between various subclasses of analytic and univalent functions, by using hypergeometric functions (see for example, [2,7,10,19,20]) and by using various distributions such as Yule-Simon distribution, Logarithmic distribution, Poisson distribution, Binomial distribution, Beta-Binomial distribution, Zeta distribution, Geometric distribution and Bernoulli distribution (see for example, [4,6,8,9,12,13,18,15]), in this paper, we determine the necessary and sufficient conditions for Φ m q (z) to be in our classes S(k, λ) and C(k, λ) and connections of these subclasses with R τ (A, B).

Finally, we give conditions for the integral operator
t dt belonging to the above classes.

Preliminary lemmas
To establish our main results, we need the following Lemmas.
The result is sharp for the function

Necessary and sufficient conditions
For convenience throughout in the sequel, we use the following identities that hold at least for m ≥ 2 and 0 ≤ q < 1: By simple calculations we derive the following relations: Unless otherwise mentioned, we shall assume in this paper that 0 < k ≤ 1, 0 ≤ λ < 1 , while m ≥ 1 and 0 ≤ q < 1.
Firstly, we obtain the necessary and sufficient conditions for Φ m q to be in the class S(k, λ).
Proof. Since in view of Lemma 2.1, it suffices to show that Writing n = (n − 1) + 1 in (8) But this last expression is bounded above by 2k if and only if (6) holds.

Inclusion Properties
Making use of Lemma 2.3, we will study the action of the Pascal distribution series on the classes S(k, λ) and C(k, λ).
Proof. In view of Lemma 2.1, it suffices to show that Since f ∈ R τ (A, B), then by Lemma 2.3, we have Thus, we have But this last expression is bounded by 2k, if (11) holds, which completes the proof of Theorem 4.1.
Applying Lemma 2.2 and using the same technique as in the proof of Theorem 4.1 we have the following result: 5. An integral operator Theorem 5.1. If m ≥ 1, then the integral operator is in C(k, λ) if and only if inequality (6) is satisfied.
Proof. According to ( (14) ) it follows that The remaining part of the proof of Theorem 5.1 is similar to that of Theorem 3.1, and so we omit the details.
Theorem 5.2. If m > 1, then the integral operator G m q (m, z)given by ( (14)) is in S(k, λ) if and only if The proof of Theorem 5.2 is lines similar to the proof of Theorem 5.1, so we omitted the proof of this theorem.

Corollaries and consequences
By specializing the parameter λ = 0 in the above theorems we obtain the following corollaries.   (A, B), then I m q ∈ S(k) if Corollary 6.4. Let m ≥ 1. If f ∈ R τ (A, B), then I m q f ∈ C(k) if Corollary 6.5. If m ≥ 1, then the integral operator G m q (m, z) given by (14) is in C(k) if and only if inequality (16) is satisfied.
If m > 1, then the integral operator G m q (m, z) given by (14) is in S(k) if and only if