Let denote the class of analytic functions of the form

and let be the subclass of consisting of functions that are univalent in .

Furthermore, for given by

the Hadamard (or convolution) product of the functions f and g is given by (see [5])

If f and g are analytic functions in , 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 , such that f(z) = g(w(z)), . Furthermore, if the function g is univalent in , then we have the following equivalence (see [3] and [13]):

Goodman in [6, 7], Rønning in [15, 16], and Bharati et al. in [2] introduced and studied the following subclasses:

• A function is said to be in the class of k-uniformly starlike functions of order β if it satisfies the condition (1.2)Re (zf′(z)f(z)-β)>k|zf′(z)f(z)-1|, z∈U,

  where 0 ≤ β < 1 and k ≥ 0.

• A function is said to be in the class of k-uniformly convex functions of order β if it satisfies the condition (1.3)Re (1+zf″(z)f′(z)-β)>k|zf″(z)f′(z)|, z∈U,

  where 0 ≤ β < 1 and k ≥ 0.

It follows from (1.2) and (1.3) that for a function we have the equivalence

For β = 0 the classes and reduce to the classes and respectively, that have been introduced and studied by Kanas and Wisniowska [10, 11].

Definition 1.1.([18])

For −1 ≤ A < B ≤ 1 and ∣θ∣ <π2, a function is said to be in the class ℒ(A,B, θ) if it satisfies the subordination condition

Using the definition of the subordination, a function belongs to the class ℒ(A,B, θ) if and only if there exists an analytic function w, satisfying w(0) = 0 and |w(z)| < 1 for all , such that

or equivalently

We note that for suitable choices of A, B and θ we obtain the following subclasses defined and studied by different authors:

• L (2α-1,1,θ)=:L (0;α) (0≤α<1,∣θ∣<π2) (see Kanas and Srivastava [9]);

• ℒ(β(2α − 1), β, 0) =: ℛ (β; α) (0 < β ≤ 1, 0 ≤ α < 1) (see Juneja and Mogra [8]);

• (see Caplinger and Causey [4]);

• L (A+(B-A)α,B,θ)=:L (A,B,θ;α) (-1≤A<B≤1,∣θ∣<π2,0≤α<1) (see Aouf [1]).

A variable x is said to have the Pascal distribution if it takes the values 0, 1, 2, 3, … with the probabilities (1−q)^r, qr(1-q)r1!, q2r(r+1)(1-q)r2!, q3r(r+1)(r+2)(1-q)r3!, …, respectively, where q, r are called the parameters, and thus

Now we introduced a power series whose coefficients are probabilities of the Pascal distribution, that is

Note that, by using ratio test we deduce that the radius of convergence of the above power series is infinity.

We will define the functions

and

and we introduce the linear operator Pqr:A→A defined by

In the present paper we established some sufficient conditions for the Pascal distribution series and other related series to be in some subclasses of analytic functions. Also, we studied similar properties for an integral transform related to this series.

Unless otherwise mentioned, we assume that −1 ≤ A < B ≤ 1, ∣θ∣ <π2, r ≥ 1, 0 ≤ q ≤ 1, λ, γ ≥ 0 and λ ≥ γ. To prove our results we need the following lemmas.

Lemma 2.1.([1, Theorem 4 for p = 1 and α = 0])

If the functionis of the form (1.1) and

then f ∈ ℒ (A,B, θ).

Lemma 2.2.([1, Theorem 1 for p = 1 and α = 0])

If the function f ∈ ℒ (A,B, θ) is of the form (1.1), then

for every n ≥ 2. The estimate is sharp.

Lemma 2.3.[10, Theorem 3.3]

If, and for some k (0 ≤ k < ∞) the following inequality holds

then. The number1k+2cannot be increased.

Lemma 2.4.([17, Theorem 2.1])

If the functionis of the form (1.1) and

then

Theorem 3.1

A sufficient condition for the functionQqrgiven by (1.4) to be in the class ℒ(A,B, θ) is

Theorem 3.2

A sufficient condition for the functionNq,λrdefined by (1.5) to be in the class ℒ(A,B, θ) is

Theorem 3.3

A sufficient condition for the functionMq,λ,γrgiven by (1.6) to be in the class ℒ(A,B, θ) is

Theorem 3.4

• If the condition(3.6)q2r(r+1)(1-q)2+3qr1-q+1-(1-q)r≤(B-A) cos θ1+∣B∣

  holds, then the operatorPqrdefined by (1.7) maps the classto the class ℒ(A,B, θ), that isPqr(S*)⊂L (A,B,θ).

• If the condition (3.1) is satisfied, then the operatorPqrmaps the classto the class ℒ(A,B, θ), that isPqr(K)⊂L(A,B,θ).

Theorem 3.5

If the condition

holds, then the operatorPqrmaps the class ℒ(A,B, θ) to the class, that isPqr(L(A,B,θ))⊂k-UCV.

Theorem 3.6

If the condition

holds, thenPqrmaps the class ℒ(A,B, θ) to the class, that isPqr(L(A,B,θ))⊂k-Sp.

Theorem 3.7

A sufficient condition for the functionIqrto be in the class ℒ(A,B, θ) is

Theorem 3.8

A sufficient condition for the functionIqrto be in the classis