KYUNGPOOK Math. J. 2019; 59(2): 301-314  
Pascal Distribution Series Connected with Certain Subclasses of Univalent Functions
Sheeza M. El-Deeb?
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
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Teodor Bulboac˘ a Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania
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Jacek Dziok
Faculty of Mathematics and Natural Sciences, University of Rzeszow, 35-310 Rzeszow, Poland
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* Corresponding Author.
Received: November 5, 2018; Revised: March 13, 2019; Accepted: March 18, 2019; Published online: June 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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The aim of this article is to make a connection between the Pascal distribution series and some subclasses of normalized analytic functions whose coefficients are probabilities of the Pascal distribution. For these functions, for linear combinations of these functions and their derivatives, for operators defined by convolution products, and for the Alexander-type integral operator, we find simple sufficient conditions such that these mapping belong to a general class of functions defined and studied by Goodman, Rønning, and Bharati et al.

1. Introduction

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]):

f(z)g(z)f(0)=g(0)and f(U)g(U).

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 Re (zf(z)f(z)-β)>k|zf(z)f(z)-1|,zU,

    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 Re (1+zf(z)f(z)-β)>k|zf(z)f(z)|,zU,

    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

eiθf(z)1+Az1+Bzcos θ+isin θ.

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

eiθf(z)=1+Aw(z)1+Bw(z)cos θ+isin θ,zU,

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)(0α<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)(-1A<B1,θ<π2,0α<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

Qqr(z):=z+n=2(n+r-2r-1)qn-1(1-q)rzn,zU         (r1,0q1).

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

Nq,λr(z):=(1-λ)Qqr(z)+λz(Qqr(z))=z+n=2(n+r-2r-1)[1+λ(n-1)]qn-1(1-q)rzn,zU,         (r1,0q1,λ0)


Mq,λ,γr(z):=(1-λ+γ)Qqr(z)+(λ-γ)z(Qqr(z))+λγz2(Qqr(z))=z+n=2(n+r-2r-1)[1+(n-1)(λ-γ+nλγ)]qn-1(1-q)rzn,zU,   (r1,0q1,λ,γ0),

and we introduce the linear operator Pqr:AA 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.

2. Preliminaries

Unless otherwise mentioned, we assume that −1 ≤ A < B ≤ 1, θ<π2, r ≥ 1, 0 ≤ q ≤ 1, λ, γ ≥ 0 and λγ. 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

n=2n(1+B)an(B-A)cos θ,

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

an(B-A)cos θn

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



3. Main Results

Theorem 3.1

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

qr1-q-(1-q)r+1(B-A)cos θ1+B.

To prove that QqrL(A,B,θ), according to Lemma 2.1 it is sufficient to show that

n=2(n+r-2r-1)n(1+B)qn-11-qr(B-A)cos θ.

In the proofs of all our results we will use the relation


and the corresponding ones obtained by replacing the value of r with r − 1, r + 1, and r + 2.

Thus, from the definition relation (1.4) combined with the assumption (3.1) we get

n=2(n+r-2r-1)n(1+B)qn-1(1-q)r=(1+B)(1-q)r[n=2(n+r-2r-1)(n-1)qn-1+n=2(n+r-2r-1)qn-1]=(1+B)(1-q)r[n=2qr(n+r-2r)qn-2+n=2(n+r-2r-1)qn-1]=(1+B)(1-q)r[qrn=0(n+rr)qn+n=1(n+r-1r-1)qn]=(1+B)(1-q)r[qrn=0(n+rr)qn+n=0(n+r-1r-1)qn-1]=(1+B)[qr1-q+1-(1-q)r](B-A)cos θ,

and from Lemma 2.1 it follows that QqrL(A,B,θ).

Theorem 3.2

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

(1+2λ)qr1-q+1-(1-q)r+λq2r(r+1)(1-q)2(B-A)cos θ1+B.

From Lemma 2.1, to prove that Nq,λrL(A,B,θ) it is sufficient to show that

n=2(n+r-2r-1)[n(1+B)][1+λ(n-1)]qn-11-qr(B-A)cos θ.



According to (3.2), from the above identity we conclude it follows that the inequality (3.3) holds, and therefore Nq,λrL(A,B,θ).

Theorem 3.3

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

(1+2λ-2γ+4λγ)qr1-q+1-(1-q)r+(λ-γ+5λγ)q2r(r+1)(1-q)2+λγq3r(r+1)(r+2)(1-q)3(B-A)cos θ1+B.

To prove that Mq,λ,γrL(A,B,θ), from Lemma 2.1 it is sufficient to show that

n=2(n+r-2r-1)n(1+B)[1+(n-1)(λ-γ+nλγ)]qn-11-qr(B-A)cos θ.

From (3.4) it follows that

n=2(n+r-2r-1)n(1+B)[1+(n-1)(λ-γ+nλγ)]qn-1(1-q)r=(1+B)(1-q)r[n=2(n+r-2r-1)(1+2λ-2γ+4λγ)(n-1)qn-1+n=2(n+r-2r-1)qn-1+n=2(n+r-2r-1)(λ-γ+5λγ)(n-1)(n-2)qn-1+n=2(n+r-2r-1)λγ(n-1)(n-2)(n-3)qn-1]=(1+B)(1-q)r[n=2qr(1+2λ-2γ+4λγ)(n+r-2r)qn-2+n=2(n+r-2r-1)qn-1+n=3(λ-γ+5λγ)q2(n+r-2r-1)(n-1)(n-2)qn-3+n=4λγq3(n+r-2r-1)qn-4(n-1)(n-2)(n-3)]=(1+B)(1-q)r[(1+2λ-2γ+4λγ)qrn=0(n+rr)qn+n=0(n+r-1r-1)qn-1+(λ-γ+5λγ)q2r(r+1)n=0(n+r+1r+1)qn+λγq3r(r+1)(r+2)n=0(n+r+2r+2)qn]=(1+B)(1-q)r[(1+2λ-2γ+4λγ)qr1(1-q)r+1+1(1-q)r-1+(λ-γ+5λγ)q2r(r+1)1(1-q)r+2+λγq3r(r+1)(r+2)1(1-q)r+3]=(1+B)[(1+2λ-2γ+4λγ)qr1-q+1-(1-q)r+(λ-γ+5λγ)q2r(r+1)(1-q)2+λγq3r(r+1)(r+2)(1-q)3](B-A)cos θ,

that is (3.5) holds, hence Mq,λ,γrL(A,B,θ).

Theorem 3.4

  • If the conditionq2r(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).


According to Lemma 2.1, to prove that Pqr(f)L(A,B,θ) it is sufficient to show that

n=2(n+r-2r-1)n(1+B)qn-11-qran(B-A)cos θ.

(i) If has the form (1.1), then the well-known inequality |an| ≤ n holds for all n ≥ 2 (see [12] and [14]), and using (3.6) we obtain that

n=2(n+r-2r-1)n(1+B)qn-1(1-q)rann=2(n+r-2r-1)n2(1+B)qn-1(1-q)r=(1+B)(1-q)r[n=2(n+r-2r-1)(n-1)(n-2)qn-1+3n=2(n+r-2r-1)(n-1)qn-1+n=2(n+r-2r-1)qn-1]=(1+B)(1-q)r[q2n=3(n+r-2r-1)(n-1)(n-2)qn-3+3qn=2(n+r-2r-1)(n-1)qn-2+n=2(n+r-2r-1)qn-1]=(1+B)(1-q)r[q2r(r+1)n=0(n+r+1r+1)qn+3qrn=0(n+rr)qn+n=1(n+r-1r-1)qn]=(1+B)(1-q)r[q2r(r+1)(1-q)r+2+3qr(1-q)r+1+1(1-q)r-1]=(1+B)[q2r(r+1)(1-q)2+3qr1-q+1-(1-q)r](B-A)cos θ,

that is (3.7) holds, hence Pqr(f)L(A,B,θ).

(ii) If is of the form (1.1), then the well-known inequality |an| ≤ 1 holds for all n ≥ 2 (see [12]), and according to (3.1) we obtain that

n=2(n+r-2r-1)n(1+B)qn-1(1-q)rann=2(n+r-2r-1)n(1+B)qn-1(1-q)r=(1+B)(1-q)r[n=2(n+r-2r-1)(n-1)qn-1+n=2(n+r-2r-1)qn-1]=(1+B)(1-q)r[n=2q(n+r-2r-1)(n-1)qn-2+n=2(n+r-2r-1)qn-1]=(1+B)(1-q)r[qrn=0(n+rr)qn+n=1(n+r-1r-1)qn]=(1+B)(1-q)r[qrn=0(n+rr)qn+n=0(n+r-1r-1)qn-1]=(1+B)[qr1-q+1-(1-q)r](B-A)cos θ,

hence (3.7) holds, and therefore Pqr(f)L(A,B,θ).

Theorem 3.5

If the condition

(B-A)qrcos θ1-qk+2

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


If f ∈ ℒ(A,B, θ) has the form (1.1), since Pqr is given by (1.7), using Lemma 2.3 we need to prove that


From Lemma 2.2 and the assumption (3.8) we obtain that

n=2(n+r-2r-1)n(n-1)qn-1(1-q)rann=2(n+r-2r-1)(n-1)(B-A)qn-1(1-q)rcos θ=(B-A)(1-q)rcos θn=2(n-1)(n+r-2r-1)qn-1=(B-A)(1-q)rqrcos θn=0(n+rr)qn=(B-A)qrcos θ1-q1k+2,

hence (3.9) holds, and consequently Pqr(f)k-UCV.

Theorem 3.6

If the condition

(B-A)cos θ[k+1-(1-q)r-k(1-q)q(r-1)(1-(1-q)r-1)]1

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


If f ∈ ℒ(A,B, θ) has the power expansion series (1.1), and Pqr is given by (1.7), according to Lemma 2.4 for β = 0 we need to prove that


From Lemma 2.2 and the assumption (3.10) we may easily show that

n=2(n+r-2r-1)[n(k+1)-k]qn-11-qrann=2(n+r-2r-1)[n(k+1)-k]qn-1(1-q)r(B-A)cos θn=(B-A)(1-q)rcos θ[n=2(k+1)(n+r-2r-1)qn-1-n=2kn(n+r-2r-1)qn-1]=(B-A)(1-q)rcos θ[(k+1)(n=0(n+r-1r-1)qn-1)-kq(r-1)(n=0(n+r-2r-2)qn-1-(r-1)q)]=(B-A)cos θ[(k+1)[1-(1-q)r]-kq(r-1)[(1-q)-(1-q)r-q(r-1)(1-q)r]]=(B-A)cos θ[k+1-(1-q)r-k(1-q)q(r-1)(1-(1-q)r-1)]1,

that is (3.11) holds, and thus Pqr(f)k-Sp.

In the following two theorems we will obtain analogues results in connection with the function Iqr defined by


Theorem 3.7

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

(1+B)[1-(1-q)r](B-A)cos θ.



to prove that IqrL(A,B,θ), according to Lemma 2.1 it is sufficient to show that

n=2(n+r-2r-1)n(1+B)nqn-11-qr(B-A)cos θ.

Using the assumption (3.12), a simple computation shows that

n=2(n+r-2r-1)n(1+B)nqn-1(1-q)r=(1+B)(1-q)rn=2(n+r-2r-1)qn-1=(1+B)(1-q)rn=1(n+r-1r-1)qn=(1+B)(1-q)r[1(1-q)r-1]=(1+B)[1-(1-q)r](B-A)cos θ.

and from (3.14) if follows that IqrL(A,B,θ).

Theorem 3.8

A sufficient condition for the functionIqrto be in the classis


Since Iqr has the form (3.13), to prove that Iqrk-Sp(β), according to Lemma 2.4 it is sufficient to prove that


Thus, from the assumption (3.15) we get


and from (3.16) we conclude that Iqrk-Sp(β).


The authors are grateful to the reviewers of this article, that gave valuable remarks, comments, and advices, in order to revise and improve the results of the paper.

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