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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(1): 99-110

Published online March 31, 2021

### Subclasses of Starlike and Convex Functions Associated with Pascal Distribution Series

Basem Aref Frasin*, Sondekola Rudra Swamy, Abbas Kareem Wanas

Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
e-mail : bafrasin@yahoo.com

Department of Computer Science and Engineering, RV College of Engineering, Bengaluru - 560 059, Karnataka, India
e-mail : mailtoswamy@rediffmail.com

Department of Mathematics, College of Science, University of Al-Qadisiyah, Iraq
e-mail : abbas.kareem.w@qu.edu.iq

Received: May 20, 2020; Revised: October 7, 2020; Accepted: October 13, 2020

In the present paper, we determine new characterisations of the subclasses TSC(α,β;γ) and TCC(α,β;γ) of analytic functions associated with Pascal distribution series Φqm(z)=z n=2n+m2m1qn1(1q)mzn. Further, we give necessary and sufficient conditions for an integral operator related to Pascal distribution series Gqmf(z)=0z Φ q m(t)tdt to belong to the above classes. Several corollaries and consequences of the main results are also considered.

Keywords: Analytic functions, Hadamard product, Pascal distribution series.

Let A denote the class of the normalized functions of the form

f(z)=z+ n=2anzn,

which are analytic in the open unit disk U={z:|z|<1} and are normalized by the conditions f(0)=f(0)1=0. Further, let T be the subclass of A consisting of functions of the form

f(z)=z n=2anzn,  zU.

A function fA is said to be starlike of order α(0α<1) if it satisfies

zf(z)f(z)>α  (zU).

Also, we say that a function fA is said to be convex of order α(0α<1) if it satisfies

1+zf(z)f(z)>α  (zU).

We denote by S(α) and C(α) the classes of functions that starlike of order α and convex of order αU, respectively. Further, TS(α) and TC(α) denote the subclasses of T consisting of functions which are starlike of order α(0α<1) and convex of order α(0α<1) with negative coefficients in U, respectively [21].

Interesting generalization of the functions classes S(α) and C(α), are classes S(α,β) and C(α,β), where

S(α,β)=fA:zf(z)βf(z)+(1β)f(z)>α,(α,β[0,1),zU)

and

C(α,β)=fA:f(z)+zf(z)f(z)+βzf(z)>α,(α,β[0,1),zU).

The classes TS(α,β)=S(α,β)T and TC(α,β)=C(α,β)T were extensively studied by Altintaş and Owa [1], Porwal [19], Moustafa [13] and Porwal and Dixit [20].

Inspired by the studies mentioned above, Topkaya and Mustafa [23] defined a unification of the functions classes S(α,β) and C(α,β) as follows.

### Definition 1.1.

A function f of the form (1.1) is said to be in the class SC(α,β;γ) if it satisfies the following condition:

zf(z)+γz2f(z)%γzf(z)+βzf(z)+(1γ)βf(z)+(1β)f(z)>α (zU),

where α,β[0,1) and γ[0,1].

Also we denote

TSC(α,β;γ)=S%C(α,β;γ)T.

Let fCC(α,β;γ) if and only if zfSC(α,β;γ) and denote the class TCC(α,β;γ) to be defined as

TCC(α,β;γ)=CC(α,β;γ)T.

In particular, the class TSC(α,β;0)=TS(α,β) and TSC(α,β;1)=TC(α,β). Also, we have TSC(α,0;0)=TS(α) and TSC(α,0;1)=TC(α).

A function fA is said to be in the class Rτ(A,B),τ\{0}, 1B<A1, if it satisfies the inequality

f(z)1(AB)τB[f(z)1]<1,zU.

This class was introduced by Dixit and Pal [4].

A variable X is said to be Pascal distribution if it takes the values 0,1,2,3, with probabilities

(1q)m,qm(1q)m1!,q2m(m+1)(1q)m2!,q3m(m+1)(m+2)(1q)m3!,,

respectively, where q and m are called the parameters, and thus

P(X=r)=r+m1m1qr(1q)m, r=0,1,2,3,.

Very recently, El-Deeb et al. [6] (see also, [2, 8, 9, 15]) introduced a power series whose coefficients are probabilities of Pascal distribution. Let

Ψqm(z):=z+ n=2n+m2m1qn1(1q)mzn,zU,

where m1, 0q1. We note that, by the ratio test, the radius of convergence of above series is infinity. We also define the series

Φqm(z):=2zΨqm(z)=z n=2n+m2m1qn1(1q)mzn,zU.

Let consider the linear operator Iqm:AA defined by the Hadamard product

Iqmf(z):=Ψqm(z)f(z)=z+ n=2n+m2m1qn1(1q)manzn,zU,

where m1 and 0q1.

There are several known results on connections between various subclasses of analytic and univalent functions using hypergeometric functions (see for example, [12, 22]) and using various distributions such as Yule-Simon distributions, Logarithmic distributions, Poisson distributions, Binomial distributions, Beta-Binomial distributions, Zeta distributions, Geometric distributions and the Bernoulli distribution (see, for example, [5, 7, 10, 14, 16, 17, 18]).

In this paper, we determine necessary and sufficient conditions for Φqm to be in the classes TSC(α,β;γ) and TCC(α,β;γ). Furthermore, we give sufficient conditions for IqmRτ(A,B)TSC(α,β;γ) and IqmRτ(A,B)TCC(α,β;γ). Finally, we provide necessary and sufficient conditions for the integral operator Gqmf(z)=0z Φ q m(t)tdt to belong to the above classes.

To establish our main results, we need the following lemmas.

### Lemma 2.1.

([23]) A function fT of the form (1.2) is in the class TSC(α,β;γ) if and only if

n=2(1+(n1)γ)nα(n1)αβan1α .

The result (2.1) is sharp.

### Lemma 2.2.

A function fT of the form (1.2) is in the class TCC(α,β;γ) if and only if

n=2n(1+(n1)γ)nα(n1)αβan1α .

The result (2.2) is sharp.

### Lemma 2.3.

([4]) If the function fRτ(A,B) is of the form (1.1), then

an(AB)τn,n{1}.

The result is sharp for the function

f(z)=0z (1+(AB)τtn1 1+Btn1 )dt, (zU;n{1}).

### 3. Necessary and Sufficient Conditions

For convenience throughout in the sequel, we use the following identities that hold for m1 and 0q<1:

n=0 n+m1 m1qn=1 (1q)m,n=0 n+m2 m2qn=1% (1q) m1,n=0 n+m mqn=1 (1q) m+1%,n=0 n+m+1 m+1qn=1% (1q) m+2.

By simple calculations we derive the following relations:

n=2n+m2m1qn1= n=0n+m1m1qn1=1(1q)m1, n=2(n1)n+m2m1qn1=qm n=0n+mmqn=qm(1q)m+1, n=3(n1)(n2) n+m2 m1qn1=q2m(m+1)n=0 n+m+1 m+1qn            =q2m(m+1) (1q) m+2.

and

n=4(n1)(n2)(n3) n+m2 m1qn1=q3m(m+1)(m+2)n=0n+m+1m+1qn=q3m(m+1)(m+2) (1q) m+3.

Unless otherwise mentioned, we shall assume in this paper that α,β[0,1), γ[0,1] and 0q<1.

Firstly, we obtain the necessary and sufficient conditions for Φqm to be in the class TSC(α,β;γ).

### Theorem 3.1.

If m1, then ΦqmTSC(α,β;γ) if and only if

γ(1αβ)q2m(m+1)(1q)m+2+γ(2α)αβ(γ+1)+1qm(1q)m+11α.

Proof. Since Φqm is defined by (1.6), in view of Lemma 2.1 it is sufficient to show that

P := n=2(1+(n1)γ)nα(n1)αβn+m2m1qn1(1q)m1α.

Writing

n=(n1)+1

and

n2=(n1)(n2)+3(n1)+1

and using (3.1)-(3.3), we get

P=n=2γ(1αβ)n2+αβ(2γ1)γ(1+α)+1n+α(1γ)(β1)×n+m2m1qn1(1q)mn=γ(1αβ)n=3(n1)(n2)n+m2m1qn1(1q)m+γ(2α)αβ(γ+1)+1n=2(n1)n+m2m1qn1(1q)m+1αn=2n+m2m1qn1(1q)m=γ(1αβ)q2m(m+1)(1q)2+γ(2α)αβ(γ+1)+1qm(1q)+1α(1(1q)m).

but this last expression is upper bounded by 1-α if and only if (3.5) holds.

### Theorem 3.2.

If m1, then ΦqmTCC(α,β;γ) if and only if

γ(1αβ)q3m(m+1)(m+2)(1q)m+3+γ(5α)αβ4γ+1+1q2m(m+1)(1q)m+2  +2γ(2α)2αβγ+1+3αqm(1q)m+11α.

Proof. In view of Lemma 2.2 it is sufficient to show that

Q := n=2n(1+(n1)γ)nα(n1)αβn+m2m1qn1(1q)m1α.

Writing

n=(n1)+1 n2=(n1)(n2)+3(n1)+1

and

n3=(n1)(n2)(n3)+6(n1)(n2)+7(n1)+1

and using (3.1)-(3.5), we get

Q=n=2γ(1αβ)n3+αβ(2γ1)γ(1+α)+1n2+α(1γ)(β1)n×n+m2m1qn1(1q)m=γ(1αβ)n=4(n1)(n2)(n3)n+m2m1qn1(1q)m+γ(5α)αβ4γ+1+1n=3(n1)(n2)n+m2m1qn1(1q)m+2γ(2α)2αβγ+1+3αn=2(n1)n+m2m1qn1(1q)m+1αn=2n+m2m1qn1(1q)m=γ(1αβ)q3m(m+1)(m+2)(1q)3+γ(5α)αβ4γ+1+1q2m(m+1)(1q)2+2γ(2α)2αβγ+1+3αqm(1q)+1α(1(1q)m).

but this last expression is upper bounded by 1-α if and only if (3.6) holds.

Making use of Lemma 2.3, we will study the action of the Pascal distribution series on the classes TSC(α,β;γ) and TCC(α,β;γ).

### Theorem 4.1.

Let m>1. If fRτ(A,B), then Iqmf(z) is in TSC(α,β;γ) if

(AB)|τ|γ(1αβ)qm(1q)+αβ(γ1)γα+1(1(1q)m)+α(1γ)(β1)q(m1)(1q)(1q)mq(m1)(1q)m1α.

Proof. In view of Lemma 2.1, it suffices to show that

Λ := n=2(1+(n1)γ)nα(n1)αβn+m2m1qn1(1q)man1α.

Since fRτ(A,B), then by Lemma 2.3,

we have

an(AB)τn.

Thus, we have

Λ=(AB)|τ| n=2γ(1αβ)n+αβ(2γ1)γ(1+α)+1      +1nα(1γ)(β1)n+m2m1qn1(1q)m=(AB)|τ| n=2γ(1αβ)(n1)+αβ(γ1)γα+1      +1nα(1γ)(β1)n+m2m1qn1(1q)m=(AB)|τ|γ(1αβ)qm(1q)+αβ(γ1)γα+1(1(1q)m)+α(1γ)(β1)q(m1)(1q)(1q)mq(m1)(1q)m.

But this last expression is bounded by 1-α, if (4.1) holds. This completes the proof of Theorem 4.1.

### Theorem 4.2.

Let m1. If fRτ(A,B), then IqmfTCC(α,β;γ) if

(AB)|τ|γ(1αβ)q2m(m+1)(1q)m+2+γ(2α)αβ(γ+1)+1qm(1q)m+11α.

Proof. According to Lemma 2.2 it is sufficient to show that

H:= n=2n(1+(n1)γ)nα(n1)αβn+m2m1qn1(1q)man1α.

Since fRτ(A,B) using Lemma we have

an(AB)τn,n{1},

therefore

H(AB)τ n=2(1+(n1)γ)nα(n1)αβn+m2m1%qn1(1q)m

The remaining part of the proof is similar to that of Theorem 3.1 and so we omit the details.

### Theorem 5.1.

Let m1. Then the integral operator

Gqm(z):=0z Φ q m(t)tdt,zU,

is in TCC(α,β;γ) if and only if the inequality (3.5) holds.

Proof. According to 1.6 it follows that

Gqm(z)=z n=2n+m2m1qn1(1q)mznn,zU.

Using Lemma 2.2, the function Gqm(z) belongs to TCC(α,β;γ) if and

only if

n=2n(1+(n1)γ)nα(n1)αβ1nn+m2m1qn1(1q)m 1α.

By a similar proof like those of Theorem 3.1 we get that GqmTCC(α,β;γ) if and only if (3.5) holds.

### Theorem 5.2.

If m>1, then the integral operator Gqm(z) given by (5.1) is in TSC(α,β;γ) if and

only if

γ(1αβ)qm(1q)+αβ(γ1)γα+1(1(1q)m)+α(1γ)(β1)q(m1)(1q)(1q)mq(m1)(1q)m1α.

The proof of Theorem Theorem 5.2 is lines similar to the proof of Theorem 4.1, so we omitted the proof of Theorem 5.2.

By taking γ=0 and β = 1 in Theorems 3.1, 4.1 and 5.2, we obtain the following necessary and sufficient conditions for Pascal distribution series to be in the classes TS(α,β) and TC(α,β).

### Corollary 6.1.

If m1, then ΦqmTS(α,β) if and only if

1αβqm(1q)m+11α.

### Corollary 6.2.

If m1, then ΦqmTC(α,β), if and only if

(1αβ)q2m(m+1)(1q)m+2+3α2αβqm(1q)m+11α.

### Corollary 6.3.

Let m>1. If fRτ(A,B), then Iqmf(z) is in TS(α,β) if

(AB)|τ|1αβ(1(1q)m)+α(β1)q(m1)(1q)(1q)mq(m1)(1q)m1α.

### Corollary 6.4.

Let m>1. If fRτ(A,B), then Iqmf(z) is in TC(α,β) if

(AB)|τ|(1αβ)qm(1q)+1α(1(1q)m)1α.

### Corollary 6.5.

If m>1, then the integral operator Gqm(z) given by (5.1) is in TS(α,β) if and only if

1αβ(1(1q)m)+α(β1)q(m1)(1q)(1q)mq(m1)(1q)m1α.

### Corollary 6.6.

If m>1, then the integral operator Gqm(z) given by (5.1) is in TC(α,β) if and only if

(1αβ)qm(1q)+1α(1(1q)m)1α.

The authors would like to thank the referees for their helpful comments and suggestions.

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