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

Published online March 31, 2021

Copyright © Kyungpook Mathematical Journal.

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

Abstract

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In the present paper, we determine new characterisations of the subclasses TS∗C(α,β;γ) and TCC(α,β;γ) of analytic functions associated with Pascal distribution series Φqm(z)=z−∑ n=2∞n+m−2m−1qn−1(1−q)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.

1. Introduction and Definitions

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

f(z)=z+∑ n=2∞anzn,

which are analytic in the open unit disk U={z∈ℂ:|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=2∞anzn,  z∈U.

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

ℜzf′(z)f(z)>α  (z∈U).

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

ℜ1+zf′′(z)f′(z)>α  (z∈U).

We denote by S∗(α) and C(α) the classes of functions that starlike of order α and convex of order α∈U, respectively. Further, TS∗(α) and TC(α) denote the subclasses of T consisting of functions which are starlike of order α(0≤α<1) and convex of order α(0≤α<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)=f∈A:ℜzf′(z)βf′(z)+(1−β)f(z)>α,  (α,β∈[0,1),z∈U)

and

C(α,β)=f∈A:ℜf′(z)+zf′′(z)f′(z)+βzf′′(z)>α,  (α,β∈[0,1),z∈U).

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 S∗C(α,β;γ) if it satisfies the following condition:

ℜzf′(z)+γz2f′′(z)%γzf′(z)+βzf′′(z)+(1−γ)βf′(z)+(1−β)f(z)>α   (z∈U),

where α,β∈[0,1) and γ∈[0,1].

Also we denote

TS∗C(α,β;γ)=S%∗C(α,β;γ)∩T.

Let f∈CC(α,β;γ) if and only if zf′∈S∗C(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 TS∗C(α,β;0)=TS∗(α,β) and TS ∗C(α,β;1)=TC(α,β). Also, we have TS∗C(α,0;0)=TS∗(α) and TS∗C(α,0;1)=TC(α).

A function f∈A is said to be in the class Rτ(A,B),τ∈ℂ\{0}, −1≤B<A≤1, if it satisfies the inequality

f′(z)−1(A−B)τ−B[f′(z)−1]<1,    z∈U.

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

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

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

P(X=r)=r+m−1m−1qr(1−q)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=2∞n+m−2m−1qn−1(1−q)mzn, z∈U,

where m≥1, 0≤q≤1. We note that, by the ratio test, the radius of convergence of above series is infinity. We also define the series

Φqm(z):=2z−Ψqm(z)=z−∑ n=2∞n+m−2m−1qn−1(1−q)mzn, z∈U.

Let consider the linear operator Iqm:A→A defined by the Hadamard product

Iqmf(z):=Ψqm(z)∗f(z)=z+∑ n=2∞n+m−2m−1qn−1(1−q)manzn, z∈U,

where m≥1 and 0≤q≤1.

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 TS∗C(α,β;γ) and TCC(α,β;γ). Furthermore, we give sufficient conditions for IqmRτ(A,B)⊂TS∗C(α,β;γ) and Iqm Rτ(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.

2. Preliminary Lemmas

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

Lemma 2.1.

([23]) A function f∈T of the form (1.2) is in the class TS∗C(α,β;γ) if and only if

∑ n=2∞(1+(n−1)γ)n−α−(n−1)αβan≤1−α .

The result (2.1) is sharp.

Lemma 2.2.

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

∑ n=2∞n(1+(n−1)γ)n−α−(n−1)αβan≤1−α .

The result (2.2) is sharp.

Lemma 2.3.

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

an≤(A−B)τn,  n∈ℕ−{1}.

The result is sharp for the function

f(z)=∫0z (1+(A−B)τtn−1 1+Btn−1 )dt,  (z∈U;n∈ℕ−{1}).

3. Necessary and Sufficient Conditions

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

∑n=0∞ n+m−1 m−1qn=1 (1−q)m, ∑n=0∞ n+m−2 m−2qn=1% (1−q) m−1,∑n=0∞ n+m mqn=1 (1−q) m+1%, ∑n=0∞ n+m+1 m+1qn=1% (1−q) m+2.

By simple calculations we derive the following relations:

∑ n=2∞n+m−2m−1qn−1=∑ n=0∞n+m−1m−1qn−1=1(1−q)m−1, ∑ n=2∞(n−1)n+m−2m−1qn−1=qm∑ n=0∞n+mmqn=qm(1−q)m+1, ∑n=3∞(n−1)(n−2) n+m−2 m−1qn−1=q2m(m+1)∑n=0∞ n+m+1 m+1qn            =q2m(m+1) (1−q) m+2.

and

∑n=4∞(n−1)(n−2)(n−3) n+m−2 m−1qn−1=q3m(m+1)(m+2)∑n=0∞n+m+1m+1qn=q3m(m+1)(m+2) (1−q) m+3.  

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

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

Theorem 3.1.

If m≥1, then Φqm∈TS∗C(α,β;γ) if and only if

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

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

P :=∑ n=2∞(1+(n−1)γ)n−α−(n−1)αβn+m−2m−1qn−1(1−q)m≤1−α.

Writing

n=(n−1)+1

and

n2=(n−1)(n−2)+3(n−1)+1

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

P=∑n=2∞γ(1−αβ)n2+αβ(2γ−1)−γ(1+α)+1n+α(1−γ)(β−1) ×n+m−2m−1qn−1(1−q)mn=γ(1−αβ)∑n=3∞(n−1)(n−2)n+m−2m−1qn−1(1−q)m +γ(2−α)−αβ(γ+1)+1∑n=2∞(n−1)n+m−2m−1qn−1(1−q)m +1−α∑n=2∞n+m−2m−1qn−1(1−q)m=γ(1−αβ)q2m(m+1)(1−q)2+γ(2−α)−αβ(γ+1)+1qm(1−q) +1−α(1−(1−q)m).

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

Theorem 3.2.

If m≥1, then Φqm∈TCC(α,β;γ) if and only if

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

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

Q :=∑ n=2∞n(1+(n−1)γ)n−α−(n−1)αβn+m−2m−1qn−1(1−q)m≤1−α.

Writing

n=(n−1)+1 n2=(n−1)(n−2)+3(n−1)+1

and

n3=(n−1)(n−2)(n−3)+6(n−1)(n−2)+7(n−1)+1

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

Q=∑n=2∞γ(1−αβ)n3+αβ(2γ−1)−γ(1+α)+1n2+α(1−γ)(β−1)n ×n+m−2m−1qn−1(1−q)m=γ(1−αβ)∑n=4∞(n−1)(n−2)(n−3)n+m−2m−1qn−1(1−q)m +γ(5−α)−αβ4γ+1+1∑n=3∞(n−1)(n−2)n+m−2m−1qn−1(1−q)m +2γ(2−α)−2αβγ+1+3−α∑n=2∞(n−1)n+m−2m−1qn−1(1−q)m +1−α∑n=2∞n+m−2m−1qn−1(1−q)m=γ(1−αβ)q3m(m+1)(m+2)(1−q)3+γ(5−α)−αβ4γ+1+1q2m(m+1)(1−q)2 +2γ(2−α)−2αβγ+1+3−αqm(1−q)+1−α(1−(1−q)m).

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

4. Inclusion Properties

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

Theorem 4.1.

Let m>1. If f∈Rτ(A,B), then Iqmf(z) is in TS∗C(α,β;γ) if

(A−B)|τ|γ(1−αβ)qm(1−q)+αβ(γ−1)−γα+1(1−(1−q)m)+α(1−γ)(β−1)q(m−1)(1−q)−(1−q)m−q(m−1)(1−q)m≤1−α.

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

Λ :=∑ n=2∞(1+(n−1)γ)n−α−(n−1)αβn+m−2m−1qn−1(1−q)man≤1−α.

Since f∈Rτ(A,B), then by Lemma 2.3,

we have

an≤(A−B)τn.

Thus, we have

Λ=(A−B)|τ|∑ n=2∞γ(1−αβ)n+αβ(2γ−1)−γ(1+α)+1      +1nα(1−γ)(β−1)n+m−2m−1qn−1(1−q)m=(A−B)|τ|∑ n=2∞γ(1−αβ)(n−1)+αβ(γ−1)−γα+1      +1nα(1−γ)(β−1)n+m−2m−1qn−1(1−q)m=(A−B)|τ|γ(1−αβ)qm(1−q)+αβ(γ−1)−γα+1(1−(1−q)m) +α(1−γ)(β−1)q(m−1)(1−q)−(1−q)m−q(m−1)(1−q)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 m≥1. If f∈Rτ(A,B), then Iqmf∈TCC(α,β;γ) if

(A−B)|τ|γ(1−αβ)q2m(m+1)(1−q)m+2+γ(2−α)−αβ(γ+1)+1qm(1−q)m+1≤1−α.

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

H:=∑ n=2∞n(1+(n−1)γ)n−α−(n−1)αβn+m−2m−1qn−1(1−q)man≤1−α.

Since f∈Rτ(A,B) using Lemma we have

an≤(A−B)τn, n∈ℕ∖{1},

therefore

H≤(A−B)τ∑ n=2∞(1+(n−1)γ)n−α−(n−1)αβn+m−2m−1%qn−1(1−q)m

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

5. An Integral Operator

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Theorem 5.1.

Let m≥1. Then the integral operator

Gqm(z):=∫0z Φ q m(t)tdt, z∈U,

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=2∞n+m−2m−1qn−1(1−q)mznn, z∈U.

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

only if

∑ n=2∞n(1+(n−1)γ)n−α−(n−1)αβ1nn+m−2m−1qn−1(1−q)m ≤1−α.

By a similar proof like those of Theorem 3.1 we get that Gqm∈TCC(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 TS∗C(α,β;γ) if and

only if

γ(1−αβ)qm(1−q)+αβ(γ−1)−γα+1(1−(1−q)m)+α(1−γ)(β−1)q(m−1)(1−q)−(1−q)m−q(m−1)(1−q)m≤1−α.

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.

6. Corollaries and Consequences

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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 m≥1, then Φqm∈TS∗(α,β) if and only if

1−αβqm(1−q)m+1≤1−α.

Corollary 6.2.

If m≥1, then Φqm∈TC(α,β), if and only if

(1−αβ)q2m(m+1)(1−q)m+2+3−α−2αβqm(1−q)m+1≤1−α.

Corollary 6.3.

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

(A−B)|τ|1−αβ(1−(1−q)m)+α(β−1)q(m−1)(1−q)−(1−q)m−q(m−1)(1−q)m≤1−α.

Corollary 6.4.

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

(A−B)|τ|(1−αβ)qm(1−q)+1−α(1−(1−q)m)≤1−α.

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

1−αβ(1−(1−q)m)+α(β−1)q(m−1)(1−q)−(1−q)m−q(m−1)(1−q)m≤1−α.

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

(1−αβ)qm(1−q)+1−α(1−(1−q)m)≤1−α.
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