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Kyungpook Mathematical Journal 2024; 64(1): 47-55

Published online March 31, 2024

A Study on Two Subclasses of Analytic and Univalent Functions with Negative Coefficients Involving the Poisson Distribution Series

Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Kingdom of Saudi Arabia
Department of Mathematics Faculty of Science, Mansoura University, Mansoura, 35516, Egypt
e-mail : alasheen@kau.edu.sa; aylashin@mans.edu.eg

Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Kingdom of Saudi Arabia

Department of Mathematics, Faculty of Science, Bishha University, Bishha, King- dom of Saudi Arabia
e-mail : fayeed@ub.edu.sa

Received: June 6, 2023; Revised: October 20, 2023; Accepted: November 3, 2023

This paper introduces two new subclasses of analytical functions with negative coefficients and derives coefficient estimates for these novel subclasses. Further, inclusion relations and necessary and sufficient conditions for the Poisson distribution series to belong to these subclasses are established.

Keywords: The Poisson distribution series, Analytic functions, Univalent functions, Convex funcions, Coefficient inequalities

1. Introduction

Let A be the class of all functions ξ with Taylor series expansion of the form

ξ(z)=z+k=2akzk,

that are analytic in the open unit disc U={zC:ertzert<1}. Further, let S denote the subclass of A consisting of normalized and univalent functions in U. Let S*(α) and K(α) denote, respectively, the two well-known subclasses of S that are starlike and convex functions of order α, 0α<1, which were introduced by Robertson [17]. It is well-known that

S*(α)=ξA:zξ(z)ξ(z)>α, zU,

and

K(α)=ξA:1+zξ(z)ξ(z)>α, zU.

Definition 1.1. Let Aβ,α denote a class of function ξA which satisfy the following condition

ertβzξ(z)ξ(z)-1+1-βz2ξ(z)ξ(z)ert<1-α,

where 0β, α<1.

Fukui [6], introduced the function class Aβ,α and he proved that Aβ,αS*(α). We note that A1,αS*(α). In [19] Singh and Singh introduced the class of functions ξA which satisfy the following condition

β1+zξ(z)ξ(z)+1-β1ξ(z)<1+2a2,

where a>12 and zU and they gave some criteria for univalence expresing by {ξ(z)}>0.

Definition 1.2. Let β,α) denote a class of functions ξA which satisfy

β1+zξ(z)ξ(z)+1-β1ξ(z)>α,

where 0β, α<1.

We note that

• F(1,α)K(α).

• F(0,α) is the class of univalent close to convex functions that satisfies

{ξ(z)}>0.

Let T be the subclass of S consisting of functions with negative coefficients of the form

ξ(z)=z-k=2akzk,ak0.

Further, we define the class Tβ,α by

Tβ,α=Aβ,αT.

Also, we define the class Rβ,α by

Rβ,α=β,α)T.

The Poisson distribution was created in 1837 by Siméon Denis Poisson. This distribution expresses the probability of a given number of events occurring in a fixed interval of time or space. Recently, Poisson, Pascal, Logarithmic, and Binomial distributions have been partially studied in Geometric Function Theory (see [1, 3, 5, 7, 12, 15]). A variable ϰ is said to have Poisson distribution if it takes the values 0,1,2,3,... with probabilities

e-m,me-m1!,m2e-m2!,m3e-m3!,...,

respectively, where m is called the parameter. Thus

P(ϰ=k)=mke-mk!,k=0,1,2,3,....

In 2014, Porwal [13] introduced a power series such that its coefficients are probabilities of the Poisson distribution

N(m,z)=z+k=2mk-1k-1!e-mzk,m>0, zU.

Also, he introduced the series

R(m,z)=2zN(m,z),=z k=2m k1 k1!emzk,  m>0, zU.

Based on earlier results using hypergeometric functions associated with various subclasses of analytic and univalent functions [4, 8, 18, 20], Porwal's recent research [14, 16], and the results using generalized Bessel functions to connect various subclasses of analytic and univalent functions [2, 9, 10, 11], this paper examines some properties and characteristics of the two classes Tβ,α and Rβ,α. We also provide necessary and sufficient conditions for the Poisson distribution to belong to these classes.

2. Main Results

Firstly, we determine the sufficient and necessary conditions for the function ξT to belong to the class Tβ,α.

Theorem 2.1. Let the function ξ be defined by (1.7). Then ξTβ,α if and only if

k=2(k-1)(k(1-β)+β)+(1-α)ak<1-α.

proof. Assume that the inequality (2.1) holds true, then we need to show that

ertβzξ(z)ξ(z)-1+1-βz2ξ(z)ξ(z)ert<1-α.

Since

β zξ(z)ξ(z)1+1βz2ξ(z)ξ(z)= k=2(k1)(k(1β)+β)akzk11 k=2akzk1 k=2(k1)(k(1β)+β)ak1 k=2ak.

Then

ertβzξ(z)ξ(z)-1+1-βz2ξ(z)ξ(z)ert<1-α,

if

k=2(k-1)(k(1-β)+β)ak<(1-α)1-k=2ak,

or equivalently

k=2((k-1)(k(1-β)+β)+(1-α))ak<1-α.

Conversely, assume that the function ξT is in the class Tβ,α. Since {z}ertzert, then we have

k=2(k-1)(k(1-β)+β)akzk-11-k=2akzk-1<1-α.

Choosing z on the real axis, then

k=2(k-1)(k(1-β)+β)akzk-11-k=2akzk-1

is real. Let z1- through real values, we get

k=2((k-1)(k(1-β)+β)+(1-α))ak<1-α

which is equivalent to (2.1). And this ends the proof of the theorem.

Corollary 2.2. Let ξT,0β1β2<1 and 0α1.If ξ(z)Tβ1,α,then ξ(z)Tβ2,α .i.e Tβ1,αTβ2,α.

proof. Let the function ξ(z)Tβ1,α. Then, by Theorem 2.1, we have

k=2((k-1)(k(1-β1)+β1)+(1-α))ak<1-α.

Since β1β2, we get

k=2((k1)(k(k1)β2)+(1α))ak k=2((k1)(k(k1)β1)+(1α))ak<1α,

which implies that ξ(z)Tβ2,α.

Corollary 2.3. All functions in the class Tβ,α are starlike.

proof. Since β<1. An application of Corollary 2.2 gives Tβ,αT1,α.

We apply the Poisson distribution function to the class Tβ,α in Theorem 2.4 below.

Theorem 2.4. Let m>0 and R(m,z) be defined by (1.13), then R(m,z)Tβ,α if and only if

(1-β)m2+(2-β)m1-αe-m.

proof. According to Theorem 2.1, it is sufficient to show that condition (2.2) is equivalent to

k=2((k-1)(k(1-β)+β)+(1-α))mk-1k-1!e-m1-α.

Thus,

k=2((k1)(k(1β)+β)+(1α))m k1 k1!em=emk=2((1β)(k1)(k2)+(k1)(2β)+(1α))m k1 k1!=em(1β)m2k=3 m k3 k3!+(2β)mk=2 m k2 k2!+(1α)k=2 m k1 k1!=(1β)m2+(2β)m+(1α)(1em)1α.

Which ends the proof.

Secondly, we determine the sufficient and necessary conditions for the function ξT to belong to the class Rβ,α.

Theorem 2.5. Let the function ξT be defined by (1.7) and let β12. Then ξRβ,α if and only if

k=2(k(βk-α))ak1-α.

proof. Let us assume inequality (2.3) holds. The goal is to prove that

ertβ1+zξ(z)ξ(z)+1-β1ξ(z)-1ert<1-α.

Since

β1+zξ(z)ξ(z)+1β1ξ(z)1= k=2(k(kβ1))akzk11 k=2kakzk1 k=2(k(kβ1))ak1 k=2kak1α.

Hence ξ satisfies the condition (2.4). Conversely, assume that the function ξT is in the class Rβ,α. Then we have

β 1+zξ(z)ξ(z)+1β1ξ(z)=1 k=2βk2akzk11 k=2kakzk1>α.

If z is chosen on the real axis, then

1-k=2βk2akzk-11-k=2kakzk-1

is real. Letting z1- through real values, we get

k=2(k(kβ-α))ak<1-α.

Which is equivalent to (2.3) and this completes the proof.

Corollary 2.6. Let ξT,12β1β2<1 and 0α1,ifξβ2,α), then ξRβ1,α. i.e Rβ2,αRβ1,α.

proof. Let the function ξβ2,α). Then, by Theorem 2.5, we have

k=2(k(β2k-α))ak1-α.

Since β1β2, we get

k=2(k(β1kα))ak k=2(k(β2kα))ak,1α,

which implies that ξ(z)Rβ1,α.

We apply the Poisson distribution function to the class Rβ,α in Theorem 2.7 below.

Theorem 2.7. Let m>0 and R(m,z) be defined by (1.13), then R(m,z)Rβ,α, if and only if

βm2+(3β-α)m+(β-α)(1-e-m)1-α.

proof. According to Theorem 2.5, we need to show that (2.6) is equivalent to the inequality

k=2(k(βk-α))mk-1k-1!e-m1-α.

Thus,

k=2(k2βkα)m k1 k1!em=emk=2(β(k1)(k2)+(3βα)(k1)+(βα))m k1 k1!=emβm2k=3 m k3 k3!+(3βα)mk=2 m k2 k2!+(βα)k=2 m k1 k1!=βm2+(3βα)m+(βα)(1em)1α.

Which completes the proof.

Our paper introduces two novel classes of analytic and univalent functions with negative coefficients and derives coefficient inequalities and inclusion relations for these two classes. Further, using a similar method to Porwal's [13], necessary and sufficient conditions for the Poisson distribution series to belong to these classes are also discussed.

Acknowledgements

The authors thank the editor and referees for their helpful comments and suggestions that helped improve the paper's presentation.

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