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

Published online March 31, 2024

Copyright © Kyungpook Mathematical Journal.

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

Abdul Moneim Yousof Lashin∗, Abeer Omard Ahmad Badghaish, Fayzah Awad Alshehri

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
e-mail : abadghaish@kau.edu.sa

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

Abstract

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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

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

ξ(z)=z+∑k=2∞akzk ,

that are analytic in the open unit disc U={z∈C: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, 0≤α<1, which were introduced by Robertson [17]. It is well-known that

S*(α)=ξ∈A: ℜzξ​​′(z)ξ(z)>α, z∈U,

and

K(α)=ξ ∈A: ℜ1+zξ​​​′′(z)ξ​​′(z)>α, z∈U.

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

ertβzξ​​′(z)ξ(z)-1+1-βz2ξ ​′′(z)ξ(z)ert<1-α,

where 0≤β, α<1.

Fukui [6], introduced the function class Aβ,α and he proved that Aβ,α⊂S*(α). We note that A1,α≡S*(α). In [19] Singh and Singh introduced the class of functions ξ∈A which satisfy the following condition

ℜβ1+zξ​​​′′(z)ξ​​′(z)+1-β1ξ ​′(z)<1+2a2, 

where a>12 and z∈U and they gave some criteria for univalence expresing by ℜ{ξ​​′(z)}>0.

Definition 1.2. Let β,α) denote a class of functions ξ∈A which satisfy

ℜβ1+zξ​​​′′(z)ξ​​′(z)+1-β1ξ ​′(z)>α,

where 0≤β, α<1.

We note that

  • F(1,α)≡K(α).

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

ℜ{ξ′(z)}>0.

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

ξ(z)=z-∑k=2∞akzk,  ak≥0.

Further, we define the class Tβ,α by

Tβ,α=Aβ,α∩T.

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=2∞mk-1k-1!e-mzk, m>0, z∈U.

Also, he introduced the series

R(m,z)=2z−N(m,z),=z−∑ k=2∞m k−1 k−1!e−mzk,  m>0, z∈U.

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

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

Theorem 2.1. Let the function ξ be defined by (1.7). Then ξ∈Tβ,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+1−βz2ξ′′(z)ξ(z)=∑ k=2∞(k−1)(k(1−β)+β)akzk−11−∑ k=2∞akzk−1≤∑ k=2∞(k−1)(k(1−β)+β)ak1−∑ k=2∞ak.

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=2∞ak,

or equivalently

∑k=2∞((k-1)(k(1-β)+β)+(1-α))ak<1-α.

Conversely, assume that the function ξ∈T is in the class Tβ,α. Since ℜ{z}≤ertzert, then we have

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

Choosing z on the real axis, then

∑k=2∞(k-1)(k(1-β)+β)akzk-11-∑k=2∞akzk-1

is real. Let z→1- 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 ξ∈T ,0≤β​1≤β​2<1 and 0≤α≤1.​If  ξ(z)∈Tβ​1,α, then ξ(z)∈Tβ​2,α .i.e Tβ​1,α⊂Tβ​2,α.

proof. Let the function ξ(z)∈Tβ​1,α. Then, by Theorem 2.1, we have

∑k=2∞((k-1)(k(1-β​1)+β​1)+(1-α))ak<1-α.

Since β​1≤β​2, we get

∑ k=2∞((k−1)(k−(k−1)β2)+(1−α))ak≤∑ k=2∞((k−1)(k−(k−1)β1)+(1−α))ak<1−α,

which implies that ξ(z)∈Tβ​2,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,α⊂T1,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)m≤1-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-m≤1-α.

Thus,

∑k=2∞((k−1)(k(1−β)+β)+(1−α))m k−1 k−1!e−m=e−m∑k=2∞((1−β)(k−1)(k−2)+(k−1)(2−β)+(1−α))m k−1 k−1!=e−m(1−β)m2∑k=3∞ m k−3 k−3!+(2−β)m∑k=2∞ m k−2 k−2!+(1−α)∑k=2∞ m k−1 k−1!=(1−β)m2+(2−β)m+(1−α)(1−e−m)≤1−α.

Which ends the proof.

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

Theorem 2.5. Let the function ξ∈T be defined by (1.7) and let β≥12. Then ξ∈Rβ,α if and only if

∑k=2∞(k(βk-α))ak≤1-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)+1−β1ξ ′(z)−1=∑ k=2∞(k(kβ−1))akzk−11−∑ k=2∞kakzk−1≤∑ k=2∞(k(kβ−1))ak1−∑ k=2∞kak≤1−α.

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

ℜ β 1+zξ′′(z)ξ′(z)+1−β1ξ ′(z)=ℜ1−∑ k=2∞βk2akzk−11−∑ k=2∞kakzk−1>α.

If z is chosen on the real axis, then

1-∑k=2∞βk2akzk-11-∑k=2∞kakzk-1

is real. Letting z→1- 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 ξ∈T,12≤β​1≤β​2<1 and  0≤α≤1,​if  ξ∈β​2,α), then ξ∈Rβ​1,α. i.e Rβ​2,α⊂Rβ​1,α.

proof. Let the function  ξ∈β​2,α). Then, by Theorem 2.5, we have

∑k=2∞(k(β​2k-α))ak≤1-α.

Since β​1≤β​2, we get

∑ k=2∞(k(β1k−α))ak≤∑ k=2∞(k(β2k−α))ak,≤1−α,

which implies that ξ(z)∈Rβ​1,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-m≤1-α.

Thus,

∑k=2∞(k2β−kα)m k−1 k−1!e−m=e−m∑k=2∞(β(k−1)(k−2)+(3β−α)(k−1)+(β−α))m k−1 k−1!=e−mβm2∑k=3∞ m k−3 k−3!+(3β−α)m∑k=2∞ m k−2 k−2!+(β−α)∑k=2∞ m k−1 k−1!=βm2+(3β−α)m+(β−α)(1−e−m)≤1−α.

Which completes the proof.

3. Conclusion

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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

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

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