# Subclass of Analytic Functions Related with Pascal Distribution Series

Authors: Amnah E. Shammaky; Basem Aref Frasin; Tamer M. Seoudy
Journal: Journal of Mathematics
 (2022)

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## Abstract

The prime purpose of this article is to derive a necessary and sufficient condition for a linear operator associated with the Pascal distribution series to be in the classTSμ,σ,δ of analytic functions. Moreover, inclusion relation and an integral operator linked to the Pascal distribution series is considered. We have also provided some results as corollaries of our theorems.


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## Body

## 1. Introduction and Class Definition

Letℂ be the set of all complex numbers. We denote by G, the set of all analytic functions in the unit disk Δ=ξ∈ℂ:ξ<1 that have the series of the form.(1)fξ=ξ+∑n=2∞anξn.For functionsf∈G given by (1) and g∈G given by gξ=ξ+∑n=2∞bnξn, we recall that the well-known Hadamard product of f and g is given by.(2)fξ∗gξ≔ξ+∑n=2∞anbnξn,ξ∈Δ.Forε∈ℂ−0 and −1≤B<A≤1, we say that a function f∈G lies in the class ℛεA,B if(3)f′ξ−1A−Bε−Bf′ξ−1<1,ξ∈Δ.The function classℛεA,B was introduced in [1].In [2], Magesh and Prameela defined the following class:Definition 1.
A functionf∈G lies in the class Sμ,σ,δ if the following inequality is satisfied.(4)ℜξf′ξ+μ2μ−1ξ2f″ξ4μ1−μξ+μ2μ−1ξf′ξ+2μ2−3μ+1fξ−σ>δξf′ξ+μ2μ−1ξ2f″ξ4μ1−μξ+μ2μ−1ξf′ξ+2μ2−3μ+1fξ−1,where δ≥0,−1≤σ<1, 0≤μ≤1, and ξ∈Δ.
We also let(5)TSμ,σ,δ=Sμ,σ,δ∩T,where T is be a subclass of G consisting of functions of the form(6)fξ=ξ−∑n=2∞anξn,ξ∈Δ.
We note that, by specializing the parametersμ,σ and δ, we obtain the following subclasses studied by various authors.(1)
TS0,σ,0=Tσ and TS1,σ,0=Kσ (Silverman [3])(2)
TS1/2,σ,0=Pσ (Al-Amiri [4], Gupta and Jain [5] and Sarangi and Uralegaddi [6])(3)
TS1/2,σ,δ=TRσ,δ (Rosy [7] and Stephen and Subramanian [8])(4)
TS0,σ,δ=TSσ,δ and TS1,σ,δ=UCVσ,δ (Bharati et al. [9])(5)
TS0,0,δ=TSpδ (Subramanian et al. [10])(6)
TS1,0,δ=UCVδ (Subramanian et al. [11])
In [12], El-Deeb et al. provided a power series expansion whose coefficients are related to the probabilities of the Pascal distribution.(7)Φrtξ=ξ+∑n=2∞n+t−2t−1rn−11−rtξn,ξ∈Δ,where t≥1;0≤r≤1. We also define the series(8)ϒrtξ=2ξ−Φrtξ=ξ−∑n=2∞n+t−2t−1rn−11−rtξn,ξ∈Δ.
Next, let us consider the linear operator(9)ℑrtf:G⟶G,defined by means of the convolution or Hadamard product.(10)ℑrtfξ=Φrtξ∗fξ=ξ+∑n=2∞n+t−2t−1rn−11−rtanξn,ξ∈Δ.
Several results on the engagements between different subclasses of analytic univalent functions and special functions or distribution series have been studied in the literature (see for example, [13–27]). Herein, we provide a necessary and sufficient condition for ϒrtξ to be in the aforementioned class TSμ,σ,δ and investigate an inclusion property of the class TSμ,σ,δ associated with the operator ℑrtf. Ultimately, we provide conditions for the integral operator Grtfξ=∫0ξϒrtt/tdt to be in the investigated class TSμ,σ,δ.
We shall need the following lemmas to state and prove our results.Lemma 1. (see [2]).
Given thatδ≥0,−1≤σ<1 and 1/2≤μ<1. A function f∈T belongs to the class TSμ,σ,δ if and only if the following inequality is satisfied.(11)∑n=2∞Ϝμ,δn2+Θμ,σ,δn+Λμ,σ,δan≤1−σ,where(12)Ϝμ,δ=2μ2−μ1+δ,Θμ,σ,δ=μ−2μ21+σ+2δ+1+δ,Λμ,σ,δ=3μ−1−2μ2σ+δ.The result is sharp for the function.(13)fξ=ξ−1−σϜμ,δn2+Θμ,σ,δn+Λμ,σ,δξn.Lemma 2. (see [1]).
Iff∈G lies in the class ℛεA,B, then(14)an≤A−Bεn,n∈ℕ−1.The result is sharp.


## 2. The Necessary and Sufficient Condition

Throughout this paper, we use significantly the following identities for0≤r<1 and t≥1.(15)∑n=2∞n+t−2t−1rn−1=11−rt−1,∑n=2∞n−1n+t−2t−1rn−1=rt1−rt+1,∑n=3∞n−1n−2n+t−2t−1rn−1=r2tt+11−rt+2.Using Lemma1, we get the following necessary and sufficient condition for ϒrtξ to be in the investigated class TSμ,σ,δ.Theorem 1.
Fort≥1, ϒrtξ∈TSμ,σ,δ if and only if(16)Ϝμ,δr2tt+11−r2+3Ϝμ,δ+Θμ,σ,δrt1−r+Ϝμ,δ+Θμ,σ,δ+Λμ,σ,δ1−1−rt≤1−σ.Proof.
From Lemma1, we only need to show that(17)∑n=2∞Ϝμ,δn2+Θμ,σ,δn+Λμ,σ,δn+t−2t−1rn−11−rt≤1−σ.
Puttingn=n−1+1 and n2=n−1n−2+3n−1+1, in (17), we have(18)∑n=2∞Ϝμ,δn2+Θμ,σ,δn+Λμ,σ,δn+t−2t−1rn−11−rt=1−rt∑n=3∞Ϝμ,δn−1n−2n+t−2t−1rn−1+∑n=2∞3Ϝμ,δ+Θμ,σ,δn−1n+t−2t−1rn−1+∑n=2∞Ϝμ,δ+Θμ,σ,δ+Λμ,σ,δn+t−2t−1rn−1=Ϝμ,δr2tt+11−r2+3Ϝμ,δ+Θμ,σ,δrt1−r+Ϝμ,δ+Θμ,σ,δ+Λμ,σ,δ1−1−rt.
But this last expression is bounded previously by1−σ if and only if (16) is satisfied.


## 3. Inclusion Properties

Next, we show thatℑrtℛεA,B⊂TSμ,σ,δ.Theorem 2.
Fort>1 and f∈ℛεA,B,ℑrtfξ∈TSμ,σ,δ if(19)A−BεϜμ,δrt1−r+Ϝμ,δ+Θμ,σ,δ11−rt−1+Λμ,σ,δrt−11−r−1−rt−rt−11−rt≤1−σ.Proof.
According to Lemma1, it suffices to prove that(20)∑n=2∞Ϝμ,δn2+Θμ,σ,δn+Λμ,σ,δn+t−2t−1rn−11−rtan≤1−σ.
Applying Lemma2, we find from (14) and (20) that(21)∑n=2∞Ϝμ,δn2+Θμ,σ,δn+Λμ,σ,δn+t−2t−1rn−11−rtan≤A−Bε1−rt∑n=2∞Ϝμ,δn+Θμ,σ,δ+Λμ,σ,δnn+t−2t−1rn−1=A−Bε1−rt∑n=2∞Ϝμ,δn−1+Ϝμ,δ+Θμ,σ,δn+t−2t−1rn−1∑n=2∞Λμ,σ,δnn+t−2t−1rn−1=A−BεϜμ,δrt1−r+Ϝμ,δ+Θμ,σ,δ11−rt−1+Λμ,σ,δrt−11−r−1−rt−rt−11−rt.
Thus, the proof is completed since the RHS of the above inequality is bounded by1−σ.


## 4. An Integral Operator

In this section, we will consider the following integral operator.(22)Grtfξ=∫0ξϒrtttdt.Theorem 3.
Fort>1, Grtfξ∈TSμ,σ,δ if and only if(23)Ϝμ,δrt1−r+Ϝμ,δ+Θμ,σ,δ11−rt−1+Λμ,σ,δrt−11−r−1−rt−rt−11−rt≤1−σ.Proof.
From (8) and (22), we easily get(24)Grtfξ=ξ−∑n=2∞n+t−2t−1rn−11−rtξnn.
Then using Lemma1, we only need to prove that.(25)∑n=2∞Ϝμ,δn2+Θμ,σ,δn+Λμ,σ,δ×1nn+t−2t−1rn−11−rt≤1−σ.
Clearly, we have(26)∑n=2∞Ϝμ,δn+Θμ,σ,δn+t−2t−1rn−11−rt+∑n=2∞Λμ,σ,δnn+t−2t−1rn−11−rt≤1−σ.
The rest of the proof can be made similar to that of Theorem2.


## 5. Corollaries and Consequences

Lettingμ=1/2 in Theorems 1–3, one can get the following corollaries.Corollary 1.
Fort≥1, ϒrtξ∈TRσ,δ if and only if(27)1+δrt1−r+1−1−rt≤1−σ.Corollary 2.
fort>1 and f∈ℛεA,B,ℑrtfξ∈TRσ,δ if(28)A−Bε1+δ11−rt−1≤1−σ.Corollary 3.
Fort>1, the integral operator Grtfξ given by (22) is in the class TRσ,δ if and only if(29)1+δ11−rt−1≤1−σ.


## 6. Conclusions

In the present paper and due the earlier works (see, for example, [12, 15, 21]), we find a necessary and sufficient condition and inclusion relation for Pascal distribution series to be in a class of analytic functions with negative coefficients. Furthermore, we consider an integral operator related to Pascal distribution series. Some interesting corollaries and applications of the results are also discussed.



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