A New Subclass of Analytic Functions Related to Mittag-Leffler Type Poisson Distribution Series

Abstract

The object of this work is to an innovation of a class in with negative coefficients, further determining coefficient estimates, neighborhoods, partial sums, convexity, and compactness of this specified class.

1. Introduction

Let be an open unit disc in . Consider the analytic class function that indicates specified on the unit disk along with normalization and has the form indicated by , the subclass of lying of functions that are univalent in A function is stated in , and , “the class of -uniformly starlike functions and convex functions of order ,” if and only if

The classes UCV and UST were introduced by Goodman [1] and studied by Ronning [2]. Due to Sakaguchi [3], the class of starlike functions w.r.t. symmetric points are defined as follows.

The function is stated to be starlike w.r.t. symmetric points in

Owa et al. [4] defined the class as complies where Here, and is named Sakaguchi function of order

In recent years, binomial distribution series, Pascal distribution series, Poisson distribution series, etc., play important role in GFT. The sufficient ways were innovated for ST, UCV for some special functions in the GFT. By the motivation of the works [5–13], we develop this work.

In [14], Porwal, Poisson distribution series, gives a gracious application on analytic functions; it exposed a new way of research in GFT. Subsequently, the authors turned on the distribution series of confluent hypergeometric, hypergeometric, binomial, and Pascal and prevail necessary and sufficient stipulation for certain classes of univalent functions.

Lately, Porwal and Dixit [15] innovate Mittag-Leffler type Poisson distribution and prevailed moments, mgf, which is an abstraction of Poisson distribution using the definition of this distribution. Bajpai [16] innovated Mittag-Leffler type Poisson distribution series and discussed about necessary and sufficient conditions.

The probability mass function for this is where

The series (7) converges for all finite values of if This suggest that the series is convergent for For further details of the study, see [17]. It is easy to see that the series (7) are reduced to exponential series for

A variable is said to have Poisson distribution if it takes the values with probabilities , respectively, where is called the parameter.

Thus,

This motivates researchers (see [15, 17, 18], etc.) to introduce a new probability distribution if it assumes nonnegative values and its probability mass function is given by (6). It is easy to see that given by (6) is the probability mass function because

It is worthy to note that for , it reduces to the Poisson distribution.

Also note that

In [18], Chakrabortya and Ong introduced and discussed about the Mittag-Leffler function distribution—a new generalization of hyper-Poisson distribution. The Mittag-Leffler type Poisson distribution series was innovated by Porwal and Dixit [15] and given as

Equation (11) is a normalization function in , since and After that, in [19], Porwal et al. discussed about the geometric properties of (11).

For given by (2) and given by their convolution, indicated by , is given by

Note that

Next, we innovate the convolution operator where

Then, using linear operator , we exemplifier a contemporary subclass of functions in

Definition 1. If is named in the class if for all for

Moreover, we named that is in the subclass if is of the compiling form

In this work, we analyze the bounds for coefficient, partial sums, and some neighborhood outcomes of the class

To claim our outcomes, we adopt lemmas [20].

Lemma 2. Let be a complex number. Then, .

Lemma 3. Suppose a complex number with real numbers . Then,

2. Coefficient Bounds

Theorem 4. A function given by (16) is in here and The result is sharp for is

Proof. By Definition 1, we have Let and
By Lemma 2, (20) becomes But Also So Conversely, suppose (18) holds. Then, we have Opting values on the +ve real axis, where , then Since , then Taking limit tends to , we obtain our needed result.

Corollary 5. If , then where and

3. Neighborhood Properties

The notion of -neighbourhood was innovated and studied by Goodman [21] and Ruscheweyh [22].

Definition 6. We define the -neighborhood of a mapping and indicate by lying of all mappings satisfies the condition where and

Theorem 7. Let and every real we get For any with , if fulfills then,

Proof. It is evident that where for some and , we obtain In other words, However, , where and since ; therefore, , which implies Now, suppose Then, by (35), which contradicts by , and thus, we arrive
If , then

4. Partial Sums

Theorem 8. If the function is of the form (2) fulfill (18) then where The estimate (38) is sharp, for every , with

Proof. Now, we define ; we can define Then, from (42), we attain Now, if It is enough to prove that the LHS of (44) is bounded above by , which implies To show that the mapping disposed by (41) gives the exact result, we notice that for Taking limit tends to , we have Hence, the proof is completed.

Theorem 9. If of the form (2) which fulfill (18) then The result is sharp with (41).

Proof. Define where This last inequality is It is enough to prove that the LHS of (51) is bounded above by , which implies This completes the proof.

Theorem 10. If of the form (2) fulfill (18) then where and is given by (40). The computation in (53) and (54) are sharp with (41).

Theorem 11. is a convex and compact subset of

Proof. Suppose , Then, for , let be given by (56). Then, Then, Therefore, is convex. Now, we have to show is compact.
For and , then we arrive Therefore, is uniformly bounded.
Let
Also, let Then, by Theorem 4, we get Assuming , then we have as
Let be the array of partial sums of the series Then, is a nondecreasing array and by (59), it is bounded above by
Thus, it is convergent and Therefore, and the class is closed.

Data Availability

Our manuscript does not contain any data.

Conflicts of Interest

The authors declare that they have no conflict of interest.

Authors’ Contributions

All authors contributed equally to this work. And all the authors have read and approved the final version of the manuscript.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.