Abstract

In this paper, we introduce new subclasses of analytic functions in the open unit disc. Furthermore, the necessary and sufficient conditions for the Poisson distribution series to be in these new subclasses are found.

1. Introduction

Let be the class of all analytic functions in the open unit disc and normalized by and . A function has the Taylor series expansion of the form

We denote by the subclass of consisting of normalized functions of the form (1) which are univalent in . Further, we denote by the subclass of consisting of functions with negative coefficients of the form

If such that is given by (1) and is given by , then, the Hadamard product is defined by

In 1837, the French mathematician Siméon Denis Poisson created the Poisson distribution which is a popular distribution expresses the probability of a given number of events occurring in a fixed interval of time or space. In [1], Porwal introduced a power series such that its coefficients are probabilities of the Poisson distribution

In addition, he introduced the series

In [2], Porwal and Kumar introduced a new linear operator defined by

In [3, 4], El-Ashwah and Kota presented the functions and as below: and where , and Suppose the functions , and are given by for every

Definition 1. Let A function is said to be in the class if the following condition is satisfied

Further, we define the class by

Indeed, we have

(1) where the class was studied by Chichra in [5]. We define the class which was introduced and studied by Orhan [6].

(2) where the class was studied by Obradovic and Joshi in [7], and the class was intoduced and studied by Ramesha et al. in [8]. We define the class which was studied by Lashin [9].

(3) where the class was studied by Ding et al. [10]. We define the class which was introduced by Hassan [11].

(4) where the classes and were introduced by Lashin et al. in [12].

In this paper, we find the necessary and sufficient conditions for the Poisson distribution series to be in the classes , and .

We first derive the sufficient and necessary conditions for the function to be in the aforementioned classes.

Theorem 2. The sufficient condition for to be in the class is

Proof. We need to show that Then, we have if condition (17) holds. This implies that which completes the proof.

Theorem 3. Let be an increasing function of and Then, the necessary and sufficient condition for the function to be in the class is

Proof. In view of Theorem 2, it suffices to show the necessary condition only. Assume that . Then which is equivalent to Choosing on the real axis, then is also real. Let through real values, we get which is equivalent to (20), and this completes the proof.

Putting and in Theorems 3, we get the following corollary due to Lashin [9].

Corollary 4 (see [9]). A function is in the class if and only if

Putting and in Theorem 3, we get the following corollary due to Hassan [11].

Corollary 5 (see [11]). A function is in the class if and only if

Putting and in Theorem 3, we get the following corollary.

Corollary 6 (see [12]). Let A function is in the class if and only if

Making use of the techniques and methodology given by Porwal [1] (see also [13–18]), we get the following theorems.

Theorem 7. The sufficient condition for to be in the class is

Also, condition (28) is necessary and sufficient for to be in the class

Proof. According to Theorem 2, we need to show that Thus, if condition (28) holds. Then, from Theorem 3, it follows that the condition (28) is necessary and sufficient for . Hence, the proof is completed.

In Theorem 7, if we put then, we get the following corollary which was obtained by Murugusundaramoorthy et al. [19].

Corollary 8. The sufficient condition for to be in the class is

Also, condition (31) is necessary and sufficient for to be in the class

Theorem 9. The sufficient condition for to be in the class is

Also, condition (32) is necessary and sufficient for to be in the class

Proof. According to Theorem 2, we need to show that Thus, if condition (32) holds. Then, from Theorem 3, it follows that condition (32) is necessary and sufficient for . Hence, the proof is completed.

In Theorem 9, if we put then, we get the following corollary which was obtained by Frasin [20].

Corollary 10. The sufficient condition for to be in the class is

Also, condition (35) is necessary and sufficient for to be in the class

Theorem 11. The sufficient condition for to be in the class is where Also, condition (36) is necessary and sufficient for to be in the class

Proof. According to Theorem 2, we need to show that Thus, if condition (36) holds. Then, from Theorem 3, it follows that the condition (36) is necessary and sufficient for . Hence, the proof is completed.

Theorem 12. The sufficient condition for to be in the class is where Also, condition (39) is necessary and sufficient for to be in the class

Proof. According to Theorem 2, we need to show that Thus, if condition (39) holds. Then, from Theorem 3, it follows that the condition (39) is necessary and sufficient for Hence, the proof is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia, for funding this project under grant no. (KEP-PhD:49-130-1443). Also, they would like to express their thanks to the referees for their helpful comments and suggestions, which improved the presentation of the paper.