### Article

Kyungpook Mathematical Journal 2021; 61(1): 99-110

**Published online** March 31, 2021 https://doi.org/10.5666/KMJ.2021.61.1.99

Copyright © Kyungpook Mathematical Journal.

### Subclasses of Starlike and Convex Functions Associated with Pascal Distribution Series

Basem Aref Frasin*, Sondekola Rudra Swamy, Abbas Kareem Wanas

Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan

e-mail : bafrasin@yahoo.com

Department of Computer Science and Engineering, RV College of Engineering, Bengaluru - 560 059, Karnataka, India

e-mail : mailtoswamy@rediffmail.com

Department of Mathematics, College of Science, University of Al-Qadisiyah, Iraq

e-mail : abbas.kareem.w@qu.edu.iq

**Received**: May 20, 2020; **Revised**: October 7, 2020; **Accepted**: October 13, 2020

### Abstract

In the present paper, we determine new characterisations of the subclasses

**Keywords**: Analytic functions, Hadamard product, Pascal distribution series.

### 1. Introduction and Definitions

Let

which are analytic in the open unit disk

A function

Also, we say that a function

We denote by

Interesting generalization of the functions classes

and

The classes

Inspired by the studies mentioned above, Topkaya and Mustafa [23] defined a unification of the functions classes

### Definition 1.1.

A function

where

Also we denote

Let

In particular, the class

A function

This class was introduced by Dixit and Pal [4].

A variable

respectively, where

Very recently, El-Deeb et al. [6] (see also, [2, 8, 9, 15]) introduced a power series whose coefficients are probabilities of Pascal distribution. Let

where

Let consider the linear operator

where

There are several known results on connections between various subclasses of analytic and univalent functions using hypergeometric functions (see for example, [12, 22]) and using various distributions such as Yule-Simon distributions, Logarithmic distributions, Poisson distributions, Binomial distributions, Beta-Binomial distributions, Zeta distributions, Geometric distributions and the Bernoulli distribution (see, for example, [5, 7, 10, 14, 16, 17, 18]).

In this paper, we determine necessary and sufficient conditions for

### 2. Preliminary Lemmas

To establish our main results, we need the following lemmas.

### Lemma 2.1.

([23])

### Lemma 2.2.

### Lemma 2.3.

([4])

### 3. Necessary and Sufficient Conditions

For convenience throughout in the sequel, we use the following identities that hold for

By simple calculations we derive the following relations:

and

Unless otherwise mentioned, we shall assume in this paper that

Firstly, we obtain the necessary and sufficient conditions for

### Theorem 3.1.

If

Writing

and

but this last expression is upper bounded by

### Theorem 3.2.

If

Writing

and

but this last expression is upper bounded by

### 4. Inclusion Properties

Making use of Lemma 2.3, we will study the action of the Pascal distribution series on the classes

### Theorem 4.1.

Let

Since

we have

Thus, we have

But this last expression is bounded by

### Theorem 4.2.

Let

Since

therefore

The remaining part of the proof is similar to that of Theorem 3.1 and so we omit the details.

### 5. An Integral Operator

### Theorem 5.1.

Let

is in

Using Lemma 2.2, the function

only if

By a similar proof like those of Theorem 3.1 we get that

### Theorem 5.2.

If

only if

The proof of Theorem Theorem 5.2 is lines similar to the proof of Theorem 4.1, so we omitted the proof of Theorem 5.2.

### 6. Corollaries and Consequences

By taking

### Corollary 6.1.

If

### Corollary 6.2.

If

### Corollary 6.3.

Let

### Corollary 6.4.

Let

### Corollary 6.5.

If

### Corollary 6.6.

If

### Acknowledgements.

The authors would like to thank the referees for their helpful comments and suggestions.

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