Subclass of Analytic Functions Related with Pascal Distribution Series

Shammaky, Amnah E.; Frasin, Basem Aref; Seoudy, Tamer M.

doi:https://doi.org/10.1155/2022/8355285

Journal of Mathematics

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 8355285 | https://doi.org/10.1155/2022/8355285

Subclass of Analytic Functions Related with Pascal Distribution Series

Amnah E. Shammaky,¹Basem Aref Frasin,²and Tamer M. Seoudy^3,4

Academic Editor: Ji Gao

Received24 Aug 2021

Accepted16 Feb 2022

Published04 Mar 2022

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 class 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.

1. Introduction and Class Definition

Let be the set of all complex numbers. We denote by , the set of all analytic functions in the unit disk that have the series of the form.

For functions given by (1) and given by , we recall that the well-known Hadamard product of and is given by.

For and , we say that a function lies in the class if

The function class was introduced in [1].

In [2], Magesh and Prameela defined the following class:

Definition 1. A function lies in the class if the following inequality is satisfied.where , , and .
We also letwhere is be a subclass of consisting of functions of the formWe note that, by specializing the parameters and , we obtain the following subclasses studied by various authors.(1) and (Silverman [3])(2) (Al-Amiri [4], Gupta and Jain [5] and Sarangi and Uralegaddi [6])(3) (Rosy [7] and Stephen and Subramanian [8])(4) and (Bharati et al. [9])(5) (Subramanian et al. [10])(6) (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.where . We also define the seriesNext, let us consider the linear operatordefined by means of the convolution or Hadamard product.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 to be in the aforementioned class and investigate an inclusion property of the class associated with the operator . Ultimately, we provide conditions for the integral operator to be in the investigated class .
We shall need the following lemmas to state and prove our results.

Lemma 1. (see [2]). Given that and . A function belongs to the class if and only if the following inequality is satisfied.where

The result is sharp for the function.

Lemma 2. (see [1]). If lies in the class , then

The result is sharp.

2. The Necessary and Sufficient Condition

Throughout this paper, we use significantly the following identities for and .

Using Lemma 1, we get the following necessary and sufficient condition for to be in the investigated class .

Theorem 1. For , if and only if

Proof. From Lemma 1, we only need to show thatPutting and , in (17), we haveBut this last expression is bounded previously by if and only if (16) is satisfied.

3. Inclusion Properties

Next, we show that .

Theorem 2. For and if

Proof. According to Lemma 1, it suffices to prove thatApplying Lemma 2, we find from (14) and (20) thatThus, the proof is completed since the RHS of the above inequality is bounded by .

4. An Integral Operator

In this section, we will consider the following integral operator.

Theorem 3. For , if and only if

Proof. From (8) and (22), we easily getThen using Lemma 1, we only need to prove that.Clearly, we haveThe rest of the proof can be made similar to that of Theorem 2.

5. Corollaries and Consequences

Letting in Theorems 1–3, one can get the following corollaries.

Corollary 1. For , if and only if

Corollary 2. for and if

Corollary 3. For , the integral operator given by (22) is in the class if and only if

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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

K. K. Dixit and S. K. Pal, “On a class of univalent functions related to complex order,” Indian Journal of Pure and Applied Mathematics, vol. 26, no. 9, pp. 889–896, 1995.
View at: Google Scholar
N. Magesh and V. Prameela, “Certain subclasses of uniformly convex functions and corresponding class of starlike functions,” Malaya Journal of Matematik, vol. 1, no. 1, pp. 18–26, 2013.
View at: Google Scholar
H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical Society, vol. 51, no. 1, 1975.
View at: Publisher Site | Google Scholar
H. S. Al-Amiri, “On a subclass of close-to-convex functions with negative coefficients,” Mathematics, vol. 31, no. 54, pp. 1–7, 1989.
View at: Google Scholar
V. P. Gupta and P. K. Jain, “Certain classes of univalent functions with negative coefficients II,” Bulletin of the Australian Mathematical Society, vol. 15, no. 3, pp. 467–473, 1976.
View at: Publisher Site | Google Scholar
S. M. Sarangi and B. A. Uralegaddi, “The radius of convexity and starlikeness of certain classes of analytic functions with negative coefficients I, atti della accademia nazionale dei lincei,” Classe di Scienze Fisiche, Matematiche e Naturali, vol. 65, no. 8, pp. 34–42, 1978.
View at: Google Scholar
T. Rosy, Study on Subclasses of Univalent Functions, University of Madras, Chennai, India, 2002.
B. A. Stephand and K. G. Subramanian, On a Subclass of Noshiro-Type Analytic Functions with Negative Coefficients, Ramanujan Mathematical Society, Tiruchirappalli, India, 1998.
R. Bharati, R. Parvatham, and A. Swaminathan, “On subclasses of uniformly convex functions and corresponding class of starlike functions,” Tamkang Journal of Mathematics, vol. 28, no. 1, pp. 17–32, 1997.
View at: Publisher Site | Google Scholar
K. G. Subramanian, T. V. Sudharsan, P. Balasubrahmanyam, and H. Silverman, “Class of uniformly starlike functions,” Publicationes Mathematicae Debrecen, vol. 53, no. 4, pp. 309–315, 1998.
View at: Google Scholar
K. G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahmanyam, and H. Silverman, “Subclasses of uniformly convex and uniformly starlike functions,” Mathematica Japonica, vol. 42, no. 3, pp. 517–522, 1995.
View at: Google Scholar
S. M. El-Deeb, T. Bulboacă, and J. Dziok, “Pascal distribution series connected with certain subclasses of univalent functions,” Kyungpook Mathematical Journal, vol. 59, pp. 301–314, 2019.
View at: Google Scholar
M. Ahmad, B. Frasin, G. Murugusundaramoorthy, and A. Al-khazaleh, “An application of Mittag-Leffler-type poisson distribution on certain subclasses of analytic functions associated with conic domains,” Heliyon, vol. 7, no. 10, Article ID e08109, 2021.
View at: Publisher Site | Google Scholar
M. K. Aouf, A. O. Mostafa, and H. M. Zayed, “Some constraints of hypergeometric functions to belong to certain subclasses of analytic functions,” Journal of the Egyptian Mathematical Society, vol. 24, no. 3, pp. 361–366, 2016.
View at: Publisher Site | Google Scholar
T. Bulboacă and G. Murugusundaramoorthy, “Univalent functions with positive coefficients involving pascal distribution series,” Communications of the Korean Mathematical Society, vol. 35, no. 3, pp. 867–877, 2020.
View at: Google Scholar
N. E. Cho, S. Y. Woo, and S. Owa, “Uniform convexity properties for hypergeometric functions,” Fractional Calculus and Applied Analysis, vol. 5, no. 3, pp. 303–313, 2002.
View at: Google Scholar
H. E. Darwish, A. Y. Lashin, and E. M. Madar, “On subclasses of uniformly starlike and convex functions defined by struve functions,” International Journal of Open Problems in Complex Analysis, vol. 8, no. 1, pp. 34–43, 2016.
View at: Google Scholar
B. A. Frasin, “An application of generalized bessel functions on general class of analytic functions with negative coefficients,” Applied Mathematics E-Notes, vol. 21, pp. 426–436, 2021.
View at: Google Scholar
B. A. Frasin and I. Aldawish, “On subclasses of uniformly spirallike functions associated with generalized bessel functions,” Journal of Function Spaces, vol. 2019, Article ID 1329462, 6 pages, 2019.
View at: Publisher Site | Google Scholar
B. A. Frasin, S. Porwal, and F. Yousef, “Subclasses of starlike and convex functions associated with Mittag-Leffler-type poisson distribution series,” Montes Taurus Journal Pure and Applied Mathematics, vol. 3, no. 3, pp. 147–154, 2021.
View at: Google Scholar
B. A. Frasin, S. R. Swamy, and A. K. Wanas, “Subclasses of starlike and convex functions associated with pascal distribution series,” Kyungpook Mathematical Journal, vol. 61, pp. 99–110, 2021.
View at: Google Scholar
B. A. Frasin, T. Al-Hawary, and F. Yousef, “Necessary and sufficient conditions for hypergeometric functions to be in a subclass of analytic functions,” Afrika Matematika, vol. 30, 2018.
View at: Publisher Site | Google Scholar
N. Magesh, S. Porwal, and C. Abirami, “Starlike and convex properties for poisson distribution series,” Studia Universitatis Babeş-Bolyai Mathematica, vol. 63, no. 1, pp. 71–78, 2018.
View at: Publisher Site | Google Scholar
E. P. Merkes and W. T. Scott, “Starlike hypergeometric functions,” Proceedings of the American Mathematical Society, vol. 12, no. 6, pp. 885–888, 1961.
View at: Publisher Site | Google Scholar
G. Murugusundaramoorthy, “Application of pascal distribution series to ronning type starlike and convex functions,” Advances in the Theory of Nonlinear Analysis and its Application., vol. 4, no. 4, pp. 243–251, 2020.
View at: Publisher Site | Google Scholar
S. Porwal, “An application of a poisson distribution series on certain analytic functions,” Journal of Complex Analysis, vol. 2014, Article ID 984135, 3 pages, 2014.
View at: Publisher Site | Google Scholar
H. Silverman, “Starlike and convexity properties for hypergeometric functions,” Journal of Mathematical Analysis and Applications, vol. 172, no. 2, pp. 574–581, 1993.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Amnah E. Shammaky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

302

Downloads

475

Citations