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.
References
- S. Topkayaa and N. MustafaO. Altintaş and S. Owa. On subclasses of univalent functions with negative coefficients, Pusan Kyongnam Math. J. 4(1988), 41-56.
- S. Çakmak, S. Yalçın, and Ş. Altınkaya. Some connections between various classes of analytic functions associated with the power series distribution, Sakarya Univ. J. Sci. 23(5)(2019), 982-985.
- N. E. Cho, S. Y. Woo, and S. Owa. Uniform convexity properties for hypergeometric functions, Fract. Cal. Appl. Anal. 5(3)(2002), 303-313.
- K. K. Dixit and S. K. Pal. On a class of univalent functions related to complex order, Indian J. Pure Appl. Math. 26(9)(1995), 889-896.
- R. M. El-Ashwah and W. Y. Kota. Some condition on a Poisson distribution series to be in subclasses of univalent functions, Acta Univ. Apulensis Math. Inform. 51(2017), 89-103.
- S. M. El-Deeb, T. Bulboača, and J. Dziok. Pascal distribution series connected with certain subclasses of univalent functions, Kyungpook Math. J. 59(2019), 301-314.
- B. A. Frasin and On certain subclasses of analytic functions associated with Poisson distribution series. Acta Univ. Sapientiae, Mathematica 11(1), 78-86 (2019).
- B. A. Frasin. Subclasses of analytic functions associated with Pascal distribution series, Adv. Theory Nonlinear Anal. Appl. 4(2)(2020), 92-99.
- B. A. Frasin. Two subclasses of analytic functions associated with Poisson distribution, Turkish J. Ineq. 4(1)(2020), 25-30.
- B. A. Frasin and I. Aldawish. On subclasses of uniformly spiral-like functions associated with generalized Bessel functions, J. Func. Spaces (2019). Art. ID 1329462, 6 pp.
- B. A. Frasin, T. Al-Hawary, and F. Yousef. Necessary and sufficient conditions for hypergeometric functions to be in a subclass of analytic functions, Afr. Mat. 30(1-2)(2019), 223-230.
- E. Merkes and B. T. Scott. Starlike hypergeometric functions, Proc. Amer. Math. Soc. 12(1961), 885-888.
- A. O. Moustafa. A study on starlike and convex properties for hypergeometric functions, J. Inequal. Pure and Appl. Math. 10(3)(2009). Article 87, 8 pp.
- G. Murugusundaramoorthy. Subclasses of starlike and convex functions involving Poisson distribution series, Afr. Mat. 28 (2017), 1357-1366.
- G. Murugusundaramoorthy, B. A. Frasin, and T. Al-Hawary. Uniformly convex spiral functions and uniformly spirallike function associated with Pascal distribution series, J. Math. Anal. Appl. 172, 574-581. arXiv:2001.07517 [math.CV].
- G. Murugusundaramoorthy, K. Vijaya, and S. Porwal. Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series, Hacet. J. Math. Stat. 45(4) (2016), 1101-1107.
- W. Nazeer, Q. Mehmood, S. M. Kang, and A. Ul Haq. An application of Binomial distribution series on certain analytic functions, J. Comput. Anal. Appl. 26(1) (2019), 11-17.
- A. T. Oladipo. Bounds for probability of the generalized distribution defined by generalized polylogarithm, Punjab Univ. J. Math. 51(7) (2019)., 19-26
- S. Porwal. An application of a Poisson distribution series on certain analytic functions, J. Complex Anal. (2014). Art. ID 984135, 3 pp.
- S. Porwal and K. K. Dixit. An application of generalized Bessel functions on certain analytic functions, Acta Univ. M. Belii Ser. Math. 21 (2013), 55-61.
- H. Silverman. Univalent functions with negative Coefficients, Proc. Amer. Math.Soc. 51 (1975), 109-116.
- H. Silverman. Starlike and convexity properties for hypergeometric functions, J. Math. Anal. Appl. 172 (1993), 574-581.
- S. Topkayaa and N. Mustafa. Some analytic functions involving Gamma function and their various properties, J. Math. Anal. Appl. 172, 574-581. arXiv:1707.05126v1 [math.CV].