• Views 173
• Citations 0
• ePub 11
• PDF 39
`Journal of Function SpacesVolume 2019, Article ID 1329462, 6 pageshttps://doi.org/10.1155/2019/1329462`
Research Article

## On Subclasses of Uniformly Spiral-like Functions Associated with Generalized Bessel Functions

1Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan
2Department of Mathematics and Statistics, College of Science, lMSIU (Imam Mohammed Ibn Saud Islamic University), P.O. Box 90950, Riyadh 11623, Saudi Arabia

Correspondence should be addressed to B. A. Frasin; moc.oohay@nisarfab

Received 10 June 2019; Accepted 25 July 2019; Published 20 August 2019

Academic Editor: Wilfredo Urbina

Copyright © 2019 B. A. Frasin and Ibtisam Aldawish. 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.

#### Abstract

The main object of this paper is to find necessary and sufficient conditions for generalized Bessel functions of first kind to be in the classes and of uniformly spiral-like functions and also give necessary and sufficient conditions for to be in the above classes. Furthermore, we give necessary and sufficient conditions for to be in provided that the function is in the class . Finally, we give conditions for the integral operator to be in the class Several corollaries and consequences of the main results are also considered.

#### 1. Introduction and Definitions

Let denote the class of the normalized functions of the formwhich are analytic in the open unit disk Further, let be a subclass of consisting of functions of the form,A function is spiral-like if for some with and for all . Also is convex spiral-like if is spiral-like.

In , Selvaraj and Geetha introduced the following subclasses of uniformly spiral-like and convex spiral-like functions.

Definition 1. A function of the form (1) is said to be in the class if it satisfies the following condition: and if and only if

We write

In particular, we note that and , the classes of uniformly spiral-like and uniformly convex spiral-like were introduced by Ravichandran et al. . For , the classes and , respectively, reduce to the classes and introduced and studied by Ronning .

For more interesting developments of some related subclasses of uniformly spiral-like and uniformly convex spiral-like, the readers may be referred to the works of Frasin [4, 5], Goodman [6, 7], Tariq Al-Hawary and Frasin , Kanas and Wisniowska [9, 10] and Ronning [3, 11].

A function is said to be in the class ,, , if it satisfies the inequality

This class was introduced by Dixit and Pal .

The generalized Bessel function (see, ) is defined as a particular solution of the linear differential equation where . The analytic function has the form Now, the generalized and normalized Bessel function is defined with the transformation where and is the well-known Pochhammer (or Appell) symbol, defined in terms of the Euler Gamma function for byThe function is analytic on and satisfies the second-order linear differential equation Using the Hadamard product, we now considered a linear operator defined by where denote the convolution or Hadamard product of two series.

The study of the generalized Bessel function is a recent interesting topic in geometric function theory. We refer, in this connection, to the works of  and others.

Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see, for example, )), and the work done in , we determine necessary and sufficient conditions for to be in and and also give necessary and sufficient conditions for to be in the function classes and . Furthermore, we give necessary and sufficient conditions for to be in provided that the function is in the class . Finally, we give conditions for the integral operator to be in the class

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

Lemma 2 (see ). (i) A sufficient condition for a function of the form (1) to be in the class is thatand a necessary and sufficient condition for a function of the form (2) to be in the class is that condition (13) is satisfied. In particular, when , we obtain a sufficient condition for a function of the form (1) to be in the class is thatand a necessary and sufficient condition for a function of the form (2) to be in the class is that condition (14) is satisfied.
(ii) A sufficient condition for a function of the form (1) to be in the class is thatand a necessary and sufficient condition for a function of the form (2) to be in the class is that condition (15) is satisfied. In particular, when , we obtain a sufficient condition for a function of the form (1) to be in the class is thatand a necessary and sufficient condition for a function of the form (2) to be in the class is that condition (16) is satisfied.

Lemma 3 (see ). If is of the form (1), then The result is sharp for the function

Lemma 4 (see ). If and , then the function satisfies the recursive relations for all .

#### 2. The Necessary and Sufficient Conditions

Unless otherwise mentioned, we shall assume in this paper that and

First we obtain the necessary condition for to be in

Theorem 5. If , , then is in if

Proof. Sinceaccording to (13), we must show thatWritingwe haveBut this last expression is bounded above by if (20) holds.

Corollary 6. If , , then is in if and only if the condition (20) is satisfied.

Proof. SinceBy using the same techniques given in the proof of Theorem 5, we have Corollary 6.

Theorem 7. If , , then is in if

Proof. We note that From (24), we get Therefore, we see that the last expression is bounded above by if (26) is satisfied.

Corollary 8. If , , then is in if and only if the condition (26) is satisfied.

Theorem 9. If , , then is in if

Proof. In view of (15), we must show thatWritingThus, we have But this last expression is bounded above by if (28) holds.

By using a similar method as in the proof of Corollary 6, we have the following result.

Corollary 10. If , , then is in if and only if the condition (28) is satisfied.

The proof of Theorem 11 (below) is much akin to that of Theorem 7, and so the details may be omitted.

Theorem 11. If , , then is in if and only if

#### 3. Inclusion Properties

Making use of Lemma 3, we will study the action of the Bessel function on the class

Theorem 12. Let , If then is in if and only if

Proof. In view of (15), it suffices to show that Since , then by Lemma 3, we getThus, we must show that The remaining part of the proof of Theorem 12 is similar to that of Theorem 5, and so we omit the details.

#### 4. An Integral Operator

In this section, we obtain the necessary and sufficient conditions for the integral operator defined byto be in

Theorem 13. If , , then the integral operator is in if and only if the condition (20) is satisfied.

Proof. Sincethen, in view of (15), we need only to show that or equivalently The remaining part of the proof is similar to that of Theorem 5, and so we omit the details.

The proofs of Theorems 14 and 15 are much akin to that of Theorem 7, and so the details may be omitted.

Theorem 14. Let , If then is in if and only if

Theorem 15. If , , then the integral operator is in if and only if the condition (32) is satisfied.

#### 5. Corollaries and Consequences

In this section, we apply our main results in order to deduce each of the following corollaries and consequences.

Corollary 16. If , ,then is in if

Corollary 17. If , , then is in if and only if the condition (42) is satisfied.

Corollary 18. If , , then is in if

Corollary 19. If , , then is in if and only if the condition (43) is satisfied.

Corollary 20. If , , then is in if

Corollary 21. If , , then is in if and only if

Corollary 22. Let , If then is in if and only if

Corollary 23. If , , then the integral operator is in if and only if the condition (42) is satisfied.

Corollary 24. Let , If then is in if and only if

Corollary 25. If , , then the integral operator is in if and only if the condition (45) is satisfied.

Remark 26. If we put in Corollary 6, we obtain Theorem 5 in  for and

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### References

1. C. Selvaraj and R. Geetha, “On subclasses of uniformly convex spirallike functions and corresponding class of spirallike functions,” International Journal of Contemporary Mathematical Sciences, vol. 5, pp. 37–40, 2010.
2. V. Ravichandran, C. Selvaraj, and R. Rajagopal, “On uniformly convex spiral functions and uniformly spirallike functions,” Soochow Journal of Mathematics, vol. 29, no. 4, pp. 393–405, 2003.
3. F. Ronning, “Uniformly convex functions and a corresponding class of starlike functions,” Proceedings of the American Mathematical Society, vol. 118, no. 1, pp. 189–196, 1993.
4. B. A. Frasin, “Quasi-Hadamard product of certain classes of uniformly analytic functions,” General Mathematics, vol. 16, no. 2, pp. 29–35, 2007.
5. B. A. Frasin, “Comprehensive family of uniformly analytic functions,” Tamkang Journal of Mathematics, vol. 36, no. 3, pp. 243–254, 2005.
6. A. W. Goodman, “On uniformly convex functions,” Annales Polonici Mathematici, vol. 56, no. 1, pp. 87–92, 1991.
7. A. W. Goodman, “On uniformly starlike functions,” Journal of Mathematical Analysis and Applications, vol. 155, no. 2, pp. 364–370, 1991.
8. T. Al-Hawary and B. A. Frasin, “Uniformly analytic functions with varying arguments,” Analele Universitatii Oradea Fasc. Matematica, Tom XXIII, vol. 23, no. 1, pp. 37–44, 2016.
9. S. Kanas and A. Wisniowska, “Conic regions and k-uniform convexity,” Journal of Computational and Applied Mathematics, vol. 105, no. 1-2, pp. 327–336, 1999.
10. S. A. Kanas and A. Wisniowska, “Conic domains and starlike functions,” Revue Roumaine de Mathématique Pures et Appliquées, vol. 45, no. 4, pp. 647–657, 2000.
11. F. Ronning, “On starlike functions associated with parabolic regions,” Annales Universitatis Mariae Curie-Sklodowska, Section A, vol. 45, no. 14, pp. 117–122, 1991.
12. 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.
13. A. Baricz, “Geometric properties of generalized Bessel functions,” Publicationes Mathematicae, vol. 73, no. 1-2, pp. 155–178, 2008.
14. A. Baricz, “Geometric properties of generalized Bessel functions of complex order,” The Mathematica Journal, vol. 48(71), no. 1, pp. 13–18, 2006.
15. A. Baricz, Generalized Bessel Functions of the First Kind, vol. 1994 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.
16. N. E. Cho, S. Y. Woo, and S. Owa, “Uniform convexity properties for hypergeometric functions,” Fractional Calculus & Applied Analysis. An International Journal for Theory and Applications, vol. 5, no. 3, pp. 303–313, 2002.
17. 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, no. 1-2, pp. 223–230, 2019.
18. E. P. Merkes and W. T. Scott, “Starlike hypergeometric functions,” Proceedings of the American Mathematical Society, vol. 12, no. 6, pp. 885–888, 1961.
19. H. Silverman, “Starlike and convexity properties for hypergeometric functions,” Journal of Mathematical Analysis and Applications, vol. 172, no. 2, pp. 574–581, 1993.
20. H. M. Srivastava, G. Murugusundaramoorthy, and S. Sivasubramanian, “Hypergeometric functions in the parabolic starlike and uniformly convex domains,” Integral Transforms and Special Functions, vol. 18, no. 7, pp. 511–520, 2007.
21. T. Al-Hawary, B. A. Frasin, and F. Yousef, Application of generalized Bessel functions to classes of analytic functions, submitted.
22. G. Murugusundaramoorthy and T. Janani, “An application of generalized bessel functions on certain subclasses of analytic functions,” Turkish Journal of Analysis and Number Theory, vol. 3, no. 1, pp. 1–6, 2015.
23. G. Murugusundaramoorthy, K. Vijaya, and M. Kasthuri, “A note on subclasses of starlike and convex functions associated with bessel functions,” Journal of Nonlinear Functional Analysis, vol. 2014, article 11, 2014.
24. S. Porwal and K. K. Dixit, “An application of generalized Bessel functions on certain analytic functions,” Acta Universitatis Matthiae Belii series Mathematics, vol. 2013, pp. 51–57, 2013.