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 [1], 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. [2]. For , the classes and , respectively, reduce to the classes and introduced and studied by Ronning [3].
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 [8], 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 [12].
The generalized Bessel function (see, [13]) 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 [13–15] and others.
Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see, for example, [16–20])), and the work done in [21–24], 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 [1]). (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 [12]). If is of the form (1), then The result is sharp for the function
Lemma 4 (see [15]). 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 [22] 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.