- Open Access
Inclusion relations for Bessel functions for domains bounded by conical domains
© Ramachandran et al.; licensee Springer. 2014
Received: 8 August 2014
Accepted: 24 October 2014
Published: 10 November 2014
In recent times, applications of Bessel differential equations have been effectively used in the theory of univalent functions. In this paper we study some subclasses of k-starlike functions, k-uniformly convex functions, and quasi-convex functions involving the Bessel function and derive their inclusion relationships. Further, certain integral preserving properties are also established with these classes. We remark here that k-starlike functions and k-uniformly convex functions are related to domains bounded by conical sections.
- (1)For in (1.2), we obtain the familiar Bessel function of the first kind of order u defined by(1.3)
- (2)For and in (1.2), we obtain the modified Bessel function of the first kind of order u defined by(1.4)
- (3)For and in (1.2), the function reduces to where is the spherical Bessel function of the first kind of order u, defined by(1.5)
- (1)Choosing in (1.20) or (1.21), we obtain the operator related with Bessel function, defined by(1.22)
- (2)Choosing and in (1.20) or (1.21), we obtain the operator related with the modified Bessel function, defined by(1.23)
- (3)Choosing and in (1.20) or (1.21), we obtain the operator related with the spherical Bessel function, defined by(1.24)
3 Main results
We study certain inclusion relationships for some subclasses of k-starlike functions, k-uniformly convex functions, and quasi-convex functions involving the Bessel equation. We reiterate that these classes of k-starlike functions and k-uniformly convex functions are related to domains bounded by conical sections.
The proof of the theorem follows now by an application of Lemma 2.1. □
Theorem 3.2 Let . If , then .
the proof of the theorem follows by Theorem 3.1 and condition (1.14). □
Theorem 3.3 Let . If , then .
and hence the proof is complete. □
Theorem 3.4 Let . If , then .
We observe that and are analytic in and .
and so the proof is complete. □
Using a similar argument to Theorem 3.4, we can prove the following theorem.
Theorem 3.5 Let . If , then .
Now we examine the closure properties of the integral operator .
Theorem 3.6 Let . If so is .
By Lemma 2.1, we have , since . This completes the proof of Theorem 3.6. □
By a similar argument we can prove Theorem 3.7 as below.
Theorem 3.7 Let . If so is .
Theorem 3.8 Let . If so is .
Let us take in (3.15) and observe that as . Now for and B as described we conclude the proof since the required conditions of Lemma 2.2 are satisfied. □
A similar argument yields the following.
Theorem 3.9 Let . If so is .
4 Concluding remarks
As observed earlier when was defined, all the results discussed can easily be stated for the convolution operators , , and , which are defined by (1.22), (1.23), and (1.24), respectively. However, we leave those results to the interested readers.
The authors sincerely thank the referee(s) for their valuable comments which essentially improved the manuscript. The work of the third author is supported by a grant from Department of Science and Technology, Government of India, vide ref: SR/FTP/MS-022/2012 under the fast track scheme.
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