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Journal of Function Spaces
Volume 2018, Article ID 2684023, 9 pages
https://doi.org/10.1155/2018/2684023
Research Article

Certain Geometric Properties of Generalized Dini Functions

1Department of Mathematics, Government College University, Faisalabad, Pakistan
2Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Pakistan
3Faculty of Science and Technology, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to Mohsan Raza; moc.oohay@679nashom

Received 20 March 2018; Accepted 25 June 2018; Published 11 July 2018

Academic Editor: Mitsuru Sugimoto

Copyright © 2018 Muhey U. Din 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.

Abstract

We are mainly interested in some geometric properties for the combinations of generalized Bessel functions of the first kind and their derivatives known as Dini functions. In particular, we study the starlikeness of order convexity of order , and close-to-convexity of order for normalized Dini function. We also study close-to-convexity with respect to certain star-like functions. Further, we obtain conditions on generalized Dini function to belong to the Hardy space

1. Introduction

Let be the class of functions of the formanalytic in the open unit disc and denote the class of all functions in which are univalent in . Let , , and denote the classes of star-like, convex, close-to-convex and strongly star-like functions of order , respectively, and they are defined as It is clear that

If and are analytic functions, then the function is said to be subordinate to , written as , if there exist a Schwarz function which is analytic with and such that Furthermore, if the function is univalent in then we have the following equivalent relation:

Special functions have great importance in pure and applied mathematics. The wide use of these functions has attracted many researchers to work on the different directions. Geometric properties of special functions such as hypergeometric functions, Bessel functions, Struve functions, Mittag-Leffler functions, Wright functions, and some other related functions are an ongoing part of research in geometric function theory. We refer to some geometric properties of these functions [110] and references therein.

Consider the second-order homogeneous differential equationwhere , and are complex numbers. The particular solution of the homogenous differential equation (6) is called the generalized Bessel functions of the first kind of order It is defined aswhere denotes the gamma function. The function unifies the Bessel, modified Bessel, and spherical Bessel functions.

Special casesFor , we have the Bessel functions of first kind of order , defined asFor , we obtain the modified Bessel functions of first kind of order whose series form is given asFor , we have the spherical Bessel functions of first kind of order , given asFor more details about these functions, see [5, 11]. Recently, Deniz et al. [12] studied the function which is defined by the relation as where Using the well-known Pochhammer symbol we obtain the series form of the function aswhere

Bessel functions are indispensable in many branches of mathematics and applied mathematics. Thus, it is important to study their properties in many aspects. We consider the generalized Dini functions defined aswhere , and . For , we obtain the normalized Dini function of order of the form By putting and , we obtain the following particular cases of normalized Dini functions

Recently Baricz et al. [13] studied the close-to-convexity of Dini functions and some monotonicity properties and functional inequalities for the modified Dini function are discussed in [5]. Further some geometric properties of Dini functions are studied in [14].

This paper studies the Dini function given by the power series (14). We determine the conditions on parameters that ensure the Dini function to be star-like of order , convex of order , and close-to-convex of order We also study the convexity in the domain Sufficient conditions on univalency of an integral operator defined by Dini function are also studied. We find the conditions on normalized Dini function to belong to the Hardy space

To prove our main results, we need the following lemmas.

Lemma 1 (see [15]). If satisfy for each , then is convex in

Lemma 2 (see [16]). Let with , and with , If satisfies then the integral operator is analytic and univalent in .

Lemma 3 (see [17]). Let be convex and univalent in the open unit disc with condition Let be analytic in the open unit disc with condition and in the open unit disc. Then, , we obtain

Lemma 4 (see [18]). If satisfies the inequality then

Lemma 5 (see [19]). If the function is analytic in and in addition or , then is close-to-convex function with respect to the convex function Moreover, if the odd function is analytic in and if or , then is univalent in .

2. Geometric Properties of Normalized Dini Functions

Theorem 6. Let , , , and with and . Then the following assertions are true:(i)If < , then (ii)If < , then (iii)If < , then (iv)If , then

Proof. We use the inequality to prove the starlikeness of order for the function So by using the well-known triangle inequality with the equality and the inequalities we obtain Furthermore, if we use reverse triangle inequality and the inequalities then we get By combining inequalities (25) and (28), we get So, is star-like function of order , where +
(ii) To prove the convexity of order of function , we have to show that By using the well-known triangle inequality with equality and the inequalities we haveMoreover, if we use reverse triangle inequality with equality and the inequalities we getBy combining inequalities (33) and (37), we have This implies that is convex function of order , where +
(iii) Using inequality (33) and Lemma 4, we have where . This shows that Therefore
(iv) To prove that , we have to show that , where By using the well-known triangle inequality with the equality and the inequality we have Therefore, for

Corollary 7. By using Theorem 6 assertions (i) and (iv), we get the following corollary:If where , then .If where , then .If where , then .If where , then .

Putting in Theorem 6, we have the following results.

Corollary 8. Let , , , and . Then the following assertions are true.
If + + , then .
If + + , then .
If + + , then .
If , then .

Corollary 9. By putting and in Corollary 8, we obtain the following results:
If , then .
If , then .
If , then .
If , then .

Remark 10. It is easy to see that our results improve the results of [14] as special cases of parameters. Some particular cases regarding Corollary 9 are given below.

Corollary 11. If , then .
If , then .
If , then .
If , then .

Theorem 12. Let , , , and . If + < , then is convex in .

Proof. By using the well-known triangle inequality with the equality , , and the inequalities we obtain In view of Lemma 1, is convex in , if , but this is true under the hypothesis.

Remark 13. When we put , our results are better than [14].
Consider the integral operator , where , , Here In the next theorem, we obtain the conditions so that is univalent in

Theorem 14. Let , , , and . Let and suppose that is a positive real number such that in the open unit disc. If then is univalent in

Proof. A calculation gives us Since , then, by the Schwarz Lemma, triangle inequality, and (29), we obtain This shows that the given integral operator satisfies Becker’s criterion for univalence, and hence is univalent in

3. Strong Starlikeness

In this section, we are mainly interested about some sufficient conditions under which the normalized Dini function belongs to the class of strong starlikeness of order

Theorem 15. Let , , , and . If , then , where and

Proof. By using the well-known triangle inequality with the equality , , and the inequalities we obtainFor and from (54), we concluded thatWith the help of Lemma 3, take with and , and we get As a result By using (56) and (58), we obtain which implies that for

Corollary 16. For , where is the positive root of , then

4. Close-to-Convexity with respect to Certain Functions

Recently many authors discuss the close-to-convexity of some special functions with respect to certain functions. Here, we are also interested in the close-to-convexity of normalized Dini function.

Theorem 17. Let , , , and . If , then is close-to-convex with respect to .

Proof. Set To prove that is close-to-convex with respect to the function we use Lemma 5. Therefore, we have to prove that is a decreasing sequence. After some computations, we obtain By using the conditions on parameters, we easily observe that for all , and thus is a decreasing sequence. By Lemma 5 it follows that is close-to-convex with respect to the function

Theorem 18. Let , , , and . If , then is close-to-convex with respect to

Proof. Set Here , and therefore we have and for all To prove our main result we will prove that is a decreasing sequence. After some computations, we obtain By using the conditions on parameters we easily observe that