Certain Geometric Properties of Generalized Dini Functions

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 [1–10] 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 as For 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 as where

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 for all , and thus is a decreasing sequence. By Lemma 5 it follows that is close-to-convex with respect to the function

5. Hardy Spaces of Dini Function

Let denote the class of all analytic functions in the open unit disk and denote the space of all bounded functions on . This is a Banach algebra with respect to the norm We denote for the space of all functions such that admits a harmonic majorant. is a Banach space if the norm of is defined to be -th root of the least harmonic majorant of for some fixed . Another equivalent definition of norm is given as follows: Let , and set Then the function if is bounded for all It is clear that For some details see [20]. It is also known [20] that, for in , then Hardy spaces of hypergeometric functions are recently studied by Ponnusamy [21]. Baricz [2] uses the idea of Ponnusamy and found the Hardy spaces of Bessel functions while Yamur and Orhan [10] studied the same problem for generalized Struve functions. Similarly Yamur [22] studied the problem for Lommel functions and Raza et al. [9] studied the same problem for Wright functions.

To prove our main results we need the following lemmas.

Lemma 19 (see [23]). , where with and the value of is best possible.

Lemma 20 (see [24]). For and , we have or equivalently .

Lemma 21 (see [25]). If the function , convex of order , where , is not of the form for some complex numbers and and for some real number then the following statements hold:
There exists , such that .
If , then there exists , such that .
If , then

The lemma defined below also plays an important role to find our main results.

Theorem 22. Let and + − < . Then

   for ;

   for .

Proof. By using the definition of Hypergeometric function we have for , , and and for On the other hand This implies that is not of the form for and for , respectively. Also, from part (ii) of Theorem 6, is convex of order Hence, by using Lemma 21, we have required result.

Theorem 23. Let and . Then the convolution is in .

Proof. Let . Then it is clear that Using Corollary 9 of part (iv), we have . Since , therefore by using Lemma 19  . It is also clear that is an entire function and therefor is entire. This implies that is bounded. Hence, we have the required result.

Theorem 24. Let with , and then , , and . If , with , then , where

Proof. Let , and then Now, from Theorem 6 of part (iv), we have . By using Lemma 19 and the fact that , we have , where Consequently, we have .

Corollary 25. Let , , and then . If, , then .

Corollary 26. Let . If, then .

By using Theorem 23 and Corollary 25, then we have the following corollary.

Corollary 27. If , then the convolutions and are in . Moreover, if , then and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work here is supported by UKM Grant no. GUP-2017-064.