Certain Geometric Properties of Normalized Wright Functions

Abstract

In this article, we find some geometric properties like starlikeness, convexity of order , close-to-convexity of order , and close-to-convexity of normalized Wright functions with respect to the certain functions. The sufficient conditions for the normalized Wright functions belonging to the classes and are the part of our investigations. We also obtain the conditions on normalized Wright function to belong to the Hardy space .

1. Introduction and Preliminaries

Let denote the class of all analytic functions in the open unit disk and denote the space of all bounded functions on . This is 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 setThen the function if is bounded for all It is clear that For some details, see [1]. It is also known [1] that in , and then Let be the class of functions of the form analytic in the open unit disc , and denote the class of all functions in which are univalent in . Let , , and denote the classes of starlike, convex, and close-to-convex functions of order , respectively, and they are defined asIt is clear that are the classes of starlike, convex, and close-to-convex functions, respectively. Also consider the subclasses and of , defined by the following relations:where The purpose of these subclasses is that when we put in (8), we get and The sufficient coefficient conditions by which a function as defined in (5) belongs to the classes and arerespectively. For some details about these classes, see [2, 3]. Recently, Baricz [4] introduced the classesFor we denote the classes and by and , respectively. Also for and , we have the classes and .

Let given by (5) and given byand then Hadamard product (or convolution) of and is defined asRecently, Prajapat [5] studied some geometric properties of Wright functionThis series is absolutely convergent in for and absolutely convergent in open unit disc for Furthermore this function is entire. The Wright functions were introduced by Wright [6] and have been used in the asymptotic theory of partitions, in the theory of integral transforms of the Hankel type and in Mikusinski operational calculus. Recently, Wright functions have been found in the solution of partial differential equations of fractional order. It was found that the corresponding Green functions can be represented in terms of the Wright function [7, 8]. For positive rational number , the Wright function can be represented in terms of generalized hypergeometric function. For some details, see [9, Section ]. In particular, the function can be expressed in terms of the Bessel functions , given asThe Wright function generalizes various functions like Array function, Whittaker function, entire auxiliary functions, and so forth. For more details, we refer to [9]. Prajapat [5] discussed some geometric properties of the Wright functions,and their normalization of the formwhere The Pochhammer (or Appell) symbol, defined in terms of Euler’s gamma functions, is given as . We refer for some geometric properties of special functions like hypergeometric functions [10, 11], Bessel functions [4, 12–14], and Struve functions [15, 16].

We need the following results to prove our results.

Lemma 1 (see [17]). If satisfies the inequalitythen

Lemma 2 (see [18]). 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 .

Lemma 3 (see [19]). , where with and the value of is the best possible.

Lemma 4 (see [20]). For and , we have or equivalently

Lemma 5 (see [21]). If the function , convex of order , where , is not of the formfor some complex numbers and and for some real number , then the following statements hold:(i)There exists such that (ii)If , then there exists such that (iii)If , then

2. Main Results

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

Proof. (i) To prove that we have to show that . By using the well-known triangle inequalitywith the inequality , , which is equivalent to , , and the inequalitywe obtainAlso consider Since , therefore by using the reverse triangle inequality and the inequality, , , we getBy combining (23) and (26), we getSo is starlike function of order , where
(ii) To prove that we have to show that . By using the well-known triangle inequalitywith the inequality , , which is equivalent to , , and the inequalitieswe have Since , therefore by using the reverse triangle inequality and the inequality, , , we getCombining (30) and (32), we haveThis implies that is convex function of order , where
(iii) Using inequality (30) and Lemma 1, we havewhere and This shows that Therefore,
(iv) To prove that we have to show that where . By using the well-known triangle inequalitywith the inequality , , and the inequalitywe haveTherefore, for .

Putting in Theorem 6, we have the following result.

Corollary 7. Let and . Then the following assertions are true:(i)If and, , then (ii)If and , then (iii)If and , then (iv)If and , then

Theorem 8. If , , and , then a sufficient condition for to be in is

Proof. Consider the identity,By using (9), we will only show thatwhere
Now,From (39), a little simplification yieldsDifferentiating (42) two times with respect to , we haveNow for , the expressions (39), (42), and (43) becomeBy using the above expressions, (41) becomesAnd it is bounded above by if (38) holds. Thus the proof is completed.

Corollary 9. The normalized Wright function is starlike of order with respect to the origin if

Theorem 10. If , , and , then a sufficient condition for to be in is

Proof. Consider the identityBy using the (10), we will only show thatwhere
Now,Now for and using (42), (43), and (44), expression (51) becomesand is bounded above by if (48) holds, which is the required result.

Corollary 11. The normalized Wright function is convex of order , , with respect to the origin if

3. Close-to-Convexity of Wright Functions with respect to Certain Functions

The work in this section is motivated by the works of Baricz, Orhan and Yagmur, and Ponnusamy and Vuorinen [13, 15, 22, 23]. In this section we will discuss some conditions on the parameters and under which the Wright functions are assured close-to-convex with respect to the functionsBy using Lemma 2, we will get the following results.

Theorem 12. If and , then is close-to-convex with respect to the function

Proof. SetWe have for all and , by using the inequality To prove that is close-to-convex with respect to the function , we use Lemma 2. Therefore, we have to prove that is a decreasing sequence. By a short computation, we obtainBy using the conditions on parameters, we can easily observe that for all , and thus is a decreasing sequence. By Lemma 2, it follows that is close-to-convex with respect to the function

Theorem 13. If and , then is close-to-convex with respect to the function

Proof. SetHere , and therefore we have and for all To prove our main result we will prove that is a decreasing sequence. By a short computation, we obtainBy using the conditions on parameters, we can easily observe that for all , and thus is a decreasing sequence. By Lemma 2 it follows that is close-to-convex with respect to the function

4. Hardy Spaces of Wright Functions

Hardy spaces of hypergeometric functions are recently studied by Ponnusamy [24]. Baricz [4] used the idea of Ponnusamy and found the Hardy spaces of Bessel functions. Yagmur and Orhan [25] studied the same problem for generalized Struve functions. Similarly, Yagmur [26] studied the problem for Lommel functions. For Hardy spaces related to some classes of analytic functions, we refer to [27–30].

Theorem 14. Let, , Then (i)for (ii)for

Proof. From the definition of Hypergeometric function, we haveNow, we havewhere , , and is any real number. Also, we haveThis implies that is not of the forms for and for , respectivelyAlso from Theorem 6(ii) is convex of order Hence, by using Lemma 5, we have the required result.

Theorem 15. Let , , and , and then the convolution is in

Proof. Let Then it is clear that . Using Corollary 7(iv), we have . Since , therefore by using Lemma 3, we have It is also clear that is an entire function and therefore is entire. This implies that is bounded. Hence, we have the required result.

Theorem 16. Let with , , , and . If , then , where

Proof. Let . Then, . Now from Theorem 6(iv), we have . By using Lemma 4 and the fact that , we have , where . Consequently, we have .

Corollary 17. Let , , and If , , then

Corollary 18. Let and . If , then

5. Particular Case

At the end of this paper, we give some particular cases of the above-mentioned theorem. When we put and in (16), we obtain the functionBy using Theorem 6 assertions (i) and (iv), we get the following corollary.

Corollary 19. (i) If , where , then
(ii) If , where , then .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of Sarfraz Nawaz Malik is supported by COMSATS Institute of Information Technology, Pakistan, Reference no. 3-64/IPF-SRG/CIIT/Wah/14/765.