Journal of Function Spaces
Volume 2016 (2016), Article ID 1896154, 8 pages
http://dx.doi.org/10.1155/2016/1896154

Research Article

Certain Geometric Properties of Normalized Wright Functions

Mohsan Raza,¹ Muhey U Din,¹ and Sarfraz Nawaz Malik²

¹Department of Mathematics, Government College University Faisalabad, Faisalabad, Pakistan
²Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan

Received 15 October 2016; Accepted 4 December 2016

Academic Editor: Adrian Petrusel

Copyright © 2016 Mohsan Raza 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.

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 set

Then 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 as

It 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 are

respectively. For some details about these classes, see [2, 3]. Recently, Baricz [4] introduced the classes

For we denote the classes and by and , respectively. Also for and , we have the classes and .

Let given by (5) and given by

and then Hadamard product (or convolution) of and is defined as

Recently, Prajapat [5] studied some geometric properties of Wright function

This 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 as

The 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 form

where 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 inequality

then

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 form

for 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 inequality

with the inequality , , which is equivalent to , , and the inequality

we obtain

Also consider

Since , therefore by using the reverse triangle inequality

and the inequality, , , we get

By combining (23) and (26), we get

So is starlike function of order , where
(ii) To prove that we have to show that . By using the well-known triangle inequality

with the inequality , , which is equivalent to , , and the inequalities

we have

Since , therefore by using the reverse triangle inequality

and the inequality, , , we get

Combining (30) and (32), we have

This implies that is convex function of order , where
(iii) Using inequality (30) and Lemma 1, we have

where and This shows that Therefore,
(iv) To prove that we have to show that where . By using the well-known triangle inequality

with the inequality , , and the inequality

we have

Therefore, 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 that

where
Now,

From (39), a little simplification yields

Differentiating (42) two times with respect to , we have

Now for , the expressions (39), (42), and (43) become

By using the above expressions, (41) becomes

And 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 identity

By using the (10), we will only show that

where
Now,

Now for and using (42), (43), and (44), expression (51) becomes

and 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 functions

By using Lemma 2, we will get the following results.

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

Proof. Set

We 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 obtain

By 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. Set

Here , 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 obtain

By 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 have

Now, we have

where , , and is any real number. Also, we have

This 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 function

By 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.

References

P. L. Duren, Theory of Hp Spaces, vol. 38 of A series of Monographs and Textbooks in Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1970.

O. Altintaş, O. Őzkan, and H. M. Srivastava, “Neighborhoods of a class of analytic functions with negative coefficients,” Applied Mathematics Letters, vol. 13, no. 3, pp. 63–67, 2000.

H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathematical Society, vol. 51, pp. 109–116, 1975.

Á. Baricz, “Bessel transforms and Hardy space of generalized Bessel functions,” Mathematica, vol. 48, no. 2, pp. 127–136, 2006.

J. K. Prajapat, “Certain geometric properties of the Wright function,” Integral Transforms and Special Functions, vol. 26, no. 3, pp. 203–212, 2015.

E. M. Wright, “On the coefficients of power series having exponential singularities,” Journal London Mathematical Society, vol. 8, pp. 71–79, 1933.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, NY, USA, 1993.

R. Gorenflo, Y. Luchko, and F. Mainardi, “Analytical properties and applications of the Wright function,” Fractional Calculus & Applied Analysis, vol. 2, no. 4, pp. 383–414, 1999.

S. S. Miller and P. T. Mocanu, “Univalence of Gaussian and confluent hypergeometric functions,” Proceedings of the American Mathematical Society, vol. 110, no. 2, pp. 333–342, 1990.

S. Ruscheweyh and V. Singh, “On the order of starlikeness of hypergeometric functions,” Journal of Mathematical Analysis and Applications, vol. 113, no. 1, pp. 1–11, 1986.

Á. Baricz, Generalized Bessel Functions of the First Kind, vol. 1994 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.

A. Baricz, “Geometric properties of generalized Bessel functions,” Publicationes Mathematicae Debrecen, vol. 73, no. 1-2, pp. 155–178, 2008.

Á. Baricz and S. Ponnusamy, “Starlikeness and convexity of generalized Bessel functions,” Integral Transforms and Special Functions. An International Journal, vol. 21, no. 9-10, pp. 641–653, 2010.

H. Orhan and N. Yagmur, “Geometric properties of generalized Struve functions,” Annals of the Alexandru Ioan Cuza University—Mathematics, 2014.

N. Yağmur and H. Orhan, “Partial sums of generalized Struve functions,” Miskolc Mathematical Notes, vol. 17, no. 1, pp. 657–670, 2016.

S. Owa, M. Nunokawa, H. Saitoh, and H. M. Srivastava, “Close-to-convexity, starlikeness, and convexity of certain analytic functions,” Applied Mathematics Letters, vol. 15, no. 1, pp. 63–69, 2002.

S. Ozaki, “On the theory of multivalent functions,” Science Reports of the Tokyo Bunrika Daigaku, vol. 2, pp. 167–188, 1935.

J. Stankiewicz and Z. Stankiewicz, “Some applications of Hadamard convolutions in the theory of functions,” Annales Universitatis Mariae Curie-Sklodowska, vol. 40, pp. 251–265, 1986.

S. Ponnusamy, “Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane,” Rocky Mountain Journal of Mathematics, vol. 28, no. 2, pp. 695–733, 1998.

P. J. Eenigenburg and F. R. Keogh, “The Hardy class of some univalent functions and their derivatives,” The Michigan Mathematical Journal, vol. 17, pp. 335–346, 1970.

S. Ponnusamy and M. Vuorinen, “Univalence and convexity properties for confluent hypergeometric functions,” Complex Variables, Theory and Application, vol. 36, no. 1, pp. 73–97, 1998.

S. Ponnusamy and M. Vuorinen, “Univalence and convexity properties for Gaussian hypergeometric functions,” Rocky Mountain Journal of Mathematics, vol. 31, no. 1, pp. 327–353, 2001.

S. Ponnusamy, “The Hardy spaces of hypergeometric functions,” Complex Variables, Theory and Application, vol. 29, no. 1, pp. 83–96, 1996.

N. Yagmur and H. Orhan, “Hardy space of generalized Struve functions,” Complex Variables and Elliptic Equations, vol. 59, no. 7, pp. 929–936, 2014.

N. Yagmur, “Hardy space of Lommel functions,” Bulletin of the Korean Mathematical Society, vol. 52, no. 3, pp. 1035–1046, 2015.

J. H. Choi, Y. C. Kim, and H. M. Srivastava, “Convex and starlike generalized hypergeometric functions associated with the Hardy spaces,” Complex Variables, vol. 31, pp. 345–355, 1996.

Y. C. Kim and H. M. Srivastava, “The Hardy space for a certain subclass of Bazilevič functions,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1201–1207, 2006.

Y. C. Kim, K. S. Lee, and H. M. Srivastava, “Certain classes of integral operators associated with the Hardy space of analytic functions,” Complex Variables, Theory and Application, vol. 20, no. 1–4, pp. 1–12, 1992.

G. Miclãus, “The Libera generalized integral operator and Hardy spaces,” Mathematical Notes. A Publication of the University of Miskolc, vol. 4, no. 1, pp. 39–43, 2003.