International Journal of Mathematics and Mathematical Sciences
Volume 2016 (2016), Article ID 3987231, 7 pages
http://dx.doi.org/10.1155/2016/3987231

Research Article

Univalent and Starlike Properties for Generalized Struve Function

Aaisha Farzana Habibullah,¹ Adolf Stephen Bhaskaran,¹ and Jeyaraman Muthusamy Palani²

¹Department of Mathematics, Madras Christian College, Tambaram, Chennai, Tamil Nadu 600 059, India
²Department of Mathematics, L. N. Government College, Ponneri, Chennai, Tamil Nadu 601 204, India

Received 10 April 2016; Accepted 22 May 2016

Academic Editor: Teodor Bulboaca

Copyright © 2016 Aaisha Farzana Habibullah 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 derive conditions on the parameters , , and so that the function where is the normalized form of generalized Struve function, belongs to the class Also, some sufficient conditions for the function to be in the class are obtained.

1. Introduction and Preliminaries

Let denote the unit disc in the complex plane and let denote the class of functions which are analytic and of the form

and normalized by the conditions and Let denote the class of functions such that

Suppose that and are two analytic functions in , and is univalent in . We say that is subordinate to , written or , if and only if and .

A function belongs to the class of starlike functions of order denoted by if

A subclass of the class of starlike functions denoted by for consists of functions for which

We also note that . In [1–3] the authors have discussed the coefficient bounds and other extremal properties of the class

For , consider the class

From [4], we have the strict inclusion . Very recently, Obradović et al. [5] have discussed the geometric behaviour of functions in . Also the class has been widely studied by many authors in [4, 6–8].

Let us consider the following second-order linear nonhomogenous differential equation (for more details see [9, 10]):

where . The function which is called the generalized Struve function of order is defined as a particular solution of (6) and has the series representation as follows:

where stands for the Euler gamma function.

Now, we consider the function defined in terms of the generalized Struve function by the transformation:

By using the well-known Pochammer symbol (or the shifted factorial) defined for in terms of the Euler -function, we have

We obtain the following series representation for the function given by (8):

where . Also, this function is analytic on and satisfies the following second-order inhomogeneous differential equation:

where

Recently, the class and its generalizations have been widely studied by many authors [4–8, 11]. By applying the admissible function method authors in [12] have obtained conditions on the triplet , , and such that is in the class , where is the confluent hypergeometric function. In [13], the authors have derived conditions on the parameters , , and such that the function is in , where is the Gaussian hypergeometric function. Moreover, in [9, 10] Yagmur and Orhan have obtained sufficient conditions for the generalized Struve function to be convex, starlike, and univalent. Most of these results were motivated by the research on geometric properties of Gaussian and confluent hypergeometric function.

Motivated by the above-mentioned works, in this paper we use the method of differential subordination to show that is in the class and also we provide sufficient conditions for the function to be in the class and hence univalent.

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

Lemma 1 (see [14]). Let (nonconstant) function be analytic in with , .
If attains its maximum value on the circle at a point , then

where is a real number and .

Lemma 2 (see [15]). If an analytic function has the form and satisfies the condition

then is univalent in

Lemma 3 (see [15]). Let be a complex number, , and let be a complex number, , , and a regular function on . If

for all , then the function

is regular and univalent in .

Lemma 4 (see [16]). Let and let be analytic and univalent on except for those for which . Suppose that satisfies the condition

where is finite, , and . If and analytic in , , , and further if

then in .
Suppose that is analytic in with and . Then the condition (16) reduces to a simple form:

whenever , , , and .

2. Main Results

Theorem 5. Let , , and such that ; then

Proof. Let , ; then is analytic in with . Since the function satisfies the differential equation (11) and , satisfies the following inhomogeneous differential equation:

Using Lemma 4, we will show that , where . For this, if we let

and , then it is sufficient to prove that whenever , , , and is real, we have that

In the last stage of the inequalities we have used the definition of and shown that

whenever , is real, and . Hence, .

Choosing in the above theorem we have the following Corollary.

Corollary 6. Let , and be such that ; then

The next results give sufficient conditions for the function to be starlike and univalent in the open unit disc.

Theorem 7. If , and such that and

then the function belongs to the class

Proof. For to be in the class we need to prove that

Upon setting

we observe that is analytic in and .
Also

Now, since satisfies the inhomogeneous differential equation (11), in terms of we see that satisfies the following equation:

where

Now, we claim that , . By using Lemma 4 with , , and , we need to show that

where is real, , and :

where

Also, when in Theorem 5 we have that if then

Also,

Now, if we show that , then we have

In order to establish that we need the following results which state that for   and   if and only if

In particular, implies that

When , we have

Using the last inequality, we have

and so we have provided that

holds, where is given by (35). And, we observe that the above condition is stated in the theorem. Thus, in and hence belongs to .

Theorem 8. Let , and . If satisfies any one of the following inequalities:

then is in and hence is univalent in .

Proof. Define a function by

and then is analytic in and Differentiating (46) gives

Hence, from (46) and (47), we have

Now suppose that there exists such that

and then from Lemma 1, we have

Therefore, letting in each of (48), we obtain that

which contradicts our assumption (42)–(45), respectively. Therefore, for all ; then from (46) we have

which implies is in the class and hence univalent.

Theorem 9. Let , , such that . If satisfies the inequality

then is univalent in .

Proof. Define a function as follows:

We see that is analytic in and . Differentiation of (54) gives

Now suppose there exists such that

Then from Lemma 1, we have

Now letting in (55), we have

where and . Since we have

This contradicts the hypothesis and therefore for all . Thus,

In view of Lemma 2 it implies that is in and hence univalent.

Theorem 10. Let , be a real number such that and let be a complex number which satisfies the inequality

If is univalent in and for all , then the function

is univalent in , where the values of the complex powers are taken with their principal values.

Proof. Define a function

Then we have .
Also

From (65), we have

From the hypothesis, we have (); then, by the Schwarz Lemma (cf [17], we obtain that

Now, since is univalent in

Using (68), we have

So, from (61) we have

Applying Lemma 3, we obtain the function defined by (62) which is univalent in .

Competing Interests

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

Acknowledgments

The work of the first author was supported by the Department of Science and Technology, India, with reference to the Sanction Order no. SR/DST-WOS A/MS-10/2013(G). The work of the second author was supported by the grant given under UGC Minor Research Project F. no. 5599/15 (MRP-SEM/UGC SERO). The work of the third author was supported by the grant given under UGC Minor Research Project, 2014-15/MRP-5591/15(SERO/UGC).

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