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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 954513, 6 pages
http://dx.doi.org/10.1155/2013/954513
Research Article

Starlikeness and Convexity of Generalized Struve Functions

1Department of Mathematics, Faculty of Science and Art, Erzincan University, 24000 Erzincan, Turkey
2Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey

Received 3 December 2012; Accepted 14 January 2013

Academic Editor: Mustafa Bayram

Copyright © 2013 Nihat Yagmur and Halit Orhan. 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 give sufficient conditions for the parameters of the normalized form of the generalized Struve functions to be convex and starlike in the open unit disk.

1. Introduction and Preliminary Results

It is well known that the special functions (series) play an important role in geometric function theory, especially in the solution by de Branges of the famous Bieberbach conjecture. The surprising use of special functions (hypergeometric functions) has prompted renewed interest in function theory in the last few decades. There is an extensive literature dealing with geometric properties of different types of special functions, especially for the generalized, Gaussian, and Kummer hypergeometric functions and the Bessel functions. Many authors have determined sufficient conditions on the parameters of these functions for belonging to a certain class of univalent functions, such as convex, starlike, and close-to-convex functions. More information about geometric properties of special functions can be found in [19]. In the present investigation our goal is to determine conditions of starlikeness and convexity of the generalized Struve functions. In order to achieve our goal in this section, we recall some basic facts and preliminary results.

Let denote the class of functions normalized by which are analytic in the open unit disk . Let denote the subclass of which are univalent in . Also let and denote the subclasses of consisting of functions which are, respectively, starlike and convex of order in . Thus, we have (see, for details, [10]), where, for convenience, We remark that, according to the Alexander duality theorem [11], the function is convex of order , where if and only if is starlike of order . We note that every starlike (and hence convex) function of the form (1) is univalent. For more details we refer to the papers in [10, 12, 13] and the references therein.

Denote by , where , the subclass of consisting of functions for which for all . A function is said to be in if .

Lemma 1 (see [4]). If and for some fixed and , and for all , then is in the class .

Lemma 2 (see [14]). Let . A sufficient condition for to be in and , respectively, is that respectively.

Lemma 3 (see [14]). Let . Suppose that . Then a necessary and sufficient condition for to be in and , respectively, is that respectively. In addition , and .

2. Starlikeness and Convexity of Generalized Struve Functions

Let us consider the second-order inhomogeneous differential equation [15, page 341] whose homogeneous part is Bessel’s equation, where is an unrestricted real (or complex) number. The function , which is called the Struve function of order , is defined as a particular solution of (8). This function has the form The differential equation which differs from (8) only in the coefficient of . The particular solution of (10) is called the modified Struve function of order and is defined by the formula [15, page 353] Now, let us consider the second-order inhomogeneous linear differential equation [16], where . If we choose and , then we get (8), and if we choose , then we get (10). So this generalizes (8) and (10). Moreover, this permits to study the Struve and modified Struve functions together. A particular solution of the differential equation (12), which is denoted by , is called the generalized Struve function [16] of order . In fact we have the following series representation for the function : Although the series defined in (13) is convergent everywhere, the function is generally not univalent in . Now, consider the function defined by the transformation By using the Pochhammer (or Appell) symbol, defined in terms of Euler’s gamma functions, by , we obtain for the function the following form: where . This function is analytic on and satisfies the second-order inhomogeneous differential equation Orhan and Yaǧmur [16] have determined various sufficient conditions for the parameters , and such that the functions or to be univalent, starlike, convex, and close to convex in the open unit disk. In this section, our aim is to complete the above-mentioned results.

For convenience, we use the notations: and .

Proposition 4 (see [16]). If , , and , then for the generalized Struve function of order the following recursive relations hold:(i);(ii); (iii); (iv); (v).

Theorem 5. If the function , defined by (15), satisfies the condition where and , then .

Proof. If we define the function by for . The given condition becomes where . By taking in Lemma 1, we thus conclude from the previous inequality that , which proves Theorem 5.

Theorem 6. If the function , defined by (15), satisfies the condition where and , then it is starlike of order with respect to .

Proof. Define the function by . Then and where and . By taking in Lemma 1, we deduce that ; that is, is starlike of order with respect to the origin for . So, Theorem 6 follows from the definition of the function , because .

Theorem 7. If for and one has for all , then is starlike of order with respect to .

Proof. Theorem 5 implies that . On the other hand, the part (v) of Proposition 4 yields Since the addition of any constant and the multiplication by a nonzero quantity do not disturb the starlikeness. This completes the proof.

Lemma 8. If , , and such that , then the function satisfies the following inequalities:

Proof. We first prove the assertion (23) of Lemma 8. Indeed, by using the well-known triangle inequality: and the inequalities , , we have Similarly, by using reverse triangle inequality: and the inequalities , , then we get which is positive if .
In order to prove assertion (24) of Lemma 8, we make use of the well-known triangle inequality and the inequalities , , and we obtain Similarly, by using the reverse triangle inequality and the inequalities , , we have which is positive if .
We now prove assertion (25) of Lemma 8 by using again the triangle inequality and the inequalities , , and we arrive at the following:
Thus, the proof of Lemma 8 is completed.

Theorem 9. If , and , then the following assertions are true.(i) If , then is convex in .(ii) If , then is starlike of order in , and consequently the function is starlike in .(iii) If , then the function is starlike of order with respect to for all .

Proof. (i) By combining the inequalities (24) with (25), we immediately see that So, for , we have This shows is convex in .
(ii) If we let and , then so that if and only if
It follows that is starlike of order if (37) holds.
From (24) and (23), we have respectively.
By combining the inequalities (38) with (39), we see that where , and the above bound is less than or equal to if and only if . It follows that is starlike of order in and is starlike in .
(iii) The part (ii) of Theorem 9 implies that for , the function is starlike of order in . On the other hand, the part (v) of Proposition 4 yields So the function is starlike of order with respect to for all .
This completes the proof.

Struve Functions. Choosing , we obtain the differential equation (8) and the Struve function of order , defined by (9), satisfies this equation. In particular, the results of Theorem 9 are as follows.

Corollary 10. Let be defined by , where stands for the Struve function of order . Then the following assertions are true.(i) If , then is convex in .(ii) If , then is starlike of order in , and consequently the function is starlike in .(iii) If , then the function is starlike of order with respect to for all .

Modified Struve Functions. Choosing   and  , we obtain the differential equation (10) and the modified Struve function of order , defined by (11). For the function defined by , where stands for the modified Struve function of order . The properties are same like for function , because we have . More precisely, we have the following results.

Corollary 11. The following assertions are true.(i) If , then is convex in .(ii) If , then is starlike of order in , and consequently the function is starlike in .(iii) If , then the function is starlike of order with respect to for all .

Example 12. If we take , then from part (ii) of Corollary 10, the function is starlike in . So the function is also starlike in . We have the image domain of illustrated by Figure 1.

954513.fig.001
Figure 1: .

Theorem 13. If , and , then a sufficient condition for to be in is Moreover, (42) is necessary and sufficient for to be in .

Proof. Since , according to Lemma 2, we need only show that We notice that This sum is bounded above by if and only if (42) holds. Since the necessity of (42) for to be in follows from Lemma 3.

Corollary 14. If and , then a sufficient condition for to be in is Moreover, (46) is necessary and sufficient for to be in .

Proof. For , the condition (42) becomes . From the part (v) of Proposition 4 we get So, if and only if . Thus, we obtain the condition (46).
Furthermore, from the proof of Theorem 13, we have necessary and sufficient condition for to be in .

Theorem 15. If and , then a sufficient condition for to be in is Moreover, (48) is necessary and sufficient for to be in .

Proof. In view of Lemma 2, we need only to show that If we let , we notice that This sum is bounded above by if and only if (48) holds. Lemma 3 implies that (48) is also necessary for to be in .

Theorem 16. If , and , then .

Proof. Since we note that if and only if .

Acknowledgment

The present paper was supported by Ataturk University Rectorship under The Scientific and Research Project of Ataturk University, Project no: 2012/173.

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