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

Nihat Yagmur¹ and Halit Orhan²

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 [1–9]. 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.

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.

References

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

E. Deniz, H. Orhan, and H. M. Srivastava, “Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions,” Taiwanese Journal of Mathematics, vol. 15, no. 2, pp. 883–917, 2011.

E. Deniz, “Convexity of integral operators involving generalized Bessel functions,” Integral Transforms and Special Functions, vol. 1, pp. 1–16, 2012.

S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canadian Journal of Mathematics, vol. 39, no. 5, pp. 1057–1077, 1987.

V. Selinger, “Geometric properties of normalized Bessel functions,” Pure Mathematics and Applications, vol. 6, no. 2-3, pp. 273–277, 1995.

H. M. Srivastava, D.-G. Yang, and N.-E. Xu, “Subordinations for multivalent analytic functions associated with the Dziok-Srivastava operator,” Integral Transforms and Special Functions, vol. 20, no. 7-8, pp. 581–606, 2009.

H. M. Srivastava, “Generalized hypergeometric functions and associated families of -uniformly convex and -starlike functions,” General Mathematics, vol. 15, no. 3, pp. 201–226, 2007.

H. M. Srivastava, G. Murugusundaramoorthy, and S. Sivasubramanian, “Hypergeometric functions in the parabolic starlike and uniformly convex domains,” Integral Transforms and Special Functions, vol. 18, no. 7-8, pp. 511–520, 2007.

D. Răducanu and H. M. Srivastava, “A new class of analytic functions defined by means of a convolution operator involving the Hurwitz-Lerch zeta function,” Integral Transforms and Special Functions, vol. 18, no. 11-12, pp. 933–943, 2007.

P. L. Duren, Univalent Functions, vol. 259 of Fundamental Principles of Mathematical Sciences, Springer, New York, NY, USA, 1983.

J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,” Annals of Mathematics, vol. 17, no. 1, pp. 12–22, 1915.

W. Kaplan, “Close-to-convex schlicht functions,” The Michigan Mathematical Journal, vol. 1, p. 169–185 (1953), 1952.

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

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

S. Zhang and J. Jin, Computation of Special Functions, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1996.

H. Orhan and N. Yağmur, “Geometric properties of generalized Struve functions,” in The International Congress in Honour of Professor Hari M. Srivastava, Bursa, Turkey, August, 2012.