## Recent Developments in Integral Transforms, Special Functions, and Their Extensions to Distributions Theory

View this Special IssueResearch Article | Open Access

# Starlikeness and Convexity of Generalized Struve Functions

**Academic Editor:**Mustafa Bayram

#### 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.

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. View at: Google Scholar | MathSciNet - 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. View at: Google Scholar | MathSciNet - E. Deniz, “Convexity of integral operators involving generalized Bessel functions,”
*Integral Transforms and Special Functions*, vol. 1, pp. 1–16, 2012. View at: Google Scholar - S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,”
*Canadian Journal of Mathematics*, vol. 39, no. 5, pp. 1057–1077, 1987. View at: Publisher Site | Google Scholar | MathSciNet - V. Selinger, “Geometric properties of normalized Bessel functions,”
*Pure Mathematics and Applications*, vol. 6, no. 2-3, pp. 273–277, 1995. View at: Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - H. M. Srivastava, “Generalized hypergeometric functions and associated families of $k$-uniformly convex and $k$-starlike functions,”
*General Mathematics*, vol. 15, no. 3, pp. 201–226, 2007. View at: Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - P. L. Duren,
*Univalent Functions*, vol. 259 of*Fundamental Principles of Mathematical Sciences*, Springer, New York, NY, USA, 1983. View at: MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - W. Kaplan, “Close-to-convex schlicht functions,”
*The Michigan Mathematical Journal*, vol. 1, p. 169–185 (1953), 1952. View at: Google Scholar | MathSciNet - S. Ozaki, “On the theory of multivalent functions,”
*Science Reports of the Tokyo Bunrika Daigaku*, vol. 2, pp. 167–188, 1935. View at: Google Scholar - H. Silverman, “Univalent functions with negative coefficients,”
*Proceedings of the American Mathematical Society*, vol. 51, pp. 109–116, 1975. View at: Google Scholar | MathSciNet - S. Zhang and J. Jin,
*Computation of Special Functions*, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1996. View at: MathSciNet - 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. View at: Google Scholar

#### Copyright

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.