Abstract and Applied Analysis

Volume 2014 (2014), Article ID 723097, 6 pages

http://dx.doi.org/10.1155/2014/723097

Research Article

Starlikeness of Functions Defined by Third-Order Differential Inequalities and Integral Operators

R. Chandrashekar,¹ Rosihan M. Ali,² K. G. Subramanian,³ and A. Swaminathan⁴

¹Department of Technology Management, Faculty of Technology Management and Business, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Batu Pahat, Johor, Malaysia

²School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

³School of Computer Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

⁴Department of Mathematics, I.I.T. Roorkee, Roorkee 247667, India

Received 3 January 2014; Accepted 12 March 2014; Published 3 April 2014

Academic Editor: Om P. Ahuja

Copyright © 2014 R. Chandrashekar 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.

Sufficient conditions are obtained to ensure starlikeness of positive order for analytic functions defined in the open unit disk satisfying certain third-order differential inequalities. As a consequence, conditions for starlikeness of functions defined by integral operators are obtained. Connections are also made to earlier known results.

1. Introduction

Let denote the class of analytic functions defined in the open unit disk . For and a positive integer, let

and , with . For , denote by the subclass of consisting of functions starlike of order satisfying

The class is the well-known subclass of starlike functions studied widely in geometric function theory.

In the sequel, we give emphasis to the class

, where . Evidently for . The class was investigated by Silverman [1], who showed that coincides with for univalent functions with negative coefficients. This class has subsequently been studied in several other works (see, e.g., [2]).

The problem of determining sufficient conditions to ensure starlikeness of functions has been widely investigated. These include conditions in terms of differential inequalities; see, for example, [2–11]. Miller and Mocanu [12], Kuroki and Owa [13], and, more recently, Ali et al. [14] determined conditions for starlikeness of functions defined by an integral operator of the form

or by the double integral operator

In this paper, conditions on certain third-order differential inequalities are found that would imply starlikeness of positive order. As a consequence, conditions on the kernel of certain integral operators are also obtained to ensure that the functions defined by these operators are starlike. Connections are also made to earlier known results.

Recall that an analytic function is subordinate to an analytic function in , written as , if there exists an analytic self-map of with satisfying .

The following lemmas will be required in the sequel.

Lemma 1 (see [15, Theorem 1, page 192] and see also [16, Theorem 3.1b, page 71]). Let be convex in with , and . If and

then

where

The function is convex and is the best -dominant.

Lemma 2 (see [17] and see also [16, Theorem 3.1d, page 76]). Let h be a starlike function with . If satisfies

then

The function is convex and is the best -dominant.

2. Main Results

The following two results are easily obtained by simple adaptations of Theorem 2.1 and Theorem 2.6 in [13]. The proofs are therefore omitted.

Lemma 3. Let , , and . If

then with an extremal function .

Lemma 4. Let , , and . If

then

is a starlike function of order .

Remark 5. Even though the conditions given in Lemmas 3 and 4 are sufficient to deduce , they are in fact sufficient to imply .

The above two lemmas are next used to obtain conditions in terms of a third-order differential inequality and a third-order integral operator to deduce starlikeness of of order .

Theorem 6. Let , , , and . Further let and satisfy

If

then . Equality is attained for .

Proof. Let

A brief computation shows that

Hence, (15) can be written in the subordination form as

It follows from Lemma 1 that

which implies

Hence on using Lemma 3.

For sharpness, it is evident that the function satisfies

Thus,

Theorem 7. Let , , , and . If

where

then

satisfies .

Proof. Let satisfy

From Theorem 6, the solution of (26) belongs to the class . Now (26) has the form

where

Equation (27) has a solution

with

Note that the function in Lemma 4 satisfies . Thus replacing the appropriate parameters in the equation

yields a solution

This completes the proof.

The next result provides a sufficient condition for starlikeness of order involving a second-order differential inequality.

Lemma 8. Let , and with . If

then with an extremal function .

Proof. Inequality (33) can be expressed in the subordination form

Writing

it follows that

Now Lemma 1 with yields

which implies

Let

Since

an application of Lemma 2 shows that

Therefore,

Combining (38) and (42) yields

which means , whence .

Remark 9. For and , Lemma 8 reduces to [12, Lemma 2.2].

The following result gives starlikeness for a function given by a double integral operator associated with Lemma 4. The proof is analogous to Theorem 2.2 of [12] and is omitted.

Lemma 10. Let , , , and . If

then

satisfies .

An application of Lemma 8 yields the following sufficient condition for starlikeness in terms of a third-order differential inequality.

Theorem 11. Let , , , , and . Further let

If

then . Equality is attained for .

Proof. Proceeding similarly as in the proof of Lemma 8, inequality (47) can be written as

Let

Then a computation yields

so that

Hence

Applying Lemma 1 yields

This implies that

and thus

which, in comparison with Lemma 8, gives the required result.

Further the result is sharp for which satisfies

Remark 12. For , the choice in Theorem 11 results in

For , this coincides with Lemma 8 at , which was also exhibited in [13, Corollary 2.4]. Further, for , (57) gives

which for and is the result given in [8, Theorem 1].

Corresponding to Theorem 11, a sufficient condition for starlikeness of order for functions defined by a triple integral operator is obtained in the following result.

Theorem 13. Let , , , , and . Further let

If

then

satisfies .

Proof. Let satisfy

From Theorem 11, we find that the solution of (62) lies in . Now (62) becomes

where

Equation (63) has a solution

with

In view of Lemma 10, the equation

has a solution

This completes the proof.

Conflict of Interests

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

Acknowledgments

The work presented here was supported in part by a research university grant from Universiti Sains Malaysia. The authors are thankful to the referees for their suggestions that helped improve the clarity of this paper.

References

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

N. Yagmur and H. Orhan, “Starlikeness and convexity of generalized Struve functions,” Abstract and Applied Analysis, vol. 2013, Article ID 954513, 6 pages, 2013.

R. M. Ali, “On a subclass of starlike functions,” The Rocky Mountain Journal of Mathematics, vol. 24, no. 2, pp. 447–451, 1994.

R. M. Ali and V. Singh, “Convexity and starlikeness of functions defined by a class of integral operators,” Complex Variables: Theory and Application, vol. 26, no. 4, pp. 299–309, 1995.

R. M. Ali, A. O. Badghaish, V. Ravichandran, and A. Swaminathan, “Starlikeness of integral transforms and duality,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 808–822, 2012.

R. M. Ali, M. M. Nargesi, and V. Ravichandran, “Convexity of integral transforms and duality,” Complex Variables and Elliptic Equations, vol. 58, no. 11, pp. 1569–1590, 2013.

R. Fournier and P. T. Mocanu, “Differential inequalities and starlikeness,” Complex Variables: Theory and Application, vol. 48, no. 4, pp. 283–292, 2003.

M. Obradović, “Simple sufficient conditions for univalence,” Matematichki Vesnik, vol. 49, no. 3-4, pp. 241–244, 1997.

R. Omar, S. A. Halim, and R. W. Ibrahim, “Differential subordination properties of certain analytic functions,” International Journal of Mathematics, vol. 24, no. 6, Article ID 1350044, 7 pages, 2013.

R. Omar and S. A. Halim, “Differential subordination properties of Sokol-Stankiewiczstarlike functions,” Kyungpook Mathematical Journal, vol. 53, no. 3, pp. 459–465, 2013.

H. Shiraishi, K. Kuroki, and S. Owa, “Some classes of order for second-order α differential inequalities,” Electronic Journal of Mathematical Analysis and Applications, vol. 1, no. 2, pp. 149–155, 2013.

S. S. Miller and P. T. Mocanu, “Double integral starlike operators,” Integral Transforms and Special Functions, vol. 19, no. 7-8, pp. 591–597, 2008.

K. Kuroki and S. Owa, “Double integral operators concerning starlike of order ,” International Journal of Differential Equations, vol. 2009, Article ID 737129, 13 pages, 2009.

R. M. Ali, S. K. Lee, K. G. Subramanian, and A. Swaminathan, “A third-order differential equation and starlikeness of a double integral operator,” Abstract and Applied Analysis, vol. 2011, Article ID 901235, 10 pages, 2011.

D. J. Hallenbeck and S. Ruscheweyh, “Subordination by convex functions,” Proceedings of the American Mathematical Society, vol. 52, pp. 191–195, 1975.

S. S. Miller and P. T. Mocanu, Differential Subordinations, Marcel Dekker, New York, NY, USA, 2000.

T. J. Suffridge, “Some remarks on convex maps of the unit disk,” Duke Mathematical Journal, vol. 37, pp. 775–777, 1970.