Abstract and Applied Analysis

Volume 2014 (2014), Article ID 738350, 5 pages

http://dx.doi.org/10.1155/2014/738350

## On the Starlikeness of Certain Class of Multivalent Analytic Functions

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China

Received 22 October 2013; Revised 5 March 2014; Accepted 5 March 2014; Published 30 March 2014

Academic Editor: Pavel Kurasov

Copyright © 2014 Lei Shi and Zhi-Gang Wang. 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

The main purpose of this paper is to determine the conditions of starlikeness for certain class of multivalent analytic functions. Relevant connections of the results presented here with those obtained in earlier works are pointed out.

#### 1. Introduction and Main Result

Let denote the class of functions of the form
which are analytic in the open unit disk . For convenience, we set . A function is said to be in the class of *-valent starlike functions of order * in , if it satisfies the following inequality:
For simplicity, we write .

In [1], Chichra introduced the class of analytic functions which satisfy the condition He proved that the members of are univalent in . Later, R. Singh and S. Singh [2] showed that . Recently, Gao and Zhou [3] considered the subclass of which is defined by They derived some mapping properties of this class. Moreover, several authors discussed some related analytic function classes associated with the class (see [4–7]). By using the method of differential subordination, Yang and Liu [8] generalized the above works and studied the subclass of which satisfies the condition where In [9], Owa et al. introduced a new subclass of which satisfies the inequality where , , and (throughout this paper unless otherwise mentioned) the parameters and are constrained as follows: The extreme points, coefficient inequalities, radius of starlikeness, and inclusion relationship for the class are derived. By setting , it is easy to see that the class reduces to the class . If we set in the class , then it reduces to the class , which was studied earlier by Silverman [10], R. Singh and S. Singh [2, 11], independently.

For some recent investigations on the starlikeness of analytic functions, one can refer to [12–20]. In the present paper, we aim at deriving the conditions of starlikeness for the class . The main result is presented below.

Theorem 1. *Let . Then*(1)* for , where is the solution of the following equation:
*(2)* for , where is the solution of the following equation:
*

*2. Preliminary Results*

*In order to establish our main theorem, we will require the following lemmas.*

*Lemma 2 (see [9]). A function if and only if f can be expressed as follows:
where is the probability measure on .*

*The proof of the following lemma is much akin to that of Theorem 1 which was obtained by Nunokawa et al. [21] (see also Liu [22] and Yang [23]). We, therefore, choose to omit the analogous details involved.*

*Lemma 3. If satisfies the inequality
then .*

*Lemma 4 (Jack's Lemma [24]). Let be a nonconstant regular function in . If attains its maximum value on the circle at , then
where is a real number.*

*We now give the lower bounds of the following continuous linear operators:
acting on the class , which played crucial role in the proof of our main result.*

*Lemma 5. If , then, for , one has the following.(1)When , then
This inequality is sharp.(2)When , then
where . The inequality is sharp.*

*Proof. *By Lemma 2, we know that
is the extreme function of the class . Thus, we only need to consider the function defined by (17); it follows that
We note that (18) can be written as follows:
Thus, we find from (19) that
Moreover, we observe that the function
is convex in , , and maps real axis to real axis; we have
Upon substituting (22) into (20) and expanding the integrand into the power series of and integrating it, we can easily get (15). The sharpness of (15) can be seen from (18).

By similarly applying the method of proof of (15), we also can prove (16) holds true. The sharpness of (16) can be found in (17).

*3. Proof of Theorem 1*

*3. Proof of Theorem 1*

*Proof. *Suppose that with . It follows from (7) that
By noting that
we get
Thus, we can easily find from (15), (23), and (25) that
We now set
Then is analytic in with . It follows from (27) that
At the same time, we can claim that . Indeed, if not, there exists a point such that
by Lemma 4, we obtain
For , by virtue of (28), we split it into two cases to prove the following.(1)When , in view of (16), we get
Let . If satisfies the condition
we have a contradiction to (26) at ; the smallest satisfies (32) is solution of (9). This implies that if and , we have . Thus, we conclude that .(2)When , by means of (15), we have
Let . If satisfies the following inequality,
we have a contradiction to (26) at ; the smallest satisfies (34) is solution of (10). This shows that if and , we have . It follows from (27) that
Therefore, by Lemma 3, we deduce that
The proof of the theorem is thus completed.

*Remark 6. *If (), we cannot find the number such that
since the continuous linear operators and acting on do not exist in sharp upper bounds.

*Putting in the first part of Theorem 1, we can get the following result.*

*Corollary 7. Let . Then for , where is the solution of the following equation:
*

*Remark 8. *Corollary 7 corrects some errors of Theorem 4 in [3].

*By setting in the first part of Theorem 1, we also can get the following criterion for starlikeness obtained by Silverman [10].*

*Corollary 9. Consider for .*

* Conflict of Interests*

*Conflict of Interests*

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

*Acknowledgments*

*Acknowledgments*

*The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008, 11226088, and 11101053, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant 14B110012 of China. The authors are grateful to the referees for their valuable comments and suggestions which essentially improved the quality of this paper.*

*References*

*References*

- P. N. Chichra, “New subclasses of the class of close-to-convex functions,”
*Proceedings of the American Mathematical Society*, vol. 62, no. 1, pp. 37–43, 1977. View at Google Scholar · View at MathSciNet - R. Singh and S. Singh, “Starlikeness and convexity of certain integrals,”
*Annales Universitatis Mariae Curie-Skłodowska A*, vol. 35, pp. 45–47, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-Y. Gao and S.-Q. Zhou, “Certain subclass of starlike functions,”
*Applied Mathematics and Computation*, vol. 187, no. 1, pp. 176–182, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. C. Kim, “Mapping properties of differential inequalities related to univalent functions,”
*Applied Mathematics and Computation*, vol. 187, no. 1, pp. 272–279, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. C. Kim and H. M. Srivastava, “Some applications of a differential subordination,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 22, no. 3, pp. 649–654, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava, N.-E. Xu, and D.-G. Yang, “Inclusion relations and convolution properties of a certain class of analytic functions associated with the Ruscheweyh derivatives,”
*Journal of Mathematical Analysis and Applications*, vol. 331, no. 1, pp. 686–700, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D.-G. Yang and J.-L. Liu, “On a class of analytic functions with missing coefficients,”
*Applied Mathematics and Computation*, vol. 215, no. 9, pp. 3473–3481, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D.-G. Yang and J.-L. Liu, “A class of analytic functions with missing coefficients,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 456729, 16 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Owa, T. Hayami, and K. Kuroki, “Some properties of certain analytic functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2007, Article ID 91592, 9 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Silverman, “A class of bounded starlike functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 17, no. 2, pp. 249–252, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Singh and S. Singh, “Convolution properties of a class of starlike functions,”
*Proceedings of the American Mathematical Society*, vol. 106, no. 1, pp. 145–152, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Nunokawa and J. Sokół, “An improvement of Ozaki's condition,”
*Applied Mathematics and Computation*, vol. 219, no. 22, pp. 10768–10776, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. Nunokawa and J. Sokół, “Remarks on some starlike functions,”
*Journal of Inequalities and Applications*, vol. 2013, article 593, pp. 1–8, 2013. View at Google Scholar - S. Siregar, “The starlikeness of analytic functions of Koebe type,”
*Mathematical and Computer Modelling*, vol. 54, no. 11-12, pp. 2928–2938, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. Sivasubramanian, M. Darus, and R. W. Ibrahim, “On the starlikeness of certain class of analytic functions,”
*Mathematical and Computer Modelling*, vol. 54, no. 1-2, pp. 112–118, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Sokół and M. Nunokawa, “On some sufficient conditions for univalence and starlikeness,”
*Journal of Inequalities and Applications*, vol. 2012, article 282, pp. 1–11, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Srivastava and A. Y. Lashin, “Subordination properties of certain classes of multivalently analytic functions,”
*Mathematical and Computer Modelling*, vol. 52, no. 3-4, pp. 596–602, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z.-G. Wang, Z.-H. Liu, and R.-G. Xiang, “Some criteria for meromorphic multivalent starlike functions,”
*Applied Mathematics and Computation*, vol. 218, no. 3, pp. 1107–1111, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z.-G. Wang, Y. Sun, and N. Xu, “Some properties of certain meromorphic close-to-convex functions,”
*Applied Mathematics Letters*, vol. 25, no. 3, pp. 454–460, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z.-G. Wang, H. M. Srivastava, and S-M. Yuan, “Some basic properties of certain subclasses of meromorphically starlike functions,”
*Journal of Inequalities and Applications*, vol. 2014, article 29, pp. 1–13, 2014. View at Google Scholar - M. Nunokawa, S. Owa, T. Sekine, R. Yamakawa, H. Saitoh, and J. Nishiwaki, “On certain multivalent functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2007, Article ID 72393, 5 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J.-L. Liu, “Remark on two results of Nunokawa,”
*Journal of Mathematical Research and Exposition*, vol. 16, no. 3, pp. 351–354, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D.-G. Yang, “Some criteria for
*p*-valent functions,”*Bulletin of the Korean Mathematical Society*, vol. 35, no. 3, pp. 571–582, 1998. View at Google Scholar · View at MathSciNet - I. S. Jack, “Functions starlike and convex of order $\alpha $,”
*Journal of the London Mathematical Society*, vol. 3, pp. 469–474, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet

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