Abstract and Applied Analysis

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

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

Research Article

On the Starlikeness of Certain Class of Multivalent Analytic Functions

Lei Shi and Zhi-Gang Wang

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

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

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

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

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.

R. Singh and S. Singh, “Starlikeness and convexity of certain integrals,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 35, pp. 45–47, 1981.

C.-Y. Gao and S.-Q. Zhou, “Certain subclass of starlike functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 176–182, 2007.

Y. C. Kim, “Mapping properties of differential inequalities related to univalent functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 272–279, 2007.

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.

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.

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.

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.

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.

H. Silverman, “A class of bounded starlike functions,” International Journal of Mathematics and Mathematical Sciences, vol. 17, no. 2, pp. 249–252, 1994.

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.

M. Nunokawa and J. Sokół, “An improvement of Ozaki's condition,” Applied Mathematics and Computation, vol. 219, no. 22, pp. 10768–10776, 2013.

M. Nunokawa and J. Sokół, “Remarks on some starlike functions,” Journal of Inequalities and Applications, vol. 2013, article 593, pp. 1–8, 2013.

S. Siregar, “The starlikeness of analytic functions of Koebe type,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2928–2938, 2011.

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.

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.

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.

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.

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.

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.

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.

J.-L. Liu, “Remark on two results of Nunokawa,” Journal of Mathematical Research and Exposition, vol. 16, no. 3, pp. 351–354, 1996.

D.-G. Yang, “Some criteria for p-valent functions,” Bulletin of the Korean Mathematical Society, vol. 35, no. 3, pp. 571–582, 1998.

I. S. Jack, “Functions starlike and convex of order ,” Journal of the London Mathematical Society, vol. 3, pp. 469–474, 1971.