Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2016, Article ID 1782916, 4 pages
http://dx.doi.org/10.1155/2016/1782916
Research Article

On Subordinations for Certain Multivalent Analytic Functions in the Right-Half Plane

1Department of Mathematics, Suqian College, Suqian 223800, China
2Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Received 2 January 2016; Accepted 16 February 2016

Academic Editor: Gestur Ólafsson

Copyright © 2016 Yi-Hui Xu and Jin-Lin Liu. 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 object of the present paper is to investigate some properties of multivalent analytic functions associated with the lemniscate of Bernoulli.

1. Introduction

Let denote the class of functions of the formwhich are analytic in the open unit disk . A function is said to be starlike of order in if it satisfies

For functions and analytic in , we say that is subordinate to and write , if there exists an analytic function in such that Furthermore, if the function is univalent in , then

Let be the class of functions defined byThus a function if lies in the region bounded by the right-half of the lemniscate of Bernoulli given by . In terms of subordination, the class consists of normalized analytic functions satisfying . This class was introduced by Sokól and Stankiewicz [1]. Several geometric properties had been investigated by Sokól and Stankiewicz [1]. They determined the radius of convexity for functions in the class . They also obtained structural formula, as well as growth and distortion theorems for these functions. Estimates for the first few coefficients of functions in were obtained in [2]. Recently, Sokól [3] determined various radii for functions belonging to the class ; these include the radii of convexity, starlikeness, and strong starlikeness of order . The -radii for certain well-known classes of functions including the Janowski starlike functions were obtained in [4]. General radii problems were also recently considered in [5, 6] wherein certain radii results for the class were obtained as special cases.

Motivated essentially by the above and some recent works [7], the main object of the present paper is to investigate some properties of multivalent analytic functions associated with the lemniscate of Bernoulli.

2. Main Results

The following lemma will be required.

Lemma 1 (see [8]). Let be univalent in the unit disk and let and be analytic in a domain containing with when . Set , . Suppose that(1)either is convex or is a starlike univalent function in ,(2) for . If is analytic in , , and satisfies then , and is the best dominant.

Theorem 2. Let . If satisfiesthen .

Proof. Let . Then is analytic in with . Condition (7) becomes thatDefine the function by with . Since is the right-half of the lemniscate of Bernoulli, is a convex set and hence is a convex function. This shows that the function is starlike with respect to the origin. We now show that if the function satisfies the differential chainFor this purpose, let the functions and be defined by and . Clearly the functions and are analytic in and . Also let and be the functions defined byIt is obvious that is a starlike univalent function in andfor . By the lemma, it follows that . To complete the proof, it is left to show thatSince , it follows thatFor , , clearlyA simple calculation shows that the minimum of the above expression is attained at . Hence Thus or . This shows that and completes the proof.

Theorem 3. Let belong to the class and satisfyThenwhere , , and is the smallest root in of the equation The result is sharp.

Proof. From (16) we can writewhere is analytic and in . Differentiating both sides of (19) logarithmically, we arrive atPut and . Then (19) implies thatWith the help of the Carathéodory inequality it follows from (21) thatwhere , , andbecause of (21). Since is a dual function of , from (23), (24), and (25), we see thatNow let us calculate the minimum value of on the closed interval . From (26) we deduce thatwhere Note that . Suppose that . ThenHence, by virtue of the mathematical induction, we have for all and . This implies thatFurther it follows from (23), (26), and (30) thatwhere Note that and . If we let denote the root in of the equation , then (31) yields the desired result (17).
To see that the bound is the best possible, we consider the functionClearly the function defined by (33) satisfies condition (16). Also it is obvious that, for , which shows that the bound can not be increased. The proof of the theorem is completed.

Setting , Theorem 3 reduces to the following.

Corollary 4. Let belong to the class and satisfy condition (16). Then is starlike of order    in The result is sharp.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (no. 11571299) and the Natural Science Foundation of Jiangsu Province (no. BK20151304). The work is also supported by the Natural Science Foundation of Suqian College (no. 2015KY25).

References

  1. J. Sokól and J. Stankiewicz, “Radius of convexity of some subclasses of strongly starlike functions,” Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka, vol. 19, pp. 101–105, 1996. View at Google Scholar · View at MathSciNet
  2. J. Sokól, “Coefficient estimates in a class of strongly starlike functions,” Kyungpook Mathematical Journal, vol. 49, no. 2, pp. 349–353, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. J. Sokól, “Radius problems in the class SL,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 569–573, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. R. M. Ali, N. K. Jain, and V. Ravichandran, “Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6557–6565, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. R. M. Ali, N. E. Cho, V. Ravichandran, and S. S. Kumar, “Differential subordination for functions associated with the lemniscate of Bernoulli,” Taiwanese Journal of Mathematics, vol. 16, no. 3, pp. 1017–1026, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. M. Ali, N. E. Cho, N. K. Jain, and V. Ravichandran, “Radii of starlikeness and convexity for functions with fixed second coefficient defined by subordination,” Filomat, vol. 26, no. 3, pp. 553–561, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. 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
  8. S. S. Miller and P. T. Mocanu, Differential Subordination, Theory and Application, Marcel Dekker, New York, NY, USA, 2000. View at MathSciNet