Journal of Complex Analysis
Volume 2017 (2017), Article ID 6104210, 5 pages
https://doi.org/10.1155/2017/6104210

Research Article

Connections between Certain Subclasses of Analytic Univalent Functions Based on Operators

R. Ezhilarasi,¹ T. V. Sudharsan,¹ Maisarah Haji Mohd,² and K. G. Subramanian³

¹Department of Mathematics, SIVET College, Chennai 600 073, India
²School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia
³Department of Mathematics, Madras Christian College, Chennai 600 059, India

Correspondence should be addressed to K. G. Subramanian; kgsmani1948@gmail.com

Received 19 August 2017; Accepted 17 October 2017; Published 15 November 2017

Academic Editor: Arcadii Z. Grinshpan

Copyright © 2017 R. Ezhilarasi 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.

In this paper, by applying the Hohlov linear operator, connections between the class ,  , and two subclasses of the class of normalized analytic functions are established. Also an integral operator related to hypergeometric function is considered.

1. Introduction

Let denote the family of functions that are analytic in the open unit disk with the normalization Let denote the subclass of functions in which are also univalent in . A well-known subclass of is the class ST (see, e.g., [1]) of starlike functions of the form (1), satisfying Re. Another class , introduced in [2], consists of functions satisfying Re Various properties of this class have been obtained in [2–4]. A related class has been recently considered in [5], initially introduced in [6]. A function of the form (1) is said to be in the class if

Theorem 1 (see [5]). A function of the form (1) is in the class if

Ponnusamy and Ronning [7] introduced and studied the class ,   of functions for which there exists a number such that . If the function of the form (1) belongs to the class , then For complex numbers ,  , and , the Gaussian hypergeometric function is defined by where is the Pochhammer symbol given by It is known that The Hadamard product (or convolution) of two functions defined by (1) and given by is defined by Hohlov [8] introduced a linear operator , corresponding to the Gaussian hypergeometric function which is defined by the convolution For a function of the form (1), we have The operator is a natural extension of several operators such as Alexander, Libera, Bernardi, and Carlson-Shaffer operators denoted, respectively, by ,  ,  , and .

Motivated by the work of Thulasiram et al. [9], in this paper, by applying the linear operator , we establish some interesting connections between the class and the classes ST and consisting of functions given by (1). Also we consider an integral operator related to the hypergeometric functions.

2. Main Results

In the sequel the function is given by (1).

Theorem 2. Let . Also let be a real number such that . If and if the inequality is satisfied, then

Proof. Let . Applying the well-known estimate due to Nevanlinna [10] for the coefficients of the functions , in view of Theorem 1, we need to prove that By virtue of the relation , and on writing and and using the fact that , we havewhich is satisfied by the hypothesis.

On setting , an improvement of the assertion of Theorem 2 is obtained as given in Theorem 3.

Theorem 3. Let . Also let be a real number such that . If , and if the inequality is satisfied, then

Theorem 4. Let . Also, let be a real number such that ,,. If and the inequality is satisfied, then .

Proof. Let be of the form (1) and let . By virtue of Theorem 1 and in view of (10), it remains to show that Using the inequality (4) and the relations and , we obtain that

On taking , an improvement of the assertion of Theorem 4 is obtained as given in Theorem 5.

Theorem 5. Let . Further let be a real number such that ,  . If and the inequality is satisfied, then

3. An Integral Operator

We now obtain results in connection with a particular integral operator [11] defined by where is given by (5).

Theorem 6. Let . Also let c be a real number such that . Let be given by (19). If the hypergeometric inequality is satisfied, then .

Proof. The function has the series representation given by In view of Theorem 1, it is enough to prove that Nowby hypothesis.

A result analogous to Theorem 6 can be stated for the class in Theorem 7.

Theorem 7. Let . Also let c be a real number such that . Let and be of the form (1). If the hypergeometric inequality is satisfied, then .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author Maisarah Haji Mohd is supported by USM Short Term Grant 304/PMATHS/6313192.

References

P. L. Duren, Univalent Functions, Springer, Berlin, Germany, 1983.

T. Rosy, B. A. Stephen, K. G. Subramanian, and H. Silverman, “Classes of convex functions,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 12, pp. 819–825, 2000.

D. Breaz, “Integral Operators on the UCD(β) class,” in Proceedings of the International conference on Theory and Applications of Mathematics and informatics (ICTAMI '03, pp. 61–65, Alba Lulia, Romania, 2003.

S. Sivasubramanian and J. Sokół, “Hypergeometric transforms in certain classes of analytic functions,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 3076–3082, 2011.

S. S. Varma and T. Rosy, “Certain properties of a subclass of univalent functions with finitely many fixed coefficients,” Khayyam Journal of Mathematics, vol. 53, no. 1, pp. 26–33, 2017.

T. Rosy, Studies on subclasses of Starlike and Convex Functions [Ph.D. thesis], University of Madras, Chennai, India, 2001.

S. Ponnusamy and F. Ronning, “Duality for Hadamard products applied to certain integral transforms,” Complex Variables, Theory and Application, vol. 32, no. 3, pp. 263–287, 1997.

Y. E. Hohlov, “Operators and operations on the class of univalent functions,” Soviet Mathematics, vol. 22, no. 10, pp. 64–69, 1978.

T. Thulasiram, K. Suchithra, T. V. Sudharsan, and G. Murugusundaramoorthy, “Some inclusion results associated with certain subclass of analytic functions inolving Hohlov operator,” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, vol. 108, no. 2, pp. 711–720, 2014.

R. Nevanlinna, “Eindeutige Analytische Funktionen,” in Översikt av Finska Vetenskaps-Soc. Förh, vol. 63, pp. 1–21, 6 edition, 1921.

H. Silverman, “Starlike and convexity properties for hypergeometric functions,” Journal of Mathematical Analysis and Applications, vol. 172, no. 2, pp. 574–581, 1993.