Research Article | Open Access
R. Ezhilarasi, T. V. Sudharsan, Maisarah Haji Mohd, K. G. Subramanian, "Connections between Certain Subclasses of Analytic Univalent Functions Based on Operators", Journal of Complex Analysis, vol. 2017, Article ID 6104210, 5 pages, 2017. https://doi.org/10.1155/2017/6104210
Connections between Certain Subclasses of Analytic Univalent Functions Based on Operators
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.
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., ) of starlike functions of the form (1), satisfying Re. Another class , introduced in , consists of functions satisfying Re Various properties of this class have been obtained in [2–4]. A related class has been recently considered in , initially introduced in . A function of the form (1) is said to be in the class if
Ponnusamy and Ronning  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  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. , 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  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.
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 .
Theorem 5. Let . Further let be a real number such that , . If and the inequality is satisfied, then
3. An Integral Operator
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.
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.
The author Maisarah Haji Mohd is supported by USM Short Term Grant 304/PMATHS/6313192.
- 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.
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