Properties for certain subclasses of analytic functions with nonzero coefficients
© Lee et al.; licensee Springer. 2014
Received: 18 July 2014
Accepted: 2 December 2014
Published: 12 December 2014
In the present paper, we obtain some mapping and inclusion properties for subclasses of analytic functions by using a linear operator defined by the Gaussian hypergeometric function.
which are analytic in the open unit disk . We also denote by the class of all functions in which are univalent in .
where A and B are complex numbers with , , and α is a positive real number.
In particular, for some real numbers A and B with and without any restriction of the coefficients () the class was introduced by Dixit and Pal . Moreover, by giving specific values t, A, B, and α in (1.2), we obtain subclasses studied by various researchers in earlier works (see [2–6]).
The classes and are introduced by Goodman [7, 8] (they are called the classes of uniformly starlike and uniformly convex functions, respectively). The classes of uniformly starlike and uniformly convex functions have been extensively studied by Ma and Minda  and Rønning .
We also recall (see [11, 12]) that the function is bounded if , and it has a pole at if . Moreover, univalence, starlikeness and convexity properties of have been extensively studied by Ponnusamy and Vuorinen  and Ruscheweyh and Singh .
where ∗ denotes the usual Hadamard product (or convolution) of power series. For a special case of the operator , we can refer to the paper by Swaminathan  and the references cited therein.
In this paper, we obtain a necessary condition for the class and sufficient conditions for the classes , , and , respectively. Moreover, we find a condition for univalency of the operator defined by (1.3).
2 Main result
Letting in (2.4), we have the inequality (2.1). Thus we complete the proof of Theorem 1. □
Therefore, by the assumption (2.5), we prove that . □
which is bounded by if the assumption (2.8) is satisfied. Thus we complete the proof of Theorem 3. □
which is bounded by if the assumption (2.9) is satisfied. Thus we complete the proof of Theorem 4. □
then , where the operator is defined by (1.3).
which completes the proof of Theorem 5. □
We introduce the following lemma which is needed for the proof of the next theorem.
Lemma 1 
Thus we complete the proof of Theorem 6. □
Therefore, by the Noshiro-Warschawski theorem , the operator is univalent in under the restrictions of Theorem 6.
The authors would like to express their thanks to the referees for many valuable advices regarding a previous version of this paper. This research was supported for the second author by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007037).
- Dixit KK, Pal SK: On a class of univalent functions related to complex order. Indian J. Pure Appl. Math. 1995,26(9):889-896.MathSciNetGoogle Scholar
- Caplinger TR, Causey WM: A class of univalent functions. Proc. Am. Math. Soc. 1973, 39: 357-361. 10.1090/S0002-9939-1973-0320294-4MathSciNetView ArticleGoogle Scholar
- Kim JA, Cho NE: Properties of convolutions for hypergeometric series with univalent functions. Adv. Differ. Equ. 2013. Article ID 101, 2013: Article ID 101Google Scholar
- Dashrath: On some classes related to spiral-like univalent and multivalent functions. PhD thesis, Kanpur University, Kanpur (1984)Google Scholar
- Padmanabhan, KS: On a certain class of functions whose derivatives have a positive real part in the unit disc. Ann. Pol. Math. 23, 73-81 (1970/1971)Google Scholar
- Ponnusamy S, Rønning F: Starlikeness properties for convolutions involving hypergeometric series. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 1998, 52: 141-155.Google Scholar
- Goodman AW: On uniformly convex functions. Ann. Pol. Math. 1991, 56: 87-92.Google Scholar
- Goodman AW: On uniformly starlike functions. J. Math. Anal. Appl. 1991, 155: 364-370. 10.1016/0022-247X(91)90006-LMathSciNetView ArticleGoogle Scholar
- Ma W, Minda D: Uniformly convex functions. Ann. Pol. Math. 1992, 57: 166-175.MathSciNetGoogle Scholar
- Rønning F: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 1993, 118: 190-196.View ArticleGoogle Scholar
- Owa S, Srivastava HM: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 1987, 39: 1057-1077. 10.4153/CJM-1987-054-3MathSciNetView ArticleGoogle Scholar
- Whitteaker ET, Watson GN: A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and Analytic Functions: With an Account of the Principal Transcendental Functions. 4th edition. Cambridge University Press, Cambridge; 1927.Google Scholar
- Ponnusamy, S, Vuorinen, M: Univalence and convexity properties for Gaussian hypergeometric functions. Preprint 82, Department of Mathematics, University of Helsinki (1995)Google Scholar
- Ruscheweyh S, Singh V: On the order of starlikeness of hypergeometric functions. J. Math. Anal. Appl. 1986, 113: 1-11. 10.1016/0022-247X(86)90329-XMathSciNetView ArticleGoogle Scholar
- Swaminathan A: Certain sufficiency conditions on Gaussian hypergeometric functions. J. Inequal. Pure Appl. Math. 2004. Article ID 83, 5: Article ID 83Google Scholar
- Jack IS: Functions starlike and convex of order α . J. Lond. Math. Soc. 1971, 3: 469-474.MathSciNetView ArticleGoogle Scholar
- Duren PL: Univalent Functions. Springer, New York; 1983.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.