- Open Access
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.
- univalent function
- uniformly starlike function
- uniformly convex function
- Gaussian hypergeometric function
- Hadamard product
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).
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).
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