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Properties for certain subclasses of analytic functions with nonzero coefficients
Journal of Inequalities and Applications volume 2014, Article number: 506 (2014)
Abstract
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.
MSC:30C45.
1 Introduction
Let denote the class of functions of the form
which are analytic in the open unit disk . We also denote by the class of all functions in which are univalent in .
A function is said to be in the class if
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 [1]. 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]).
A function is said to be in the class if
Furthermore, a function is said to be in the class if
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 [9] and Rønning [10].
Let us consider the Gaussian hypergeometric function defined by
where is the Pochhammer symbol (or the shifted factorial) defined (in terms of the gamma function) by
We note that and
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 [13] and Ruscheweyh and Singh [14].
For , we define the operator by
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 [15] 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
Theorem 1 Let a function f of the form (1.1) be in the class with (), then
Proof From the definition of , we have
and so
If we take , then we see that
Then, by using (2.3) to (2.2), we have
or, equivalently,
Letting in (2.4), we have the inequality (2.1). Thus we complete the proof of Theorem 1. □
Theorem 2 Let a function f of the form (1.1) be in the class . If
where A and B are complex numbers with , , and α is a positive real number, then . The result is sharp for the function defined by
Proof In view of the definition of , it suffices to prove that
From (2.6), we have
Hence it is sufficient to show that
which is equivalent to the relation
Letting in (2.7), we have
Therefore, by the assumption (2.5), we prove that . □
Theorem 3 Let a function f of the form (1.1) be in the class . If
then . The result is sharp for the function
Proof It suffices to show that
We have
which is bounded by if the assumption (2.8) is satisfied. Thus we complete the proof of Theorem 3. □
Theorem 4 Let a function f of the form (1.1) be in the class . If
then . The result is sharp for the function
Proof It suffices to show that
We have
which is bounded by if the assumption (2.9) is satisfied. Thus we complete the proof of Theorem 4. □
Theorem 5 Let and . If with , , and
then , where the operator is defined by (1.3).
Proof We want to prove from Theorem 2 that
where
Since, from Theorem 1,
which completes the proof of Theorem 5. □
We introduce the following lemma which is needed for the proof of the next theorem.
Lemma 1 [16]
Let ω be regular in the unit disk with . Then, if attains a maximum value on the circle () at a point , we can write
Theorem 6 Let a function f of the form (1.1) be in the class . Assume
for some real α (), , and . Then
Proof Let us define ω by
Then it follows that ω is analytic in with . By (2.10), we have
Suppose that there exists a point such that
Then, by Lemma 1, we can put
Hence, we obtain
which contradicts the condition (2.12). This shows that
Thus we complete the proof of Theorem 6. □
Remark 1 The condition (2.11) in Theorem 6 implies that
Therefore, by the Noshiro-Warschawski theorem [17], the operator is univalent in under the restrictions of Theorem 6.
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Acknowledgements
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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Lee, H.J., Cho, N.E. & Owa, S. Properties for certain subclasses of analytic functions with nonzero coefficients. J Inequal Appl 2014, 506 (2014). https://doi.org/10.1186/1029-242X-2014-506
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DOI: https://doi.org/10.1186/1029-242X-2014-506
Keywords
- univalent function
- uniformly starlike function
- uniformly convex function
- Gaussian hypergeometric function
- Hadamard product