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Properties of convolutions for hypergeometric series with univalent functions
Advances in Difference Equations volume 2013, Article number: 101 (2013)
Abstract
The purpose of the present paper is to investigate various mapping and inclusion properties involving subclasses of analytic and univalent functions for a linear operator defined by means of Hadamard product (or convolution) with the Gaussian hypergeometric function.
MSC:30C45, 30C55, 33C20.
1 Introduction
Let denote the class of functions of the form
which are analytic in the open unit disk . Denote by the class of all functions in which are univalent in .
A function is said to be in the class if
Clearly, a function f belongs to if and only if there exists a function w regular in satisfying and () such that
The class was introduced by Dixit and Pal [1]. By giving specific values to t, A and B in (1.2), we obtain the following subclasses studied by various researchers in earlier works:
-
(i)
For (), () and , we obtain the class of functions f satisfying the condition:
(1.4)
In this case, the class is equivalent to the class which is studied by Ponnusamy and Rønning [2]. Here, is the class of functions satisfying the condition:
-
(ii)
For (), we obtain the class of functions satisfying the condition
which was studied by Dashrath [3].
-
(iii)
For , and (), we obtain the class of functions f satisfying the condition:
which was studied by Caplinger and Cauchy [4] and Padmanabhan [5].
Let and denote the subclasses of consisting of starlike and convex functions of order α () in , respectively. It is well known that , and . For , define
and
It is a known fact that a sufficient condition for of the form (1.1) to belong to the class is that . A simple extension of this result is the following [6]:
For , this was previously proved by Schild [2]. Since if and only if , we have a corresponding results for ,
Now we introduce the class (resp., UCV) of uniformly starlike (resp., convex) functions. We say [7, 8] that is in (resp., ) if for each and each circular arc γ in with center η, the image arc is starlike with respect to (resp., is a convex curve).
In this paper, we 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 [4]) that the function is bounded if , and has a pole at if . Moreover, univalence, starlikeness and convexity properties of have been studied extensively in Ponnusamy and Vuorinen [9] and Ruscheweyh and Singh [10].
For , we define the operator by
where ∗ denotes the usual Hadamard product (or convolution) of power series. If f equals to the convex function , then the operator becomes . For a survey of special cases of this operator and also more general operators, we can refer to the article by Srivastava [11–13] and Swaminathan [14], where also a long list of other references can be found. Thus, the operator and hence the Gaussian hypergeometric function is a natural object for studying inclusion properties related to the convolution product. In the present paper, we find a condition for univalency of the operator . We also investigate conditions such that (, , and ), whenever .
2 A set of lemmas
Now we introduce several lemmas which are needed for the proof of our main results.
Lemma 2.1 [1]
Let a function f of the form (1.1) be in . Then
The result is sharp for the function
Lemma 2.2 [1]
Let a function f of the form (1.1) be in . If
then . The result is sharp for the function
Lemma 2.3 [15]
Let be regular in the unit disk with . Then, if attains a maximum value on the circle () at a point z, we can write
where m is real and .
Lemma 2.4 [2]
-
(i)
For and with ,
-
(ii)
For with and and ,
Lemma 2.5 [16]
A function f of the form (1.1) is in if
Lemma 2.6 [16]
A function f of the form (1.1) is in if
3 Main results
Theorem 3.1 Let . If
then is univalent in .
Proof
We note that
in . Define w by
for . Then it follows that w is analytic in with . By (3.1),
Suppose that there exists a point such that
Then, by Lemma 2.3, we can put
Therefore, we obtain
which contradicts the condition (3.2). This shows that
which implies that for . Therefore, by the Noshiro-Warschawski theorem [17], is univalent in . □
Theorem 3.2 Let and . Suppose that and satisfy the condition
Then the operator maps into .
Proof Let , and suppose that . Then, by Lemma 2.2, it suffices to show that
where
From Lemma 2.1 and the fact that , we have
Using the formula (1.7) and the assumption, we find that
which implies that the operator maps into .
If, in the proof of Theorem 3.2, we take , then we have the following theorem under a weaker condition on the parameter c. □
Theorem 3.3 Let and . Suppose that and satisfy the condition
Then the operator maps into .
Proof The proof of Theorem 3.3 follows in the similar lines on the proof of Theorem 3.2 and so we omit the details. □
Theorem 3.4 Let and . Suppose that , , and such that and satisfy the condition
Then the operator maps into .
Proof Let and with , and . Suppose that . Then, by (1.5), it is sufficient to show that
By using Lemma 2.1 and (i) of Lemma 2.4, we observe that
by (3.5), which completes the proof of Theorem 3.4. □
Taking and in Theorem 3.4, we have the following result.
Corollary 3.1 Let and . Suppose that and satisfy the condition
Then .
By using the same method as in the proof of Theorem 3.4, we have the following result.
Theorem 3.5 Let and , and . Suppose that
Then the operator maps into .
Proof Let and . Suppose . To show that the operator belongs to , from (1.6), it is enough to show that
From Lemma 2.1 and (1.7), we find that
by (3.6) and the conclusion follows. □
Similarly, taking and in Theorem 3.5, we have the following result.
Corollary 3.2 Let , and . Suppose that
Then .
By using Lemma 2.5 and Lemma 2.6, we have the following theorem for and .
Theorem 3.6 Let , and . Suppose that
Then the operator maps into .
Proof Let and . Suppose that . By Lemma 2.5, we need only to show that
Then, from (1.7) and , we have
by (3.7), and so we have Theorem 3.6. □
Theorem 3.7 Let , with , , and and . Suppose that
Then the operator maps into .
Proof Let and with , and . Suppose that . By Lemma 2.6, it suffices to show that
Then, from (1.7) and , we have
by (3.8), which completes the proof of Theorem 3.7. □
Next, we give the condition on the parameters a, b and c that the convolution of the odd function and belongs to .
Theorem 3.8 Let , with and and . Suppose that
Then the operator .
Proof Let and with , . Suppose that . We note that
By Lemma 2.2, it is enough to show that
Then, by a similar proof as Theorem 3.7, we get
by (3.9), and hence we have the result. □
Finally, we establish the condition on the parameters a, b and c that the function belongs to the class .
Theorem 3.9 Let and . Suppose that
Then the function .
Proof
By Lemma 2.2, it is sufficient to show that
Then, by (ii) of Lemma 2.1, we observe that
by (3.10). This completes the proof of Theorem 3.9. □
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors would like to express their thanks to the editor and the referees for many valuable advices regarding a previous version of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).
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Kim, J.A., Cho, N.E. Properties of convolutions for hypergeometric series with univalent functions. Adv Differ Equ 2013, 101 (2013). https://doi.org/10.1186/1687-1847-2013-101
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DOI: https://doi.org/10.1186/1687-1847-2013-101