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Conditions for starlikeness of the Libera operator
Journal of Inequalities and Applications volume 2014, Article number: 135 (2014)
Abstract
Let denote the class of functions f that are analytic in the unit disc and normalized by . In this paper some conditions are determined for starlikeness of the Libera integral operator .
MSC:30C45, 30C80.
1 Introduction
Let ℋ be the class of functions analytic in the unit disk , and let us denote by the class of functions with the normalization of the form
with .
Let denote the class of strongly starlike functions of order β, ,
which was introduced in [1] and [2], and is the well-known class of starlike functions in . Functions in the class
where are called functions with bounded turning. The Libera transform , , where
is the Libera integral operator, which has been studied by several authors on different counts. In [3] Mocanu considered the problem of starlikeness of F and proved the following result.
Theorem 1.1 [3]
If is analytic and in the unit disc and if the function F is given in (1.1), then .
This result may be written briefly as follows:
where . In 1995 Mocanu [4] improved (1.2) by showing that
In 2002 Miller and Mocanu [5] showed that a subcase of this last result can be sharpened to
The problem of strongly starlikeness of for was consider also in [6] where it is shown that
The above inclusion relationship is equivalent to the following differential implication:
or equivalently
where F is given by (1.1).
In [7] Ponnusamy improved (1.2) by showing that
On the order of starlikeness of convex functions was considered also in the recent paper [8].
2 Main result
In this paper we go back to the problem of starlikeness of Libera transform. We need the following lemmas.
Lemma 2.1 [[9], p.73]
Let n be a positive integer, and let be the positive root of the equation
In addition, let
for . If is analytic in , then
implies the following subordination:
If in Lemma 2.1 we put , , then the solution of (2.1) satisfies , so we may take , which gives .
Corollary 2.2 Assume that . If
then
Note that if , then a sufficient condition for is ; see [[5], p.96].
Lemma 2.3 [10]
Let be of the form
with in . If there exists a point , , such that
for some , then we have
where
and
where
Let be analytic in the unit disc . If
then
and
where lies between 0.911621904 and 0.911621907.
Theorem 2.5 Let be analytic in and suppose that
for some . If is analytic and in with and such that
then we have
Proof If there exists a point , , for which
and
then from Nunokawa’s Lemma 2.3, we have
where
and
For the case , we have
where , and
Let us put
then it is easy to see that
Putting
we have
because . Therefore, for we get , so from (2.11) we have
and so
Therefore, we have the following inequality from (2.10):
This contradicts the hypothesis and for the case , applying the same method as above, we also have (2.12). This is also a contradiction and it completes the proof. □
Corollary 2.6 Assume that
and
for some , where is given in (1.1). Then we have
hence is strongly starlike of order β.
Proof If we set
then
If we let , then by (2.14) and (2.15) the assumptions of Theorem 2.5 are satisfied. Therefore,
□
Theorem 2.7 Let be analytic in , with and satisfy
If is analytic in , with and if
where is given in (2.8), then we have
Proof By Lemma 2.4, we have
If there exists a point , , such that
and
then from Nunokawa’s Lemma 2.3, we have
where
and
For the case , we have
Therefore, for the case , we have
Moreover, by (2.16)
Therefore, we can write
This contradicts the hypothesis and for the case , applying the same method as above, we have
This is also a contradiction and it completes the proof. □
Corollary 2.8 Assume that
then we have
where is Libera integral given in (1.1).
Proof Because
by Corollary 2.2 and by (2.18) we obtain
Therefore, if we let , then
If we set
then
The assumptions of Theorem 2.7 are satisfied. Therefore, (2.19) holds. □
Corollary 2.8 is an extension of Mocanu’s result (1.2) from the paper [3] because in (2.18) we have , while in (1.2) we have the stronger assumption that .
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Acknowledgements
The authors would like to express their thanks to the referees for valuable advice 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. 2011-0007037).
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Nunokawa, M., Sokół, J., Cho, N.E. et al. Conditions for starlikeness of the Libera operator. J Inequal Appl 2014, 135 (2014). https://doi.org/10.1186/1029-242X-2014-135
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DOI: https://doi.org/10.1186/1029-242X-2014-135