Conditions for starlikeness of the Libera operator
© Nunokawa et al.; licensee Springer. 2014
Received: 31 December 2013
Accepted: 7 March 2014
Published: 31 March 2014
KeywordsNunokawa’s lemma starlike functions strongly starlike functions
is the Libera integral operator, which has been studied by several authors on different counts. In  Mocanu considered the problem of starlikeness of F and proved the following result.
Theorem 1.1 
where F is given by (1.1).
On the order of starlikeness of convex functions was considered also in the recent paper .
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 [, p.73]
If in Lemma 2.1 we put , , then the solution of (2.1) satisfies , so we may take , which gives .
Note that if , then a sufficient condition for is ; see [, p.96].
Lemma 2.3 
where lies between 0.911621904 and 0.911621907.
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. □
hence is strongly starlike of order β.
This is also a contradiction and it completes the proof. □
where is Libera integral given in (1.1).
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  because in (2.18) we have , while in (1.2) we have the stronger assumption that .
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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