- Open Access
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
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 .
- Nunokawa’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 
If is analytic and in the unit disc and if the function F is given in (1.1), then .
where F is given by (1.1).
On the order of starlikeness of convex functions was considered also in the recent paper .
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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