Conditions for Carathéodory Functions
© N. E. Cho and I. H. Kim 2009
Received: 12 April 2009
Accepted: 13 October 2009
Published: 15 October 2009
The purpose of the present paper is to derive some sufficient conditions for Carathéodory functions in the open unit disk. Our results include several interesting corollaries as special cases.
respectively. We note that (1.3) and (1.4) can be expressed, equivalently, by the argument functions. The classes and were introduced by Brannan and Kirwan  and studied by Mocanu  and Nunokawa [3, 4]. Also, we note that if , then coincides with , the well-known class of starlike(univalent) functions with respect to origin, and if , then consists only of bounded starlike functions , and hence the inclusion relation is proper. Furthermore, Nunokawa and Thomas  (see also ) found the value such that .
In the present paper, we consider general forms which cover the results by Mocanu  and Nunokawa and Thomas . An application of a certain integral operator is also considered. Moreover, we give some sufficient conditions for univalent (close-to-convex) and (strongly) starlike functions (of order ) as special cases of main results.
2. Main Results
To prove our results, we need the following lemma due to Nunokawa .
With the help of Lemma 2.1, we now derive the following theorem.
in Theorem 2.2, we can see that (2.4) is satisfied. Therefore, the result follows from Theorem 2.2.
By a similar method of the proof in Theorem 2.2, we have the following theorem.
in Theorem 2.5, we have Corollary 2.6 immediately.
If we combine Corollaries 2.4 and 2.6, then we obtain the following result obtained by Nunokawa and Thomas .
the conclusion of Corollary 2.8 immediately follows.
Letting in Corollary 2.8, we have the result obtained by Miller and Mocanu .
The proof of the following theorem below is much akin to that of Theorem 2.2 and so we omit for details involved.
in Theorem 2.10. Then (2.36) is satisfied and so the result follows.
By applying Theorem 2.10, we have the following result obtained by Mocanu .
which completes the proof of Corollary 2.12.
Finally, we have the following result.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2009-0066192).
- Brannan DA, Kirwan WE: On some classes of bounded univalent functions. Journal of the London Mathematical Society 1969, 1: 431–443.MathSciNetView ArticleMATHGoogle Scholar
- Mocanu PT: On strongly-starlike and strongly-convex functions. Studia Universitatis Babes-Bolyai—Series Mathematica 1986,31(4):16–21.MathSciNetMATHGoogle Scholar
- Nunokawa M: On the order of strongly starlikeness of strongly convex functions. Proceedings of the Japan Academy, Series A 1993,69(7):234–237. 10.3792/pjaa.69.234MathSciNetView ArticleMATHGoogle Scholar
- Nunokawa M, Thomas DK: On convex and starlike functions in a sector. Journal of the Australian Mathematical Society (Series A) 1996,60(3):363–368. 10.1017/S1446788700037873MathSciNetView ArticleMATHGoogle Scholar
- Mocanu PT: Alpha-convex integral operator and strongly-starlike functions. Studia Universitatis Babes-Bolyai—Series Mathematica 1989,34(2):19–24.MathSciNetMATHGoogle Scholar
- Mocanu PT: Some starlikeness conditions for analytic functions. Revue Roumaine de Mathématiques Pures et Appliquées 1988,33(1–2):117–124.MathSciNetMATHGoogle Scholar
- Miller SS, Mocanu PT: Univalent solutions of Briot-Bouquet differential equations. Journal of Differential Equations 1985,56(3):297–309. 10.1016/0022-0396(85)90082-8MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.