- Open Access
On differential subordinations in the complex plane
© Nunokawa et al.; licensee Springer 2015
- Received: 3 November 2014
- Accepted: 17 December 2014
- Published: 9 January 2015
The purpose of this work is to present a new approach to solve some problems in differential subordination theory. We also discuss the new results closely related to the generalized Briot-Bouquet differential subordination.
- analytic functions
- convex functions
- convex of order alpha
- univalent functions
- differential subordination
The purpose of the present paper is to investigate interesting new results in connection with differential subordination and to improve some results obtained by Miller and Mocanu . Also we remark that the reader may refer to the recent results obtained by Sokół and Nunokawa  as applications of differential subordination.
The following lemma will be required in our present investigation.
The authors would like to express their thanks to the referees for their 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).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
- Study, E: Konforme Abbildung einfach-zusammenhängender Bereiche. Teubner, Leipzig (1913) MATHGoogle Scholar
- Robertson, MS: On the theory of univalent functions. Ann. Math. 37, 374-408 (1936) View ArticleGoogle Scholar
- Miller, SS, Mocanu, PT: Differential subordinations and univalent functions. Mich. Math. J. 28, 151-171 (1981) MathSciNetGoogle Scholar
- Miller, SS, Mocanu, PT: Differential Subordinations: Theory and Applications. Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 225. Dekker, New York (2000) Google Scholar
- Miller, SS, Mocanu, PT: Differential subordination and inequalities in the complex plane. J. Differ. Equ. 67(2), 199-211 (1987) View ArticleMATHMathSciNetGoogle Scholar
- Sokół, J, Nunokawa, M: On the subordination under Bernardi operator. Proc. Jpn. Acad., Ser. A, Math. Sci. 89, 11-14 (2013) View ArticleMATHGoogle Scholar
- Nunokawa, M: On properties of non-Carathéodory functions. Proc. Jpn. Acad., Ser. A, Math. Sci. 68(6), 152-153 (1992) View ArticleMATHMathSciNetGoogle Scholar
- Nunokawa, M: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad., Ser. A, Math. Sci. 69(7), 234-237 (1993) View ArticleMATHMathSciNetGoogle Scholar