- Open Access
New extensions concerned with results by Ponnusamy and Karunakaran
© Nunokawa et al.; licensee Springer 2013
- Received: 11 January 2013
- Accepted: 22 April 2013
- Published: 9 May 2013
A subclass of analytic functions in the open unit disk is introduced. By means of the result due to Fukui and Sakaguchi (Bull. Fac. Edu. Wakayama Univ. Natur. Sci. 30:1-3, 1980), some interesting properties of in concerned with Ponnusamy and Karunakaran (Complex Var. Theory Appl. 11:79-86, 1989) are discussed.
- Jack’s lemma
which are analytic in the open unit disk . For two functions and belonging to the class , Sakaguchi  proved the following result.
It is the purpose of the present paper to discuss Theorem B applying the lemma due to Fukui and Sakaguchi . To discuss our problems, we need the following lemmas.
which shows that is a positive real number.
This completes the proof of Lemma 2. □
With the help of Lemma 2, we derive the following theorem.
which contradicts (2.5). This completes the proof of the theorem. □
Remark 1 If and satisfy in Theorem 1, then Theorem 1 becomes Theorem B given by Ponnusamy and Karunakaran . We also have the following theorem.
which contradicts our condition (2.7).
which contradicts (2.7). □
Dedicated to Professor Hari M Srivastava.
- Sakaguchi K: On a certain univalent mapping. J. Math. Soc. Jpn. 1959, 11: 72-75. 10.2969/jmsj/01110072MathSciNetView ArticleGoogle Scholar
- Srivastava HM, Owa S (Eds): Current Topics in Analytic Function Theory. World Scientific, Singapore; 1992.Google Scholar
- Ponnusamy S, Karunakaran V: Differential subordination and conformal mappings. Complex Var. Theory Appl. 1989, 11: 79-86. 10.1080/17476938908814326MathSciNetView ArticleGoogle Scholar
- Fukui S, Sakaguchi K: An extension of a theorem of S. Ruscheweyh. Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 1980, 30: 1-3.MathSciNetGoogle Scholar
- Jack IS: Functions starlike and convex of order α . J. Lond. Math. Soc. 1971, 2: 469-474.MathSciNetView ArticleGoogle Scholar
- Nunokawa M: On properties of non-Carathéodory functions. Proc. Jpn. Acad. 1992, 68: 152-153. 10.3792/pjaa.68.152MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.