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.
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. □
2 Main results
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.
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