On some differential inequalities in the unit disk with applications
© Attiya; licensee Springer. 2014
Received: 13 December 2013
Accepted: 6 January 2014
Published: 24 January 2014
In this paper we obtain a number of interesting relations associated with some differential inequalities in the open unit disk, . Some applications of the main results are also obtained.
Definition 1.1 Let and be analytic functions. The function is said to be subordinate to , written , if there exists a function analytic in , with and , and such that . If is univalent, then if and only if and .
and for . Further, let .
In this paper we obtain some interesting relations associated with some differential inequalities in . These relations extend and generalize the Carathéodory functions in which have been studied by many authors e.g., see [1–14].
2 Main results
To prove our results, we need the following lemma due to Miller and Mocanu [, p.24].
such that and , .
where E is defined by (2.2). This is in contradiction to (2.1). Then we obtain . □
such that and , .
This is in contradiction to (2.6). Then we obtain . □
3 Applications and examples
Putting (; real) in Theorem 2.1 we have the following corollary.
Putting in Corollary 3.1, we obtain the following corollary.
where () and E is defined by (2.2) with .
Since , which gives , therefore, . This completes the proof of the corollary. □
Putting () and in Theorem 2.1, we have Theorem 1 due to Kim and Cho .
Putting (), (; real) and in Theorem 2.1, we have Corollary 1 due to Kim and Cho .
Putting and in Theorem 2.1, we have the result due to Nunokawa et al. .
Putting (), and in Theorem 2.1, we have Corollary 2 due to Kim and Cho .
Putting in Theorem 2.2, we have the following corollary.
The author would like to express his gratitude to the referee(s) for the valuable advices to improve this paper.
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