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Argument Property for Certain Analytic Functions
Let P be the class of functions p(z) of the form which are analytic in the open unit disk . The object of the present paper is to derive certain argument inequalities of analytic functions p(z) in P.
Let be the class of functions of the form which are analytic in the open unit disk . For functions and in the class , we say that is subordinate to if there exists an analytic function in with , , and such that . We denote this subordination by If is univalent in , then this subordination is equivalent to and .
Recently, several authors investigated various argument properties of analytic functions (see, e.g., [1–6]). The object of the present paper is to discuss some argument inequalities for in the class .
Throughout this paper, we let
In order to prove our main result, we well need the following lemma.
2. Main Result
Our main theorem is given by the following.
Proof. By taking in Lemma 1.1, we find that if satisfies
then (2.4) holds true.
For , and , we get
We consider the following two cases.
(i) If then from (2.7), and (2.6), we have and so where ,, , ,
We now calculate the maximum value of . It is easy to verify that and that
Set then . Noting that we easily have Hence, , and it follows from (2.11) to (2.16) that where denotes . Thus, by using (2.1), (2.10), and (2.17), we arrive at
(ii) If , then we obtain which leads to where ,, , ,
Now, we have Let then , , Hence, we deduce that and where . Further, by using (2.1), (2.20), and (2.25), we find that In view of , we conclude from (2.18) and (2.26) that properly contains the angular region in the complex -plane. Therefore, if satisfies (2.2), then the subordination relation (2.5) holds true, and thus we arrive at (2.4).
Furthermore, for the function defined by (1.8), we have Hence, by using (2.18) and (2.25), we see that the bounds and in (2.2) are best possible.
The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped them to improve the paper.
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Copyright © 2012 Qing Yang and Jin-Lin Liu. 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.