Abstract
By using the method of differential subordinations, we derive some properties of multivalent analytic functions. All results presented here are sharp.
This paper is dedicated to Professor Miodrag Mateljević on the occasion of his 65th birthday
1. Introduction
Let denote the class of functions of the form which are analytic in the open unit disk . Let and be analytic in . Then, we say that is subordinate to in , written as , if there exists an analytic function in , such that and . If is univalent in , then the subordination is equivalent to and . Let be analytic in . Then, for , it is clear that if and only if
Recently, a number of results for argument properties of analytic functions have been obtained by several authors (see, e.g., [1–5]). The objective of the present paper is to derive some further interesting properties of multivalent analytic functions. The basic tool used here is the method of differential subordinations.
To derive our results, we need the following lemmas.
Lemma 1 (see [6, Theorem 1, page 776]). Let be analytic and starlike univalent in with . If is analytic in and , then
Lemma 2 (see [5, Theorem 1, page 1814]). Let , , , and . Also let If is analytic in with and where is (close-to-convex) univalent in , then The bounds and in (9) are sharp for the function defined by
Remark 3 (see [5, Lemma 2, page 1813]). The function defined by (10) is analytic and univalent convex in and
2. Main Results
Our first result is contained in the following.
Theorem 4. Let and . If satisfies and where is the smallest positive root of the equation then The bound is sharp for each .
Proof. Let
We can see easily that (13) has two positive roots. Since and , we have
Put
Then, from the assumption of the theorem, we can see that is analytic in with and for all . Taking the logarithmic differentiations in both sides of (17), we get
for all . Thus, inequality (12) is equivalent to
By using Lemma 1, (20) leads to
or to
According to (16), (22) can be written as
Now, by taking and in (2) and (3), we have
for all because of . This proves (14).
Next, we consider the function defined by
for all . It is easy to see that
for all . Since
it follows from (3) that
Hence, we conclude that the bound is the best possible for each .
Next, we derive the following.
Theorem 5. If satisfies and where then The bound in (31) is sharp.
Proof. Let
Then, from the assumption of the theorem we can see that is analytic in with and for all . According to (32) and (29), we have immediately
that is,
Now, by using Lemma 1, we obtain
Since the function is convex univalent in and
from (35), we get inequality (31).
To show that the bound in (31) cannot be increased, we consider
It is easy to verify that the function satisfies inequality (29). On the other hand, we have
as . Now, the proof of the theorem is complete.
Finally, we discuss the following theorem.
Theorem 6. Let . If satisfies and for all , where then The bound in (39) is sharp.
Proof. Define the function by (17). For , it follows from (17) and (18) that
for all . Putting
in Lemma 2 and using (42), we see that if
where
then (41) holds true.
Letting and , we deduce that
Making use of (46), we obtain that
Therefore, if satisfies (39), then the subordination (44) holds, and, thus, we obtain (41).
For the function
we find that
where is defined by (45). In view of (46) and (49), we conclude that the bound in (39) is the largest number such that (41) holds true. This completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to express their sincere thanks to the referees for careful reading and suggestions which helped them improve the paper.