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On the multivalency of certain analytic functions
Journal of Inequalities and Applications volume 2014, Article number: 357 (2014)
Abstract
We prove several relations of the type for functions satisfying some geometric conditions.
MSC:30C45, 30C80.
1 Introduction
Let p be positive integer and let be the class of functions
which are analytic in the unit disk and denote .
The subclass of consisting of p-valently starlike functions is denoted by . An analytic description of is given by
The subclass of consisting of p-valently and strongly starlike functions of order α, is denoted by . An analytic description of is given by
The subclass of consisting of p-valently convex functions and p-valently strongly convex functions of order α, , are denoted by and , respectively. The analytic descriptions of and are given by
and
For the classes and become the well known classes and of strongly starlike and strongly convex functions of order α, respectively. The concept of strongly starlike and strongly convex functions of order α was introduced in [1] and [2] with their geometric interpretation. For the classes and become the classes and of starlike and convex functions; see for example [3]. In this paper, we need the following lemma.
Lemma 1.1 Assume that with in . Assume also that for all θ, , the function f satisfies the following condition:
where , . Then we have
Proof First we note that from
the implication
follows by mathematical induction.
For the case , , we have
Let , , . By (1.1) is an increasing function with respect to ρ, thus
Therefore, by (1.2) and by (1.4), we have
Using (1.5) in (1.3), we obtain
or we have
for and .
For the case , from the hypothesis (1.1), we find that is an decreasing function with respect to ρ, thus
and
Therefore, in a similar way to above, we obtain
and we also have
for and . From (1.7) and (1.8), we have
It completes the proof of Lemma 1.1. □
Corollary 1.2 Assume that with in . Assume also that satisfies the following condition:
then we have
Proof The conditions (1.1) and (1.9) are equivalent. If , then with . By (1.7), we have also
If , then with . By (1.8), we have also
In both cases, we have (1.10). □
The inequality (1.10) can be written in the equivalent form
Recall that if is analytic in and in , then f is called typically real function; see [[4], Chapter 10]. Therefore, Corollary 1.2 says that if is a typically real function, then is a typically real function, too.
2 Main result
Theorem 2.1 Let . Assume that for all θ, , satisfies the following condition:
where , , moves on the segment from to and
Then is starlike in or .
Proof From the hypothesis (2.1) and the hypothesis (2.2) and applying Lemma 1.1, we have
This shows that is starlike in . □
Applying the same method as in the proof of Lemma 1.1, we have the following lemma.
Lemma 2.2 Let . Assume that for all θ, , satisfies the following condition:
where , moves on the segment from to . Then we have
Applying Lemma 2.2, we have the following theorem.
Theorem 2.3 Let . Suppose that for all θ, , satisfies the following condition:
where , , moves on the segment from to and
Then we have or is 2-valently convex in .
Proof From the hypothesis (2.3) and (2.4) and applying Lemma 2.2, we have
Therefore, we have
It completes the proof. □
Applying the same method as in the proof of Lemma 1.1 and Lemma 2.2, we can generalize Theorem 2.1 and Theorem 2.3 as follows.
Lemma 2.4 Let . Suppose that for all θ, , satisfies the following condition:
where , moves on the segment from to . Then we have
Proof For the case , from the hypothesis (2.5), we have
and therefore we have
for and .
For the case , applying the same method as above and in the proof of Lemma 1.1 and Lemma 2.2, we have
for and . From (2.6) and (2.7), we have
It completes the proof of Lemma 2.4. □
Thus, we have the following theorems.
Theorem 2.5 Let . Assume that for all θ, , satisfies the following condition:
where , , , moves on the segment from to and suppose that
where . Then we have or is p-valently and strongly starlike of order α in .
Theorem 2.6 Let , . Assume that for all θ, , satisfies the following condition:
where , moves on the segment from to and suppose that
where . Then we have or is p-valently and strongly convex of order α.
Lemma 2.7 Let be analytic in and suppose that it satisfies the following condition:
where . Then for we have
while for we have
where , .
Proof Let , . Then it follows that
This proves (2.9) and (2.10) and it shows that the function is an increasing function with respect to ρ, , and , and that the function is a decreasing function with respect to ρ, , and . □
Theorem 2.8 Let be analytic in and suppose that it satisfies the following condition:
where and
Then is starlike in .
Proof From Lemma 2.7 and (2.12), for the case , we have
This shows that
where , , and .
For the case , applying the same method as above, we have
where , , and . Applying (2.12), (2.13), and (2.14), we have
This completes the proof. □
Remark 2.9 The functions satisfy the conditions of Theorem 2.8 whenever .
References
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Brannan DA, Kirwan WE: On some classes of bounded univalent functions. J. Lond. Math. Soc. 1969,1(2):431-443.
Duren PL: Univalent Functions. Springer, New York; 1983.
Goodman AW: Univalent Functions. Vols. I and II. Mariner Publishing Co., Tampa; 1983.
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Nunokawa, M., Sokół, J. On the multivalency of certain analytic functions. J Inequal Appl 2014, 357 (2014). https://doi.org/10.1186/1029-242X-2014-357
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DOI: https://doi.org/10.1186/1029-242X-2014-357