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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 494917, 12 pages
A New Class of Meromorphic Functions Associated with Spirallike Functions
School of Mathematics and Statistics, Anyang Normal University, Henan, Anyang 455002, China
Received 22 September 2012; Revised 26 October 2012; Accepted 31 October 2012
Academic Editor: Alberto Cabada
Copyright © 2012 Lei Shi et al. 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.
We introduce a new class of meromorphic functions associated with spirallike functions. Such results as subordination property, integral representation, convolution property, and coefficient inequalities are proved.
Let denote the class of functions of the form which are analytic in the punctured open unit disk
Let denote the class of functions given by which are analytic in and satisfy the condition
Let , where is given by (1.1) and is defined by then the Hadamard product (or convolution) is defined by
For two functions and , analytic in , we say that the function is subordinate to in and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence:
A function is said to be in the class of meromorphic starlike functions of order if it satisfies the inequality
For the real number , we know that If the complex number satisfies the condition it can be easily verified that
We now introduce and investigate the following class of meromorphic functions.
Definition 1.1. A function is said to be in the class if it satisfies the inequality
Remark 1.2. For , the class is the familiar class of meromorphic starlike functions of order .
Remark 1.3. If , then the condition (1.16) is equivalent to
which implies that belongs to the class of meromorphic spirallike functions. Thus, the class of meromorphic spirallike functions is a special case of the class .
For some recent investigations on spirallike functions and related functions, see, for example, the earlier works [1–9] and the references cited in each of these earlier investigations.
Remark 1.4. The function
belongs to the class .
It is clear that Then, for the function given by (1.18), we know that which implies that .
In this paper, we aim at deriving the subordination property, integral representation, convolution property, and coefficient inequalities of the function class .
2. Preliminary Results
In order to derive our main results, we need the following lemmas.
Lemma 2.1. Let be a complex number. Suppose also that the sequence is defined by Then
Lemma 2.2 (Jack's Lemma ). Let be a nonconstant regular function in . If attains its maximum value on the circle at , then for some real number .
3. Main Results
We begin by deriving the following subordination property of functions belonging to the class .
Theorem 3.1. A function if and only if
Proof. Suppose that
We easily know that , which implies that
where is analytic in with and .
It follows from (3.3) that which is equivalent to the subordination relationship (3.1).
On the other hand, the above deductive process can be converse. The proof of Theorem 3.1 is thus completed.
Theorem 3.2. Let . Then where is analytic in with and .
Proof. For , by Theorem 3.1, we know that (3.1) holds true. It follows that
where is analytic in with and .
We now find from (3.6) that which, upon integration, yields The assertion (3.5) of Theorem 3.2 can be easily derived from (3.8).
Theorem 3.3. Let . Then
Proof. Assume that . By Theorem 3.1, we know that (3.1) holds, which implies that It is easy to see that the condition (3.10) can be written as follows: We note that Thus, by substituting (3.12) into (3.11), we get the desired assertion (3.9) of Theorem 3.3.
Theorem 3.4. Let . If , then The inequality (3.13) is sharp for the function given by
Proof. Suppose that
We easily know that .
If we put it is known that From (3.15), we have We now set It follows from (3.18) that Combining (1.1), (3.16), and (3.20), we obtain In view of (3.21), we get From (3.17) and (3.22), we obtain Moreover, we deduce from (3.17) and (3.23) that
Next, we define the sequence as follows:
In order to prove that we make use of the principle of mathematical induction. By noting that Therefore, assuming that Combining (3.25) and (3.26), we get Hence, by the principle of mathematical induction, we have as desired.
By means of Lemma 2.1 and (3.26), we know that Combining (3.31) and (3.32), we readily get the coefficient estimates asserted by Theorem 3.4.
For the sharpness, we consider the function given by (3.14). A simple calculation shows that Thus, the function belongs to the class . Since , we have Then becomes This completes the proof of Theorem 3.4.
Theorem 3.5. If satisfies the inequality then .
Proof. To prove , it suffices to show that
which is equivalent to
From (3.36), we know that
Now, by the maximum modulus principle, we deduce from (1.1) and (3.39) that which implies that the assertion of Theorem 3.5 holds.
Theorem 3.6. If satisfies the condition then .
Proof. Define the function by
Then we see that is analytic in with .
It follows from (3.42) that By differentiating both sides of (3.43) logarithmically, we obtain From (3.41) and (3.44), we find that Next, we claim that . Indeed, if not, there exists a point such that By Lemma 2.2, we have Moreover, for , we find from (3.44) and (3.47) that But (3.48) contradicts to (3.45). Therefore, we conclude that , that is which shows that .
The present investigation was supported by the National Natural Science Foundation under Grants 11101053 and 11226088, the Key Project of Chinese Ministry of Education under Grant 211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under Grant 10B002, the Open Fund Project of the Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002 of the People’s Republic of China. The authors would like to thank the referees for their careful reading and valuable suggestions which essentially improved the quality of this paper.
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