1. Introduction
Let
Σ denote the class of functions
f of the form
(1.1)
which are
analytic in the
punctured open unit disk
(1.2)
Let
𝒫 denote the class of functions
p given by
(1.3)
which are analytic in
𝕌 and satisfy the condition
(1.4)
Let
f,
g ∈
Σ, where
f is given by (
1.1) and
g is defined by
(1.5)
then the Hadamard product (or convolution)
f*
g is defined by
(1.6)
For two functions
f and
g, analytic in
𝕌, we say that the function
f is subordinate to
g in
𝕌 and write
(1.7)
if there exists a Schwarz function
ω, which is analytic in
𝕌 with
(1.8)
such that
(1.9)
Indeed, it is known that
(1.10)
Furthermore, if the function
g is univalent in
𝕌, then we have the following equivalence:
(1.11)
A function
f ∈
Σ is said to be in the class
ℳ𝒮*(
β) of
meromorphic starlike functions of order β if it satisfies the inequality
(1.12)
For the real number
β (0 <
β < 1), we know that
(1.13)
If the complex number
α satisfies the condition
(1.14)
it can be easily verified that
(1.15)
We now introduce and investigate the following class of meromorphic functions.
Definition 1.1. A function f ∈ Σ is said to be in the class ℳ𝒮α if it satisfies the inequality
(1.16)
Remark 1.2. For 0 < α < 1, the class ℳ𝒮α is the familiar class of meromorphic starlike functions of order α.
Remark 1.3. If α = |α|eiψ (−π/2 < ψ < π/2), then the condition (1.16) is equivalent to
(1.17)
which implies that
f belongs to the class of meromorphic spirallike functions. Thus, the class of meromorphic spirallike functions is a special case of the class
ℳ𝒮α.
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
(1.18)
belongs to the class
ℳ𝒮α.
It is clear that
(1.19)
Then, for the function
f given by (
1.18), we know that
(1.20)
which implies that
f ∈
ℳ𝒮α.
In this paper, we aim at deriving the subordination property, integral representation, convolution property, and coefficient inequalities of the function class ℳ𝒮α.
2. Preliminary Results
In order to derive our main results, we need the following lemmas.
Lemma 2.1. Let λ be a complex number. Suppose also that the sequence is defined by
(2.1)
Then
(2.2)
Proof. From (2.1), we know that
(2.3)
By virtue of (
2.3), we find that
(2.4)
Thus, for
k≧1, we deduce from (
2.4) that
(2.5)
By virtue of (
2.1) and (
2.5), we get the desired assertion (
2.2) of Lemma
2.1.
Lemma 2.2 (Jack′s Lemma [10]). Let ϕ be a nonconstant regular function in 𝕌. If |ϕ| attains its maximum value on the circle |z| = r < 1 at z0, then
(2.6)
for some real number
t (
t≧1).
3. Main Results
We begin by deriving the following subordination property of functions belonging to the class ℳ𝒮α.
Theorem 3.1. A function f ∈ ℳ𝒮α if and only if
(3.1)
Proof. Suppose that
(3.2)
We easily know that
h ∈
𝒫, which implies that
(3.3)
where
ω is analytic in
𝕌 with
ω(0) = 0 and |
ω(
z)| < 1
(
z ∈
𝕌).
It follows from (3.3) that
(3.4)
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 f ∈ ℳ𝒮α. Then
(3.5)
where
ω is analytic in
𝕌 with
ω(0) = 0 and |
ω(
z)| < 1
(
z ∈
𝕌).
Proof. For f ∈ ℳ𝒮α, by Theorem 3.1, we know that (3.1) holds true. It follows that
(3.6)
where
ω is analytic in
𝕌 with
ω(0) = 0 and |
ω(
z)| < 1
(
z ∈
𝕌).
We now find from (3.6) that
(3.7)
which, upon integration, yields
(3.8)
The assertion (
3.5) of Theorem
3.2 can be easily derived from (
3.8).
Theorem 3.3. Let f ∈ ℳ𝒮α. Then
(3.9)
Proof. Assume that f ∈ ℳ𝒮α. By Theorem 3.1, we know that (3.1) holds, which implies that
(3.10)
It is easy to see that the condition (
3.10) can be written as follows:
(3.11)
We note that
(3.12)
Thus, by substituting (
3.12) into (
3.11), we get the desired assertion (
3.9) of Theorem
3.3.
Theorem 3.4. Let λ = [Re(1/α) − 1]|α|. If f ∈ ℳ𝒮α, then
(3.13)
The inequality (
3.13) is sharp for the function given by
(3.14)
Proof. Suppose that
(3.15)
We easily know that
h ∈
𝒫.
If we put
(3.16)
it is known that
(3.17)
From (
3.15), we have
(3.18)
We now set
(3.19)
It follows from (
3.18) that
(3.20)
Combining (
1.1), (
3.16), and (
3.20), we obtain
(3.21)
In view of (
3.21), we get
(3.22)
(3.23)
From (
3.17) and (
3.22), we obtain
(3.24)
Moreover, we deduce from (
3.17) and (
3.23) that
(3.25)
Next, we define the sequence as follows:
(3.26)
In order to prove that
(3.27)
we make use of the principle of mathematical induction. By noting that
(3.28)
Therefore, assuming that
(3.29)
Combining (
3.25) and (
3.26), we get
(3.30)
Hence, by the principle of mathematical induction, we have
(3.31)
as desired.
By means of Lemma 2.1 and (3.26), we know that
(3.32)
Combining (
3.31) and (
3.32), we readily get the coefficient estimates asserted by Theorem
3.4.
For the sharpness, we consider the function f given by (3.14). A simple calculation shows that
(3.33)
Thus, the function
f belongs to the class
ℳ𝒮α. Since 0 <
α < 1, we have
(3.34)
Then
f becomes
(3.35)
This completes the proof of Theorem
3.4.
Theorem 3.5. If f ∈ Σ satisfies the inequality
(3.36)
then
f ∈
ℳ𝒮α.
Proof. To prove f ∈ ℳ𝒮α, it suffices to show that
(3.37)
which is equivalent to
(3.38)
From (
3.36), we know that
(3.39)
Now, by the maximum modulus principle, we deduce from (1.1) and (3.39) that
(3.40)
which implies that the assertion of Theorem
3.5 holds.
Theorem 3.6. If f ∈ Σ satisfies the condition
(3.41)
then
f ∈
ℳ𝒮α.
Proof. Define the function φ by
(3.42)
Then we see that
φ is analytic in
𝕌 with
φ(0) = 0.
It follows from (3.42) that
(3.43)
By differentiating both sides of (
3.43) logarithmically, we obtain
(3.44)
From (
3.41) and (
3.44), we find that
(3.45)
Next, we claim that |
φ(
z)| < 1. Indeed, if not, there exists a point
z0 ∈
𝕌 such that
(3.46)
By Lemma
2.2, we have
(3.47)
Moreover, for
z =
z0, we find from (
3.44) and (
3.47) that
(3.48)
But (
3.48) contradicts to (
3.45). Therefore, we conclude that |
φ(
z)| < 1, that is
(3.49)
which shows that
f ∈
ℳ𝒮α.
Acknowledgments
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.
