A Class of Analytic Functions with Missing Coefficients
Ding-Gong Yang1 and Jin-Lin Liu2
1Department of Mathematics, Soochow University, Jiangsu, Suzhou 215006, China
2Department of Mathematics, Yangzhou University, Jiangsu, Yangzhou 225002, China
Academic Editor: Paul Eloe
Received01 Mar 2011
Accepted09 May 2011
Published03 Jul 2011
Let and denote the class of functions of the form
which are analytic in the open unit disk and satisfy
the following subordination condition , for, for. We obtain sharp bounds on ,
and coefficient estimates for functions
belonging to the class
Conditions for univalency and starlikeness, convolution properties, and the radius of convexity
are also considered.
Let denote the class of functions of the form
which are analytic in the open unit disk . Let and denote the subclasses of whose members are univalent and starlike, respectively.
For functions and analytic in , we say that is subordinate to in and we write , if there exists an analytic function in such that
Furthermore, if the function is univalent in , then
Throughout our present discussion, we assume that
We introduce the following subclass of .
Definition 1.1. A function is said to be in the class if it satisfies
have been studied by several authors (see [1–5]). Recently, Gao and Zhou  showed some mapping properties of the following subclass of :
For further information of the above classes (with ) and related analytic function classes, see Srivastava et al. , Yang and Liu , Kim , and Kim and Srivastava .
In this paper, we obtain sharp bounds on , and coefficient estimates for functions belonging to the class . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also presented. One can see that the methods used in  do not work for the more general class than .
2. The bounds on , , and in
In this section, we let
With (2.1), it is easily seen that the function given by (1.6) can be expressed as
Theorem 2.1. Let . Then, for ,
The bounds in (2.4) are sharp for the function defined by
Proof. The analytic function given by (1.6) is convex (univalent) in (cf. ) and satisfies . Thus, for ,
Let . Then, we can write
where is analytic and for . By the Schwarz lemma, we know that . It follows from (2.7) that
which leads to
we deduce from (2.6) and (2.10) that
Now, by using (2.3) and (2.12), we can obtain (2.4). Furthermore, for the function defined by (2.5), we find that
Hence, and from (2.13), we see that the bounds in (2.4) are the best possible. Hereafter, we write
Corollary 2.2. Let . Then, for ,
The results are sharp.
Proof. For , it follows from (2.12) (used in the proof of Theorem 2.1) that
for . From these, we have the desired results.
The bounds in (2.16) and (2.17) are sharp for the function
Theorem 2.3. Let . Then, for ,
The results are sharp.
Proof. Noting that
an application of Theorem 2.1 yields (2.20). Furthermore, the results are sharp for the function defined by (2.5).
Corollary 2.4. Let . Then, for ,
The results are sharp for the function defined by (2.19).
Proof. For , it follows from (2.6) and (2.10) (with ) that
for and . Making use of (2.21) and (2.23), we can obtain (2.22).
Theorem 2.5. Let and and ). If
then , where the symbol stands for the familiar Hadamard product (or convolution) of two analytic functions in .
Proof. Since and ), it follows from Corollary 2.4 (with ) and (2.24) that
Thus, has the Herglotz representation
where is a probability measure on the unit circle and . For , we have
In view of the function is convex (univalent) in , we deduce from (2.26) to (2.28) that
This shows that .
Corollary 2.6. Let , and
Proof. By taking , and , (2.24) in Theorem 2.5 becomes
Hence, the desired result follows as a special case from Theorem 2.5.
R. Singh and S. Singh [4, Theorem 3] proved that, if and belong to , then . Obviously, for
Corollary 2.6 generalizes and improves Theorem 3 in .
Theorem 2.8. Let and . Then, for ,
The result is sharp, with the extremal function defined by (2.5).
Proof. It is well known that for and ,
Since , we have and so (2.35) leads to
By virtue of (1.6), (2.10), and (2.36), we have
for and . Now, by using (2.3), (2.21) and (2.37), we can obtain (2.34).
Theorem 2.9. Let
The result is sharp for each .
Proof. It is known (cf. ) that, if
where is analytic in and is analytic and convex univalent in , then . By (2.38), we have
and is given by (1.6). Since the function is analytic and convex univalent in , it follows from (2.41) that
which gives (2.39). Next, we consider the function
It is easy to verify that
The proof of Theorem 2.9 is completed.
3. The Univalency and Starlikeness of
Theorem 3.1. if and only if
Proof. Let and (3.1) be satisfied. Then, by (2.16) in Corollary 2.2, we see that . Thus, is close-to-convex and univalent in . On the other hand, if
then the function defined by (2.19) satisfies and
as . Hence, there exists a point such that . This implies that is not univalent in and so the theorem is proved.
Theorem 3.2. Let (3.1) in Theorem 3.1 be satisfied. If and
Proof. We first show that
Equation (3.5) is obvious when . For , we have
is a monotonically decreasing sequence. Therefore, the inequality (3.5) follows from (3.6). Let . Then,
Define in by
In view of (3.1) in Theorem 3.1 is satisfied, the function is univalent in , and so is analytic in . Also it follows from (3.10) that
We want to prove now that for . Suppose that there exists a point such that
Then, applying a result of Miller and Mocanu [13, Theorem 4], we have
For , we deduce from Corollaries 2.2 and 2.4, (3.1), (3.5), (3.11), (3.13), and (3.4) that
But this contradicts (3.9) at . Therefore, we must have and the proof of Theorem 3.2 is completed.
Remark 3.3. In [6, Theorem 4(ii)], the authors gave the following: if and is the solution of the equation
then for . However, this result is not true because the series in (3.15) diverges.
4. The Radius of Convexity
Theorem 4.1. Let belong to the class defined by
and . Then,
where is the root in of the equation
The result is sharp.
Proof. For , we can write
where is analytic and in . Differentiating both sides of (4.4) logarithmically, we arrive at
Put and . Then, (4.4) implies that
With the help of the Carathéodory inequality
it follows from (4.5) and (4.6) that
because of (4.6) and (4.7). In view of (4.10) and (4.11), we see that
Let us now calculate the minimum value of on the closed interval . Noting that
and (4.7), we deduce from (4.12) that
and . Suppose that . Then,
Hence, by virtue of the mathematical induction, we have for all and . This implies that
In view of (4.14) and (4.18), we see that
Further it follows from (4.9), (4.12), and (4.19) that
Note that and . If we let denote the root in of the equation , then (4.20) yields the desired result (4.2).
To see that the bound is the best possible, we consider the function
It is clear that for ,
which shows that the bound cannot be increased.
Setting , Theorem 4.1 reduces to the following result.
Corollary 4.2. Let and . Then, is convex of order in
The result is sharp.
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