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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 274985, 17 pages
http://dx.doi.org/10.1155/2012/274985
Research Article

Generalized k-Uniformly Close-to-Convex Functions Associated with Conic Regions

Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan

Received 24 January 2012; Accepted 31 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Khalida Inayat Noor. 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.

Abstract

We define and study some subclasses of analytic functions by using a certain multiplier transformation. These functions map the open unit disc onto the domains formed by parabolic and hyperbolic regions and extend the concept of uniformly close-to-convexity. Some interesting properties of these classes, which include inclusion results, coefficient problems, and invariance under certain integral operators, are discussed. The results are shown to be the best possible.

1. Introduction

Let denote the class of analytic functions defined in the unit disc and satisfying the condition . Let and be the subclasses of consisting of functions which are univalent, starlike of order , convex of order , and close-to-convex of order , respectively, . Let and .

For analytic functions and , by we denote the convolution (Hadamard product) of and , defined by

We say that a function is subordinate to a function and write if and only if there exists an analytic function , for such that

If is univalent in , then

For , define the domain as follows, see [1]: For fixed represents the conic region bounded, successively, by the imaginary axis , the right branch of hyperbola , a parabola .

Related with , the domain is defined in [2] as follows:

The functions which play the role of extremal functions for the conic regions are denoted by with , and are univalent, map onto , and are given as

It has been shown [3, 4] that is continuous as regards to and has real coefficients for all .

Let be the class of functions which are analytic in with such that for . It can easily be seen that , where is the class of Caratheodory functions of positive real part.

The class is defined in [5] as follows.

Let be analytic in with . Then if and only if, for ,,,

For ,, the class coincides with the class introduced by Pinchuk in [6]. Also .

The generalized Harwitz-Lerch Zeta function [7] is given as

Using (1.7), the following family of linear operators, see [79], is defined in terms of the Hadamard product as where ,

and is given by (1.7).

From (1.7) and (1.8), we can write

For the different permissible values of parameters and , the operator has been studied in [3, 4, 7, 1012].

We observe some special cases of the operator (1.10) as given below

(i) ,

(ii) ,

(iii) .

We remark that is the well-known Libera operator and is the generalized Bernardi operator, see [13, 14]. Also represents the operator closely related to the multiplier transformation studied by Flett [3].

We define the operator assee [15]. This gives us

From (1.12), the following identity can easily be verified

Remark 1.1. (i)For , we note that the domain given by (1.3) represents the following hyperbolic region:
The extremal function , for , can be written aswhere , in a simplified form, is given below and the branch of is chosen such that .
It is easy to see that, for . That isand the order is sharp with the extremal function , where is given by (1.16).(ii) For , the extremal function maps conformally onto the parabolic region .
It can easily be verified that and, in this case, the order is sharp.

We now define the following.

Definition 1.2. Let and let the operator be defined by (1.12). Then for , and if and only if
We note the following.
(i)For ,, and , the class reduces to , and gives us the class of uniformly starlike functions, see [2, 16].(ii) is the class of functions of bounded radius rotation, see [13, 14].(iii)We denote as , see [5].(iv)Let . Then implies that and we note that, for , .

Definition 1.3. Let . Then if and only if there exists such that

Special Cases.
(i). (ii)For , we obtain the class introduced and discussed in [17].(iii)When we take and , then , the class of uniformly close-to-convex functions, see [2].

2. Preliminary Results

We need the following results in our investigation.

Lemma 2.1 (see [18]). Let be convex in and with ,. If , analytic in with , satisfies then

In the following, one gives an easy extension of a result proved in [1].

Lemma 2.2 (see [5]). Let and let be any complex numbers with and . If is analytic in , and satisfies and is an analytic solution of then is univalent, and is the best dominant of (2.3).

Lemma 2.3 (see [19]). If ,, then for each analytic in with , where denotes the convex hull of .

Lemma 2.4 (see [18]). Let , and let be a complex-valued function satisfying the conditions:(i) is continuous in a domain ,(ii) and ,(iii), whenever and .If is a function analytic in such that and for , then in .

Lemma 2.5 (see [20]). Let ,. Then, with ,, one has(i), (ii).

Lemma 2.6 (see [5]). Let . Then there exist such that

Lemma 2.7. Let and . Then where

Proof. Let . Then,
Now the proof follows immediately by using the well-known Rogosinski’s result, see [21].

3. Main Results

We shall assume throughout, unless stated otherwise, that ,,,, and .

Theorem 3.1. Let . Let, for , Then, in .

Proof. Set We note is analytic in with .
From (3.1), we have
That is
Logarithmic differentiation of (3.4) and simple computations give us
Define then, with ,, we have
From (3.2), (3.5), and (3.7), it follows that
On applying Lemma 2.2, we obtainwhere is the best dominant and is given as
Consequently it follows, from (3.2), that and in .

For ,, we have the following special case.

Corollary 3.2. Let and let be defined by (3.1). Then, , where

Proof. We write and proceeding as in Theorem 3.1, we obtain
We construct the functional by taking ,  , as
The first two conditions of Lemma 2.4 can easily be verified. For condition (iii), we proceed as follows: where
The right-hand side of (3.15) is less than equal to zero when and . From , we obtain as given by (3.11), and ensures that .
This shows that all the conditions of Lemma 2.4 are satisfied and therefore . This implies and consequently . That is as required.

By taking , and , we obtain a well-known result that every convex function is starlike of order . Also, for ,,, and , we obtain from (3.1) the Libera operator and in this case we obtain a known result with for starlike functions, see [18].

Assigning permissible values to different parameters, we obtain several new and known results from Theorem 3.1 and Corollary 3.2.

Theorem 3.3. Let and let be defined by (3.1). Then .

Proof. We can write (3.1) as where is convex in .
Let . Then there exists some such that
From Theorem 3.1, it follows that . We can write (3.19) as where , given by (3.17), is convex in .
Since , so . It can easily be shown that and are in the class .
Now, from (3.1), we have
We use Lemma 2.3 with , to have
Thus from (3.19), (3.21), and (3.22) we obtain the required result that . This completes the proof.

As a special case we note that, for , the subclass is invariant under the integral operator defined by (3.1).

Theorem 3.4. One has

Proof. Let and let where is analytic in and is defined by (3.2).
Then, from (1.13), we have
Applying similar technique used before, we have from (3.2) and (3.7) for
Thus, using Lemma 2.2, it follows that and , consequently in and this completes the proof.

As special cases, we have the following.(i)Let ,. Then, from Theorem 3.4, it easily follows that(ii)Let and . Then implies , that is, is a function of bounded radius rotation in .

Theorem 3.5. One has

Proof. Let . Then, for , for some .
We define an analytic function in such that where . We shall show that in .
Since and , we have
Now, on using (1.13), we have
Differentiation of (3.30) gives us and using (3.33) in (3.32), we obtain
Since , we have with , and thus, applying Lemma 2.1, we have in . This shows in and consequently .

As a special case, we note that is a close-to-convex function for .

Theorem 3.6. Let and let be convex in . Then for .

Proof. We have
Now , and is a convex in , we use Lemma 2.3 to (3.36) and conclude that for . This completes the proof.

Remark 3.7. Following the similar technique, we can easily extend Theorem 3.6 to the class , that is, , is invariant under convolution with convex function.

3.1. Applications of Theorem 3.6

The classes and are preserved under the following integral operators:

(1) , where ,

(2) , where ,

(3) , where ,

(4) , where ,

The proof is immediate since is convex in for .

With essentially the same method together with Lemma 2.7, we can easily prove the following sharp coefficient results.

Theorem 3.8. Let and let it be given by Then where is Pochhamer symbol defined, in terms of Gamma function , by and is as given by (2.9).
As special case, one notes that
(i),, then one has see [2].(ii)Let ,. Then,
This coefficient bound is well known for , see [2].

Using Theorem 3.8 with , the following result can easily be proved.

Theorem 3.9. Let . Then, for where is as given by (2.9).

By assigning different permissible values to the parameters, we obtain several known results, see [2, 22].

We now prove the following.

Theorem 3.10. Let with . Then, for , where is a constant.

Proof. Let ,. Since , we can write
That is
Now, , and it follows from a result proved in [5] that there exist such thatThus, for and , where and in .
Let . Then, by a result [23], there exists a with such that, for , We now use (3.48), distortion theorems for starlike functions for , and Lemma 2.5 with ,, and obtain from (3.47), From (3.48), we easily obtain the required result given by (3.43), .
This completes the proof.

Using the similar technique, we can easily prove the following.

Theorem 3.11. Let . Then where is a constant depending on , and only. The exponent is best possible.

Acknowledgment

The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities and environment.

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