Abstract and Applied Analysis

Volume 2013 (2013), Article ID 847287, 6 pages

http://dx.doi.org/10.1155/2013/847287

Research Article

Certain Properties of a Class of Close-to-Convex Functions Related to Conic Domains

Wasim Ul-Haq and Shahid Mahmood

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan

Received 30 September 2012; Revised 3 March 2013; Accepted 17 March 2013

Academic Editor: Nikolaos Papageorgiou

Copyright © 2013 Wasim Ul-Haq and Shahid Mahmood. 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 aim to de…fine a new class of close-to-convex functions which is related to conic domains. Many interesting properties such as sufficiency criteria, inclusion results, and integral preserving properties are investigated here. Some interesting consequences of our results are also observed.

1. Introduction

Let be the class of functions

which are analytic in the open unit disc . Let and be analytic in , and we say that is subordinate to , written as if there exists a Schwarz function , which is analytic in with and   , such that . In particular, when is univalent, then the above subordination is equivalent to and ; see [1].

Kanas and Wisniowska [2, 3] introduced and studied the classes of -uniformly convex functions denoted by and the corresponding class related by the Alexander-type relation. Later, the class -uniformly close-to-convex functions denoted by defined as

was considered by Acu [4]; for study details on these classes, we refer to [5–7]. All these above mentioned classes were generalized to the classes , , and by Shams et al. [8] and Srivastava et al. [9], respectively. The classes and are defined as

where , , . The class known as -uniformly close-to-convex functions of order type is the class of all those functions which satisfies the following condition:

for some .

Motivated by the work of Noor et al. [10–13], we define the following.

Definition 1. Let . Then, is in the class if and only if, for , ,

for some , where

Special Cases

(i); see [9].(ii) and , the classes of uniformly close-to-convex and quasiconvex functions introduced and investigated in [14].(iii), the class of alpha quasiconvex functions, introduced and studied in [11].(iv), the class of close-to-convex functions of order type , [15].(v), the class of quasiconvex functions of order type , [16].

The conditions , , on the parameters are assumed throughout the entire paper unless otherwise mentioned.

Geometric Interpretation. A function is in the class if and only if the functional takes all the values in the conic domain defined as follows:

Extremal functions for these conic regions are denoted by , which are analytic in and map onto such that and . These functions are given as:

where , and is chosen such that , where is Legendre's complete elliptic integral of the first kind and is complementary integral of ; see [2, 3].

2. Preliminaries Result

We require the following results which are essential in our investigations.

Lemma 2 (see [17, page 70]). Let be convex function in and with , . If is analytic in with , then

Lemma 3 (see [17, page 195]). Let be convex function in with and . Suppose that   and that ,  , and are analytic in and satisfy

for . If is analytic in with and the following subordination relation holds:

then

Lemma 4 (see [12]). If and , then for ,

3. Main Results

First, we prove the following sufficiency criteria for the functions in the class .

Theorem 5. A function is said to be in the class , if

where

Proof. Let us assume that relation (6) holds. Now, it is sufficient to show that

First, we consider

Using (1) and the series in (17), we have

Now,

The last inequality is bounded above by 1, if

Hence,

where is given by (15). This completes the proof.

When we take  , , and in the above theorem, we obtain the following sufficient condition for the functions to be in the class which is proved in [14].

Corollary 6 (see [14]). A function   is said to be in the class if

Corollary 7 (see [14]). A function   is said to be in the class if

The above corollary is obtained when we take , ,  and    in Theorem 5.

Theorem 8. Let and . Then, .

Proof. Let

where is analytic and . Now differentiating (24), we have

where . Using (24) and (25) in relation (6), we obtain

where

Now, since , we have

Replacing by and by , the above subordination is equivalent to

where . Using Lemma 3 with , we obtain

This implies that

Hence, . This completes the proof.

Corollary 9. Let . Then, . That is, ,  .

The above result is well-known inclusion proved in [11].

For , consider the following integral operator defined by

This operator was given by Bernardi [18] in 1969. In particular, the operator was considered by Libera [19]. Now let us prove the following.

Theorem 10. Let . Then, .

Proof. Let the function be such that (6) is satisfied. It can easily be seen that according to [4], the function , and from (32), we deduce

If we let and , then simple computations yield us

Let

where is analytic and . From (36), we have

where . Using (36) and (37) in (6), we have

where

Now proceeding in the similar manner as in the proof of Theorem 5 and using Lemma 3 with , we obtain

From (36), it implies that

By employing Lemma 2, we immediately obtain the desired result.

Theorem 11. For ,

Proof. Let . Then, consider

After some simple computations, we have

Now, since , we have . Also, Theorem 8 gives us that . The use of Lemma 4 leads us to the required relation; that is, . This completes the proof.

Acknowledgment

The authors would like to thank the reviewer of this paper for his/her valuable comments on the earlier version of this paper. They would also like to acknowledge Prof. Dr. Ehsan Ali, VC AWKUM, for the financial support on the publication of this article.

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