Abstract
For and , we consider a liner operator on the class of analytic functions in the unit disk defined by the convolution , where , and introduce a certain new subclass of using this operator. Several interesting properties of these classes are obtained.
1. Introduction
Let denote the class of functions of the form which are analytic in the unit disk .
If satisfies then is said to be strongly starlike of order and type in , and denoted by .
If satisfies then is said to be strongly convex of order and type in , and denoted by . It is obvious that belongs to if and if . Further, we note that and which are, respectively, starlike and convex univalent functions of order .
Let denote the class of functions of the form analytic in which satisfy the condition .
For functions given by (1.1) and , let denote the Hadamard product (or convolution) of and , defined by If and are analytic in , we say that is subordinate to , written or , if there exists a Schwarz function in such that [1].
Let . Denote by the operator defined by It is obvious that , , and The operator is called the th-order Ruscheweyh derivative of . Recently, K. I. Noor [2] and K. I. Noor and M. A. Noor [3] defined and studied an integral operator , analogous to as follows.
Let , and be defined such that Then We note that , . The operator is called the Noor integral of th order of (see [4, 5]), which is an important tool in defining several classes of analytic functions. In recent years, it has been shown that Noor integral operator has fundamental and significant applications in the geometric function theory.
For real or complex numbers , , other than , the hypergeometric series is defined by where is Pochhammer symbol defined by
We note that the series (1.9) converges absolutely for all so that it represents an analytic function in . Also an incomplete beta function is related to Gauss hypergeometric function as and we note that , where is Koebe function. Using a convolution operator [6], was defined by Carlson and Shaferr. Furthermore, Hohlov [7] introduced a convolution operator using .
N. Shukla and P. Shukla [8] studied the mapping properties of a function to be as given in and investigated the geometric properties of an integral operator of the form
Kim and Shon [9] considered linear operator defined by .
We now introduce a function given by and obtain the following linear operator: The operator is known as the generalized integral operator. For in (1.14), , which was introduced by K. I. Noor [10].
Now we find the explicit form of the function . It is well known that Putting (1.9) and (1.16) in (1.14), we get Therefore, the function has the following form:
Now we note that
From (1.19), we note that
Also it can easily be verified that Now we introduce the following classes in term of the new operator . For , , , and , let be the class of functions satisfying Observe that and . Also, for , , , and , let be the class of functions satisfying Observe that and .
Clearly, if and only if .
Note that , , , and .
Finally, let be the class of functions satisfying for , , , , and , where We also note that and are the classes of quasiconvex and close-to-convex functions of order and type , respectively, introduced and studied by Noor and Alkhorasani [11] and Silverman [12].
2. Main Results
In order to give our results, we need the following lemmas.
Lemma 2.1 (see [13]). Let , be complex numbers. Let be convex univalent in with and , and with , . If is analytic in , then implies
Lemma 2.2 (see [14]). Let , be complex numbers. Let be convex univalent in with and , . If is analytic in , then implies
Lemma 2.3 (see [15]). Let be convex univalent in and let . Suppose is analytic in with , If is analytic in , then implies
Theorem 2.4. Let be convex univalent in with and . If satisfies the condition then for , , and .
Proof. Let where . By using (1.21) in (2.9) and then differentiating, we get where and . Hence by applying Lemma 2.1, we obtain the required result.
Theorem 2.5. Let be convex univalent in with and . If satisfies the condition then for , , and .
Proof. By using the same technique in the proof of Theorem 2.4 and using (1.22) and applying Lemma 2.2, we obtain the required result.
Taking in Theorem 2.4 and in Theorem 2.5, we have the following.
Corollary 2.6. It holds that for , , , and .
Also, by taking in Theorem 2.4 and in Theorem 2.5, we have the following.
Corollary 2.7. It holds that for , , , and .
Corollary 2.8. For , , , and , one has
Proof. We will proof the first relation and by the same method we can proof the second relation
Theorem 2.9. Let be convex univalent in with and . If satisfies the condition then where be the integral operator defined by
Proof. From (2.19), we have Now, let where . Then by using (2.20), we get Differentiating both sides of (2.22) logarithmically, we obtain Then, by Lemma 2.2, we obtain that
Now, by letting in Theorem 2.9, we have the following.
Corollary 2.10. For , , , and . If , then , where given by (2.19).
Also, by taking in Theorem 2.9, we have the following.
Corollary 2.11. For , , , , and . If , then .
Corollary 2.12. For , , , , and . If , then .
Proof. It holds that
Theorem 2.13. Let . Then for , , , , and .
Proof. Let , then by the definition, we can write
for some .
Letting and , we observe that , . Now by Corollary 2.6, and so . Also, note that
Differentiating
both sides in (2.28) yields
Now by using
the identity (1.21), we obtain
From (2.27),
(2.28), and (2.30), we conclude that
Letting and , we obtain
The above
inequality satisfies the conditions required by Lemma 2.3. Hence and so the proof is complete.
Theorem 2.14. Let . Then for , , , , and .
Proof. By using the same technique as in the proof of Theorem 2.13, we get By letting and , we obtain Then, by applying Lemma 2.3, we obtain the required result.
Theorem 2.15. Let , , , , and . If , then , where is given by (2.19).
Proof. Also, by using the same technique as in the proof of Theorem 2.13, we get By letting and , we obtain Then, applying Lemma 2.3, we obtain the required result.
Acknowledgment
The work presented here was supported by Fundamental Research Grant Scheme UKM-ST-01-FRGS0055-2006.