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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 834064, 9 pages
On Starlike and Convex Functions with Respect to -Symmetric Points
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600 Selangor, Malaysia
Received 29 January 2011; Revised 13 March 2011; Accepted 19 March 2011
Academic Editor: Stanisława R. Kanas
Copyright © 2011 Afaf A. Ali Abubaker and Maslina Darus. 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.
We introduce new subclasses and of analytic functions with respect to -symmetric points defined by differential operator. Some interesting properties for these classes are obtained.
Let denote the class of functions of the form which are analytic in the unit disk .
Also let be the class of analytic functions with , which are convex and univalent in and satisfy the following inequality: A function is said to be starlike with respect to symmetrical points in if it satisfies This class was introduced and studied by Sakaguchi in 1959 . Some related classes are studied by Shanmugam et al. .
Recall that the function is subordinate to if there exists a function , analytic in , with and , such that , . We denote this subordination by . If is univalent in , then the subordination is equivalent to and .
A function is in the class satisfying where , is a fixed positive integer, and is given by the following: The classes of starlike functions with respect to -symmetric points and of convex functions with respect to -symmetric points were considered recently by Wang et al. . Moreover, the special case imposes the class , which was studied by Gao and Zhou , and the class was studied by Ma and Minda .
Let two functions given by and be analytic in . Then the Hadamard product (or convolution) of the two functions , is defined by and for several function , The theory of differential operators plays important roles in geometric function theory. Perhaps, the earliest study appeared in the year 1900, and since then, many mathematicians have worked extensively in this direction. For recent work see, for example, [8–12].
We now define differential operator as follows: where , , for , and is the Pochhammer symbol defined by Here can also be written in terms of convolution as where and .
Note that and . When , we get the Sǔlǔgean differential operator , when , we obtain the Ruscheweyh operator , when , , we obtain the Al-Shaqsi and Darus , and when , we obtain the Al-Oboudi differential operator .
In this paper, we introduce new subclasses of analytic functions with respect to -symmetric points defined by differential operator. Some interesting properties of and are obtained.
Applying the operator where is a fixed positive integer, we now define classes of analytic functions containing the differential operator.
Definition 1.1. Let denote the class of functions in satisfying the condition where .
Definition 1.2. Let denote the class of functions in satisfying the condition where .
In order to prove our results, we need the following lemmas.
Lemma 1.3 (see ). Let , and let be the integral operator defined by , where Let be a convex function, with and in . If and , then , where is univalent and satisfies the differential equation
Lemma 1.4 (see ). Let , be complex numbers. Let be convex univalent in with and , , and let with and . If with , then
Lemma 1.5 (see ). Let and , respectively, be in the classes convex function and starlike function. Then, for every function , one has where denotes the closed convex hull of .
2. Main Results
Theorem 2.1. Let . Then defined by (1.5) is in .
Proof. Let , then by Definition 1.1 we have Substituting by , where () in (2.1), respectively, we have According to the definition of and , we know for any , and summing up, we can get Hence there exist in such that for since is convex. Thus .
Theorem 2.2. Let and . Then
and the operator can be written as .
Then from the definition of the differential operator , we can verify Thus if and only if .
Corollary 2.3. Let . Then defined by (1.5) is in .
Theorem 2.4. Let , with . If , then where and is the univalent solution of the differential equation
Proof. Let . Then in view of Theorem 2.1, , that is, From the definition of , we see that which implies that Using (2.10) and (2.12), we see that Lemma 1.3 can be applied to get (2.8), where and with and satisfies (2.9). We thus complete the proof of Theorem 2.4.
Theorem 2.5. Let and . Then
Proof. Let . Then Set where is analytic function with . By using the equation we get and then differentiating, we get Hence Applying (2.16) for the function we obtain Using (2.20) with , we obtain Since , then by using (2.14) in (2.21) we get the following. We can see that , hence applying Lemma 1.4 we obtain the required result.
Corollary 2.6. Let and . Then
Now we prove that the class , , is closed under convolution with convex functions.
Theorem 2.7. Let , , and is a convex function with real coefficients in . Then .
Corollary 2.8. Let , , and is a convex function with real coefficients in . Then .
The work presented here was partially supported by UKM-ST-06-FRGS0244-2010, and the authors would like to thank the anonymous referees for their informative and critical comments on the paper.
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