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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 259054, 10 pages
On an Integral Transform of a Class of Analytic Functions
Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148106, India
Received 5 June 2012; Accepted 14 September 2012
Academic Editor: Josip E. Pecaric
Copyright © 2012 Sarika Verma et al. 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.
For and , let denote the class of all normalized analytic functions in the open unit disc such that , for some . It is known (Noshiro (1934) and Warschawski (1935)) that functions in are close-to-convex and hence univalent for . For , we consider the integral transform , where is a nonnegative real-valued integrable function satisfying the condition . The aim of present paper is, for given , to find sharp values of such that (i) whenever and (ii) whenever .
Let denote the class of analytic functions defined in the open unit disc with the normalizations , and let be the subclass of consisting of functions univalent in . For any two functions and in , the Hadamard product (or convolution) of and is the function defined by For , Fournier and Ruscheweyh  introduced the integral operator where is a nonnegative real-valued integrable function satisfying the condition . This operator contains some well-known operators such as Libera, Bernardi, and Komatu as its special cases. Fournier and Ruscheweyh  applied the famous duality theory to show that for a function in the class the linear integral operator is univalent in . Since then, this operator has been studied by a number of authors for various choices of . In another remarkable paper, Barnard et al. in  obtained conditions such that whenever is in the class with . Note that for , functions in satisfy the condition in and thus are close-to-convex in . A domain in is close-to-convex if its compliment in can be written as union of nonintersecting half lines.
In 2008, Ponnusamy and Rønning  discussed the univalence of for the functions in the class In a very recent paper, Ali et al.  studied the class where and . In this paper, they obtained sufficient conditions so that the integral transform maps normalized analytic functions into the class of starlike functions. It is evident that , and .
In the present paper, we shall mainly tackle the following problems. (1)For given , find sharp values of such that whenever . (2)For given , find sharp values of such that whenever . To prove one of our results, we shall need the generalized hypergeometric function , so we define it here.
Let and be complex numbers with . Then the generalized hypergeometric function is defined by where is the Pochhammer symbol, defined in terms of the Gamma function, by In particular, is called the Gaussian hypergeometric function. We note that the series in (1.7) converges absolutely for if and for if .
We shall also need the following lemma.
Lemma 1.1 (see ). Let , , and . Then, for analytic in with , the conditions and imply , where .
2. Main Results
Theorem 2.1. Let , satisfy (2.1). Further, let be given, and define by If , then . The value of is sharp.
Proof. The case corresponds to Theorem 1.5 in . So we assume that .
Define Writing , it follows that It is a simple exercise to see that Let , where is defined by (1.2). Then for , we can write
Since , it follows that for some . Now, for each , we first claim that which, by Lemma 1.1, implies that . Therefore, it suffices to verify the inequality (2.7). Using the identity (which can be checked by comparing the coefficients of on both sides) it follows that Thus, Therefore, for , we have in the view of .
To prove the sharpness, let be the function determined by Using a series expansion, we see that we can write Then, where . Equation can be restated as Finally, which for takes the value This shows that the result is sharp.
Letting and in Theorem 1.1, we obtain the following result of Ruscheweyh .
Corollary 2.2. Let , and define by If , then . The value of is sharp.
Theorem 2.3. Let and , and define by If , then . The value of is sharp.
Proof. The idea of the proof is similar to the one used to prove Theorem 2 in .
Let . Clearly, Since, , so with we have , where .
For , Putting this value in (2.20), Equivalently, Thus In the case when , Since This leads to, which is clearly (2.25) with .
Further if and only if . Now using (2.25), we obtain Since for some , it follows by duality principle [8, page 23] that if, and only if, Using , we get By using (2.19), we have Thus, which implies that Thus, we deduce, using duality principle, that is contained in a half plane not containing the origin. So, and hence .
To prove the sharpness, let . Further, gives or Further, assume that Since ,
so, Therefore, for , This shows that the result is sharp.
Corollary 2.4. Let and , and define by
If , then . The value of is sharp.
Upon setting with , we have the following corollary.
Corollary 2.5. Let , , and be given, and let be defined by Suppose that , then , where The constant is sharp.
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