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ISRN Mathematical Analysis
Volume 2011 (2011), Article ID 149830, 11 pages
General Properties for Volterra-Type Operators in the Unit Disk
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor Darul Ehsan 43600, Malaysia
Received 26 October 2010; Accepted 5 December 2010
Academic Editors: A. L. Sasu, B. Kaltenbacher, and J. Colliander
Copyright © 2011 Rabha W. Ibrahim 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.
The object of this paper is to study general properties such as boundedness, compactness, and geometric properties for two integral operators of Volterra-Type in the unit disk.
Let be the class of analytic functions in . Suppose that is a holomorphic map, . The integral operator, called Volterra-type operator, was introduced by Pommerenke in . Another natural integral operator is defined as follows: The importance of the operators and comes from the fact that where the multiplication operator is defined by Furthermore, Volterra integral equations arise in many physical applications (see [2–4]).
In the past few years, many authors focused on the boundedness and compactness of Volterra-type integral operator between several spaces of holomorphic functions. In , Pommerenke showed that is a bounded operator on the Hardy space . The boundedness and compactness of and between some spaces of analytic functions, as well as their -dimensional extensions, were investigated in [5–11].
For functions , the integral operators and contain well-known integral operators in the analytic function theory and geometric function theory such as the generalized Bernardi-Libera-Livingston linear integral operator (cf. [12–14]) and the Srivastava-Owa fractional derivative operators (cf. [15, 16]). Recently, Breaz and Breaz introduced two integral operators of analytic functions taking the form (1.1) and (1.2) (see ). Further, the integral operators of Volterra-Type involving the integral operators were studied in [18–22]. Finally, these operators are involving the Cesáro integral operator (see [23–25]).
A function is called in the class if and only if it has the norm (see ) Note that the fraction is called pre-Schwarzian derivative which is usually used to discuss the univalency of analytic functions (see ). Moreover, the norm in (1.5) is a modification to one defined in .
The purpose of this paper is to study the boundedness, compactness, and some geometric properties of the integral operators and for the functions and is an analytic function on the open unit disk.
2. The Boundedness and Compactness
In this section, we consider the boundedness and compactness of the operators and on the classes .
Consider the space of all functions which are satisfying
Theorem 2.1. Assume that is an analytic function on . Then, for functions , is bounded if and only if .
Proof. Assume that is bounded. Taking the function given by , we see that .
Conversely, assume that , we have By taking the supremum for the last assertion over and using the fact that the quantity is finite, the boundedness of the operator follows.
Theorem 2.2. Assume that is an analytic function on . Then, is bounded if and only if , where
Proof. Assume that . Then, we obtain
By taking the supremum for the last assertion over , the boundedness of the operator follows.
Conversely, assume that is bounded, then there is a positive constant such that for every . Set for such that . Then, we have Thus, and then It is clear that the relation (2.9) is finite when , hence . Setting therefore, we have
Now letting for such that . Then, we obtain Thus, we conclude that In the same manner of the previous case, we have Consequently, we have
From (2.12) and (2.17), we have for all . Also, we have From (2.18) and (2.19), we obtain , as desired.
In the following results, we study the compactness of the integral operators and in an open disc.
Theorem 2.3. Assume that is an analytic function on . Then, for functions , the integral operator is compact if and only if .
Proof. If is compact, then it is bounded, and by Theorem 2.1 it follows that .
Now assume that , that is a sequence in , and uniformly on as . Now for every , there is such that where . Since is arbitrary, then we can chose for and Since for on we have , and that is an arbitrary positive number, by letting in the last inequality, we obtain that . Therefore, is compact.
Theorem 2.4. Assume that is an analytic function on . Then, the integral operator is compact if and only if is a constant defer from zero.
Proof. Assume that is a constant without loss of generality and assume that . Then, it is clear that is compact.
Conversely, assume that is compact. Let , be a sequence in such that as . Our aim is to show that as , then by the maximum modulus theorem, we have is a constant. In fact, setting Then, we obtain Consequently, we have Similar to the proof of Theorem 2.2, we see that uniformly on . Since is compact, then we get Thus, Implies that and consequently is a constant as desired.
3. Some Geometric Properties
In this section, we introduce some geometric properties for analytic function . A function which normalized as denoted this class by . Recall that a function is said to be star-like of order in if it satisfies Also, a function is called convex in if it satisfies It follows that
In the next result, we discuss the convexity of the integral operators and .
Theorem 3.1. Assume that . If and such that , then the function is convex of order .
Proof. Assume that . Then, we obtain Consequently, we get Hence, .
Theorem 3.2. Assume that . If and such that , then the function is convex of order .
Proof. Assume that . Then, we have Consequently, we get Hence, .
Theorem 3.3. Assume that . If and such that , then the multiplication operator is star-like of order .
Proof. Assume that . Then, we obtain
The next result comes directly from the definition of the class and the fact that if and only if is uniformly locally univalent (see ).
Theorem 3.4. Assume that is an analytic function on and . Then, the functions and are in the class if and only if is locally univalent in .
The work presented here was supported by the MOHE Grant no. UKM-ST-06-FRGS0107-2009, Malaysia.
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