%0 Journal Article
%T Norm of a Volterra Integral Operator on Some Analytic Function Spaces
%A Hao Li
%A Songxiao Li
%J ISRN Mathematical Analysis
%D 2013
%R 10.1155/2013/492052
%X Let be an analytic function in the unit disc . The Volterra integral operator is defined as follows: In this paper, we compute the norm of on some analytic function spaces. 1. Introduction Let be the unit disk of complex plane and the class of functions analytic in . Denote by the normalized Lebesgue area measure in and the Green function with logarithmic singularity at ; that is, , where is the M£¿bius transformation of . Let . The is the space of all functions such that From [1, 2], we see that = BMOA, the space of all analytic functions of bounded mean oscillation. When , the space is the same and equal to the Bloch space , which consists of all for which See [3, 4] for the theory of Bloch functions. For , the -Bloch space, denoted by , is the space of all such that It is clear that for . Let and let . The mean Lipschitz space consists of those functions for which It is obvious that is just the Bloch space , which is contained in for every . Note that increases with . We refer to [5] for more information of mean Lipschitz spaces. For , we say that an belongs to the growth space if It is easy to see that . For , an is said to belong to the space if For , the Besov space is defined to be the space of all analytic functions in such that Let . The Volterra integral operators and are defined as follows: It is easy to see that where denotes the multiplication operator; that is, . If is a constant, then all results about , , or are trivial. In this paper, we assume that is a nonconstant. Both operators have been studied by many authors. See [6¨C23] and the references therein. Norms of some special operators, such as composition operator, weighted composition operator, and some integral operators, have been studied by many authors. The interested readers can refer [13, 24¨C32], for example. Recently, Liu and Xiong studied the norm of integral operators and on the Bloch space, Dirichlet space, BMOA space, and so on in [13]. In this paper, we study the norm of integral operator on some function spaces in the unit disk. 2. Main Results In this section, we state and prove our main results. In order to formulate our main results, we need some auxiliary results which are incorporated in the following lemmas. Lemma 1 (see [5, page 144]). If £¿£¿ , then , , and the inequality is sharp for each fixed . Lemma 2. Let and . For any , the following one has: where is any point in . Proof. For any , taking and the subharmonicity of , we get and so For any , let . Replacing by and applying the change of variable formula give the following: The proof is complete. Theorem 3. Let .
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2013/492052/