Abstract

Let be an open unit disc in the complex plane and let as well as be analytic maps. For an analytic function on the weighted fractional differentiation composition operator is defined as , where , , and . In this paper, we obtain a characterization of boundedness and compactness of weighted fractional differentiation composition operator from mixed-norm space to weighted-type space .

1. Introduction

The classical/Gaussian hypergeometric series is defined by the power series expansion Here , , are complex numbers such that , , and is Pochhammer’s symbol/shifted factorial defined by Appel’s symbol and for .

Obviously, . Many properties of the hypergeometric series including the Gauss and Euler transformations are found in standard textbooks such as [1, 2].

For any two analytic functions and represented by their power series expansion, in , the Hadamard product (or convolution) of and denoted by and is defined by in . Moreover, In particular, if , we have If and , then the fractional derivative (see [3]) of order is defined by In terms of convolution, we also have

For , we define

It is obvious to find that the fractional derivative and the ordinary derivative satisfy

Let be fixed and let   be nonconstant analytic self-map of . For and , we can define an operator on , called a weighted fractional differentiation composition operator, by We can regard this operator as a generalization of a multiplication operator and a weighted composition operator. In this paper we study the boundedness and the compactness of weighted fractional differentiation composition operator from mixed-norm spaces to weighted-type spaces. Recall that a positive continuous function on is called normal if there is and and with such that

Let be the normalized Lebesgue area measure on and let    be the space of all analytic functions on .

For , , and is normal we denote by the space of all functions such that where and where

When and , , then (classical weighted Bergman space), defined by Further, when then the natural limit to the weighted Bergman space is the Hardy space ; that is, . For more details on Hardy space, see [4].

Suppose is normal and radial; that is, . The weighted-type space consists of all such that A little version of is denoted by as the subset of consisting of all such that Note that is a closed subspace of . For the space , see [5].

For , we have the space of bounded analytic functions , where

For ,   equals the weighted composition operator defined by , , which reduces to the composition operator for . During the last century, composition operators were studied between different spaces of analytic functions. There are many papers about these operators. Among other things, they deal with boundedness and compactness of these operators. But we will discuss mostly the mixed-norm spaces. For some recent papers on weighted composition operators on some -type spaces, see [5–14].

If , we get the operator which for gives which for gives and gives . For particular choices of , , and , we obtain many operators which are product, addition, and composition of multiplication and differentiation operators. For more details of these types of operators see [15–21].

Recall that, in [22], Stević characterized the boundedness and compactness of weighted differentiation composition operators from mixed-norm spaces to the weighted-type spaces. Our results can be viewed as generalizations of their results.

Throughout this paper, denotes a positive constant which may vary for different lines. The notation means there is a positive constant such that .

2. Some Lemmas

We collect some basic lemmas which are useful in the proof of the main results.

Lemma 1 (see [23, Lemma 6]). For and one has

Lemma 2 (see [24, Proposition ]). For one has

Lemma 3 (see [22, Lemma 1]). Assume , , is normal, and . Then, for every , there is a positive constant independent of such that

3. Main Results

In the following results we show boundedness of the operator from mixed-norm space to weighted-type space . As a consequence we obtain many new results in the form of corollaries.

Lemma 3 is true for the th order derivative of . Here we will give a more general result which involves fractional derivative of .

Proposition 4. Assume , , is normal, and . Then, for every , there is a positive constant independent of such that

Theorem 5. For , , , , and are normal, and is an analytic self-map of . Then is bounded if and only if Moreover, if is bounded, then the following asymptotic relation holds:

Proof of Proposition 4. Let .
Using integral representation of convolution we have, for , Since , Minkowski’s inequality gives where .
For , we have Putting , we have, for any , Since therefore For Putting , we have By using the following asymptotic relations: we have A simple calculation using Cauchy’s integral formula gives The above inequality can be written as the following: When , from inequality (31) we have For we have and so . Therefore the proof is similar to the case, .