Weighted Fractional Differentiation Composition Operators from Mixed-Norm Spaces to Weighted-Type Spaces

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