Some Inclusion Relations Associated with Generalized Fractional Differintegral Operator

A generalized fractional differintegral operator is used to define some new subclasses of analytic functions in the open unit disk . For each of these new function classes, several inclusion relationships are established.


Introduction and Definitions
Let A be the class of normalized functions  of the form which are analytic in the open unit disk U = { ∈ , || < 1}.
If  ∈ A is given by (1) and  ∈ A given by () =  + ∑ ∞  = 2     in  ∈ U, then the Hadamard product (or convolution) of  and  is defined by ( Let   () denote the class of functions ℎ() analytic in the unit disk U, satisfying the properties ℎ(0) = 1 and ( This class   () has been introduced in [1].Note that, for  = 0, we obtain the class   defined and studied in [2], and for  = 2, we have the class () of functions with positive real part greater than .In particular, (0) is the class  of functions with positive real part.From (3), we can easily deduce that ℎ ∈   () if and only if ) ℎ 2 () , ℎ 1 , ℎ 2 ∈  () .
We recall here the following family of generalized fractional integral operators due to Srivastava et al. [8] and generalized fractional derivative operators due to Raina and Nahar [9] (see also [10,11]).
Definition 2. For  − 1 ≦  <  and ,  ∈ R and  ∈ N, the fractional derivative operator J ,, 0, is defined by where the multiplicity ( − ) − is removed as in the above definition.
The operators I ,, 0, and J ,, 0, include the Riemann-Liouville and Erdelyi-Kober operators of fractional calculus (see, e.g., [10]).Using the hypotheses of Definitions 1 and 2 the generalized fractional differintegral operator is defined by (see also [12][13][14][15]) It is easily seen from ( 12) that for a function  of the form (1), we have We note that the operator S ,, 0, is a generalization of many other operators, for example, (i) S 0,0,0 0,  () =  () , which is a multiplier transformation operator Ω   studied by Jung et al. [16], which is the generalized Bernardi-Libra-Livingston integral operator.It can easily be verified from (12) that International Journal of Analysis In this paper we establish some inclusion relationships and some other interesting properties for these subclasses.