#### Abstract

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.

#### 1. Introduction and Definitions

Let be the class of normalized functions of the form which are analytic in the open unit disk . If is given by (1) and given by in , then the Hadamard product (or convolution) of and is defined by Let denote the class of functions analytic in the unit disk , satisfying the properties and This class has been introduced in [1]. Note that, for , we obtain the class defined and studied in [2], and for , we have the class of functions with positive real part greater than . In particular, is the class of functions with positive real part. From (3), we can easily deduce that if and only if Following the recent investigations [3] (see also [4, 5]), we have the following subclasses: We note that the class and are, respectively, the subclasses of , consisting of functions which are starlike of order and convex of order in . The class was considered by Noor [6], and is the class of quasiconvex univalent functions which was first introduced and studied in [7]. It can be easily seen from the above definition that 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 1. *Let and . Then the fractional integral operator is defined by
where the function is analytic in a simply connected region of the complex -plane containing the origin, with the order
and the multiplicity is removed by requiring to be real when .

*Definition 2. *For and and , the fractional derivative operator is defined by
where the multiplicity is removed as in the above definition.

The operators and 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 [12–15]) It is easily seen from (12) that for a function of the form (1), we have

We note that the operator is a generalization of many other operators, for example, which is a multiplier transformation operator studied by Jung et al. [16], which is the generalized Bernardi-Libra-Livingston integral operator. It can easily be verified from (12) that For the generalized fractional differintegral operator , we now define the following subclasses of .

*Definition 3. *Let . Then if and only if , for .

*Definition 4. *Let . Then if and only if , for .

*Definition 5. *Let . Then if and only if , for .

*Definition 6. *Let . Then if and only if , for .

*Definition 7. *Let . Then if and only if , for .

In this paper we establish some inclusion relationships and some other interesting properties for these subclasses.

#### 2. Main Inclusion Relationships

We recall first the following necessary lemma.

Lemma 8 (see [17]). *Let and , and let be a complex-valued function satisfying the conditions:*(i)* is continuous in ,*(ii)* and ,*(iii)* whenever and .*

If is a function analytic in such that and , for , then for .

Our first main inclusion relationship is given by the theorem below.

Theorem 9. *Let , , , and ; then
**
where
**
and .*

*Proof. *Let . Then upon setting
we see that the function is analytic in , with in . Using the identity (16) in (19) and differentiating with respect to , we get
Let
Then, by convolution technique, (see [18]), we have
and this implies that
We want to show that , where is given by (18), and this will show that for . Let
Then in view of (23) and (24), we obtain, for ,
We now form a function by choosing and in (25). Thus
We can easily see that the first two conditions of Lemma 8 are easily satisfied as is continuous in , , and . Now for we obtain
where , , and .

We note that if and only if and . From , we obtain as given by (18), and gives us . This completes the proof.

Theorem 10. *Let , , , and ; then
**
where is given by (18).*

*Proof. *To prove the inclusion relationship, we observe (in view of Theorem 9) that
which establishes Theorem 10.

Theorem 11. *Let , , , , and ; then
**
where is given by (18) and is defined in the proof.*

*Proof. *Let . Then there exists such that
Since , therefore we have
where . Next consider,
and proceed on the same technique of Theorem 9 (see also [5]); we obtain , where

In view of relation (7), an application of Theorem 11 is obtained as follows.

Theorem 12. *Let , , , , and ; then
**
where and are as in Theorem 11.*

Lemma 13 (see [19]). *Let be analytic in with and , . Then, for and (complex),
**
where is given by and , and this result is the best possible.*

Theorem 14. *Let , . Then for , where
**
This radius is exact.*

*Proof. *Let
where and in . Using a similar argument as in Theorem 9, we obtain
Applying Lemma 13, we get
where is given by (37). This completes our proof.

Lemma 15 (see [20]). *Let be analytic in with , and is a complex number satisfying , ; then , implies , where is given by
**
which is an increasing function of and . The estimate is sharp in the sense that bound cannot be improved.*

Theorem 16. *Let . Then
**
where
**
which is an increasing function of and .*

*Proof. *Let
where is analytic and in . Identity (16) gives us
This implies that
Now using Lemma 15, we get the desired result.

Lemma 17 (see [21]). *Let be convex and starlike in . Then, for analytic in with , is contained in the convex hull of . By convex hull of a set , one means the intersection of all convex sets that contain .*

Theorem 18. *Let be a convex function and . Then .*

*Proof. *Let , and let
Then
Also, . Therefore, . By logarithmic differentiation of (48), we have
where
is analytic in , and . From Lemma 17, we see that is contained in the convex hull of . Since is analytic in and , then lies in . This implies that .

This is application of Theorem 18.

Theorem 19. *The class is invariant under the following differintegral operators. That is if , then so does , where are given to us:*(i)*,
*(ii)*,
*(iii)*, , ,*(iv)*, .*

The proof immediately follows from Theorem 18. Since we can write with each is convex for , and 4.

#### Acknowledgment

The authors would like to express their thanks to the referee for the careful reading and suggestions made for the improvement of the paper.