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.
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 . Note that, for , we obtain the class defined and studied in , 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  (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 , and is the class of quasiconvex univalent functions which was first introduced and studied in . 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.  and generalized fractional derivative operators due to Raina and Nahar  (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., ). 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. , 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 ). 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
Then, by convolution technique, (see ), 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).
Theorem 11. Let , , , , and ; then where is given by (18) and is defined in the proof.
Theorem 12. Let , , , , and ; then where and are as in Theorem 11.
Lemma 13 (see ). 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.
Lemma 15 (see ). 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 .
Lemma 17 (see ). 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.
The authors would like to express their thanks to the referee for the careful reading and suggestions made for the improvement of the paper.
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