Abstract
Let be the class of functions in the open unit disk with and . Also, let be a Möbius transformation in for some . Applying the Möbius transformations, we consider some properties of fractional calculus (fractional derivatives and fractional integrals) of . Also, some interesting examples for fractional calculus are given.
1. Introduction
Let denote the class of functions of the form which are analytic in the open unit disk .
If satisfiesfor some real , then is said to be starlike of order in . We denote by the class of all starlike functions of order in and . Furthermore, if satisfies for some real , then we say that is convex of order in . We also denote by the class of all such functions and . In view of definitions for the classes and , we know that(i) if and only if ;(ii) if and only if .
Further, MacGregor [1] and Wilken and Feng [2] have the sharp inclusion relation that for each with For , Marx [3] and Strohhäcker [4] showed that . Also, by Robertson [5], we know that the extremal function for the class is and the extremal function for the class is
For , we apply the following Möbius transformation:for a fixed . This Möbius transformation maps onto itself and to .
2. Fractional Calculus
From among the various definitions for fractional calculus (i.e., fractional derivatives and fractional integrals) given in the literature, we have to recall here the following definitions for fractional calculus which are used by Owa [6, 7] and by Owa and Srivastava [8].
Definition 1. The fractional integral of order is defined, for , by where and the multiplicity of is removed by requiring to be real when .
Definition 2. The fractional derivative of order is defined, for , bywhere and the multiplicity of is removed as in Definition 1 above.
Definition 3. Under the hypotheses of Definition 2, the fractional derivative of order is defined by where and .
Remark 4. In view of definitions for the fractional calculus of , we see that for and .
Therefore, we can write thatfor any real number .
Using the fractional calculus (13), we defineIf we take in (14), thenimplies the Libera integral operator defined by Libera [9]. Therefore, given by (14) is the generalization operator of Libera integral operator.
Let us give two examples for the fractional operator defined in (14).
Example 5. Let us define byThen, we have thatwhere . If we definethenThis shows us thatThat is, Therefore, .
For , given by (16), becomesThen, we see that .
Next, let us consider the function given byfor a fixed , where is given by (7). Then, it is easy to see that . Taking in (23), we have thatLettingwe obtain thatThis shows that , , and . Therefore, there exists some such that for . It follows thatThus, we say that Consequently, we say that , for given by (16).
If , thenThe open unit disk is mapped on the starlike domain of order in Figure 1.
If , thenThus, maps onto the starlike domain of order in Figure 2.
Example 5 means that there is some function such that and .
Next, we consider the following.
Example 6. Let a function be given by Then, we have thatwhere . Defining by we have that which shows us that Thus, we obtain thatThis gives us that .
For , given by (31), becomesThen, it is easy to see that .
For this , we consider defined by (23). If we take for , we have that If we write that thenSince , , and , there exists some such that for . This gives us that It follows thatTherefore, we say that , for .
If , thenmaps onto the convex domain of order in Figure 3.
If , then This function maps onto the convex domain of order in Figure 4.
Example 6 says that there exists some function such that and .
In view of Examples 5 and 6, we introduce the following.
Definition 7. Let , with and let be defined by (23) for a fixed . Then, we say that(i) if is univalent in ,(ii) if ,(iii) if .Also, we write that and when .
In order to discuss our classes , , and , we need the following lemma due to Robertson [5] (also see Duren [10]).
Lemma 8. If , thenwith the equality in (45) with given by (5). If , thenwith the equality in (46) with given by (6).
We also need the following lemma.
Lemma 9. If is defined byfor a fixed for , thenfor , where is given by (7).
Proof. We use the mathematical induction to prove (48). For , the right-hand side of (48) becomes Therefore, (48) holds true for .
Assume that relation (48) is true for a fixed positive integer . Then, some calculations lead us to This means that relation (48) holds true for . Thus, by applying the mathematical induction, we complete the proof of the lemma.
Taking in Lemma 9, we have the following.
Corollary 10. If is defined by (47) for , then one hasfor . Furthermore, we have
Applying Corollary 10, we have the following.
Theorem 11. Let be defined by (14) for with .(i)If , then with the equality for given by (ii)If , then with the equality for given by (iii)If , then with the equality for given by
Proof. Note that for . Therefore, Corollary 10 gives us that According to Lemma 8, we have if , then we calculate that because If , thenby means of (45). This implies inequality (55) for . Furthermore, if , then satisfieswhich implies inequality (57). Consequently, we complete the proof of the theorem.
Since , letting in Theorem 11, we have the following.
Corollary 12. Let with .(i)If , then for .(ii)If , then for .(iii)If , then for .
Taking in Corollary 12, we have the following.
Corollary 13. If , thenif , then and if , then
3. Univalence of Fractional Calculus
Let and be analytic in . Then, is said to be subordinate to , written , if there exists a function analytic in with and (), such that . Furthermore, if is univalent in , then the subordination is equivalent to and (cf. Miller and Mocanu [11]).
To discuss the univalence of fractional calculus given by (14), we need the following lemma due to Miller and Mocanu [12] (or due to Jack [13]).
Lemma 14. Let the function be analytic in with . If there exists a point such thatthenwhere .
Now, we derive the following.
Theorem 15. If defined by (14) for satisfies for some real which satisfies , then
Proof. We define the function bywith . Then, we see that is analytic in with and thatIf there exists a point such that then Lemma 14 gives us that and . This implies thatLettingwe have thatfor . Thus, Therefore, satisfiesfor . This contradicts our condition (74) for . Thus, satisfies for all . With this reason above, we conclude that
Next, we show the following.
Theorem 16. If defined by (14) for satisfies for some real , then
Proof. Let us define the function by Then,Suppose that there exists a point such that then we can write thatand . Therefore, we have that which contradicts our condition (86). Thus, we say that there is no such that . Consequently, letting for all , we prove the theorem.
Taking in Theorem 16, we have
Corollary 17. If defined by (14) for satisfiesthen
Remark 18. In view of the result for the univalence of analytic functions due to Ozaki and Nunokawa [14], we see that satisfying inequality (94) is univalent in .
Example 19. Let us consider the function given by Then, we have thatTherefore, satisfiesFor such a function , we see thatBy using the computer, we know that . Indeed, the function satisfying (97) implies thatThis shows us that .
Competing Interests
The authors declare that they have no competing interests.