Journal of Function Spaces
Volume 2016 (2016), Article ID 6086409, 9 pages
http://dx.doi.org/10.1155/2016/6086409

Research Article

Fractional Calculus of Analytic Functions Concerned with Möbius Transformations

Nicoleta Breaz,¹ Daniel Breaz,¹ and Shigeyoshi Owa²

¹Department of Mathematics-Informatics, Faculty of Sciences, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania
²Department of Mathematics, Faculty of Education, Yamato University, Katayama 2-5-1, Suita, Osaka 564-0082, Japan

Received 17 October 2015; Accepted 3 March 2016

Academic Editor: Simone Secchi

Copyright © 2016 Nicoleta Breaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 satisfies

for 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 , by

where 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 that

for any real number .
Using the fractional calculus (13), we define

If we take in (14), then

implies 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 by

Then, we have that

where . If we define

then

This shows us that

That is,

Therefore, .
For , given by (16), becomes

Then, we see that .
Next, let us consider the function given by

for a fixed , where is given by (7). Then, it is easy to see that . Taking in (23), we have that

Letting

we obtain that

This shows that , , and . Therefore, there exists some such that for . It follows that

Thus, we say that

Consequently, we say that , for given by (16).
If , then

The open unit disk is mapped on the starlike domain of order in Figure 1.
If , then

Thus, maps onto the starlike domain of order in Figure 2.

Figure 1

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 that

where . Defining by

we have that

which shows us that

Thus, we obtain that

This gives us that .
For , given by (31), becomes

Then, it is easy to see that .
For this , we consider defined by (23). If we take for , we have that

If we write that

then

Since , , and , there exists some such that for . This gives us that

It follows that

Therefore, we say that , for .
If , then

maps onto the convex domain of order in Figure 3.
If , then

This function maps onto the convex domain of order in Figure 4.

Figure 3

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 , then

with the equality in (45) with given by (5). If , then

with the equality in (46) with given by (6).

We also need the following lemma.

Lemma 9. If is defined by

for a fixed for , then

for , 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 has

for . 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 , then

by means of (45). This implies inequality (55) for . Furthermore, if , then satisfies

which 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 , then

if , 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 that

then

where .

Now, we derive the following.

Theorem 15. If defined by (14) for satisfies

for some real which satisfies , then

Proof. We define the function by

with . Then, we see that is analytic in with and that

If there exists a point such that

then Lemma 14 gives us that

and . This implies that

Letting

we have that

for . Thus,

Therefore, satisfies

for . 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 that

and . 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 satisfies

then

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 that

Therefore, satisfies

For such a function , we see that

By using the computer, we know that . Indeed, the function satisfying (97) implies that

This shows us that .

Competing Interests

The authors declare that they have no competing interests.

References

T. H. MacGregor, “A subordination for convex functions of order α,” Journal of the London Mathematical Society, vol. 9, no. 4, pp. 530–536, 1975.

D. R. Wilken and J. Feng, “A remark on convex and starlike functions,” Journal of the London Mathematical Society, vol. 21, no. 2, pp. 287–290, 1980.

A. Marx, “Untersuchungen über schlichte Abbildungen,” Mathematische Annalen, vol. 107, no. 1, pp. 40–67, 1933.

E. Strohhäcker, “Beiträge zur Theorie der schlichten Funktionen,” Mathematische Zeitschrift, vol. 37, no. 1, pp. 356–380, 1933.

M. S. Robertson, “On the theory of univalent functions,” Proceedings of the American Mathematical Society, vol. 37, no. 2, pp. 374–408, 1936.

S. Owa, “On the distortion theorems,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59, 1978.

S. Owa, “On applications of the fractional calculus,” Mathematica Japonica, vol. 25, no. 2, pp. 195–206, 1980.

S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canadian Journal of Mathematics, vol. 39, no. 5, pp. 1057–1077, 1987.

R. J. Libera, “Some classes of regular univalent functions,” Proceedings of the American Mathematical Society, vol. 16, no. 4, pp. 755–758, 1965.

P. L. Duren, Univalent Functions, Springer, New York, NY, USA, 1983.

S. S. Miller and P. T. Mocanu, Differential Subordinations, vol. 225 of Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.

S. S. Miller and P. T. Mocanu, “Second-order differential inequalities in the complex plane,” Journal of Mathematical Analysis and Applications, vol. 65, no. 2, pp. 289–305, 1978.

I. S. Jack, “Functions starlike and convex of order α,” Journal of the London Mathematical Society, vol. 2-3, no. 3, pp. 469–474, 1971.

S. Ozaki and M. Nunokawa, “The Schwarzian derivative and univalent functions,” Proceedings of the American Mathematical Society, vol. 33, pp. 392–394, 1972.