Chinese Journal of Mathematics
Volume 2017 (2017), Article ID 4674782, 4 pages
https://doi.org/10.1155/2017/4674782

Research Article

New Subclasses concerning Some Analytic and Univalent Functions

Maslina Darus¹ and Shigeyoshi Owa²

¹School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia
²Department of Mathematics, Faculty of Education, Yamato University, Katayama 2-5-1, Suita, Osaka 564-0082, Japan

Correspondence should be addressed to Maslina Darus; maslina@ukm.edu.my

Received 4 April 2017; Accepted 17 July 2017; Published 20 August 2017

Academic Editor: Tomasa Calvo

Copyright © 2017 Maslina Darus and Shigeyoshi Owa. 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.

Considering a function which is analytic and starlike in the open unit disc and a function which is analytic and convex in we introduce two new classes and concerning . The object of the present paper is to discuss some interesting properties for functions in the classes and

1. Introduction and Preliminaries

Let be the class of functions which are analytic in the open unit disk with and .

Let denote the subclass of consisting of functions which are univalent in . Also, let be the subclass of consisting of which are starlike of order in . Further, we say that if satisfies . A function is said to be convex of order in (cf. [1–3]).

With the above definitions for classes , , , and , it is known that

and if and only if

The function given by

is in the class and the function given by

is in the class .

If we consider the function given by

for some real , we discuss some properties between functions in (2) and (3), where we consider the principal value for .

With the function given by (4), we introduce a class of analytic functions with series expansion in such that

for some real , where we take the principal value for If satisfies

for some real , then we say that

Also, if satisfies

for some real , then we say that

With the above definitions for the classes and , we have that if and only if and that if and only if .

2. Some Properties

In this section, we consider some properties of functions with series expansion given by (4).

Theorem 1. If is given by (4), then for and for .

Proof. For given by (4), we see that for and

for This shows that for Further, we have that for and

for Letting

we have that

Thus, we see that

for . This completes the proof of the theorem.

Corollary 2. A function

belongs to the class and

Next, we discuss some properties of functions for

Theorem 3. If given by (5) satisfies

for some , then
The equality holds true for given by

Proof. Let the function be given by (5); then, we have that

if satisfies (14). This shows that Further, if we consider a function given by (15), then we see that

Theorem 4. If given by (5) satisfies

for some , then
The equality in (18) holds true for given by

Further, we obtain the following.

Theorem 5. Let be given by (5) with Then, if and only if

for some The equality holds true for

Proof. Theorem 3 implies that if satisfies (20), then Next, we suppose that Then,

If we consider , then we have that

Then, we obtain that

This gives us

that is,

Thus, if and only if the coefficient inequality (20) holds true.

Further, for the class , we have the following.

Theorem 6. Let be given by (5) with Then, if and only if

for some The equality holds true for

3. Radius Problems

In this section, we consider

for some real . Then, we say that and for any real

Now, we derive the following.

Theorem 7. If is given by (29) with , then

Proof. For given by (29), we have that

for . This gives us

Letting

we see that This gives us

Corollary 8. If is given by (29) with , then

for

Proof. If we consider

then

Remark 9. If in (35), then

and if , then

4. Partial Sums

Finally, we consider the partial sums of given by (5). In view of (5), we write

for some real Recently, Darus and Ibrahim [4] and Hayami et al. [5] have shown some interesting results for some partial sums of analytic functions.

Now, we derive the following.

Theorem 10. Let be given by (40) with Then,

Proof. It follows that

where and . This gives us

Defining by

we have that with
Thus, we obtain

Making in (46), we see (41). Also letting in (46), we see (42).

Corollary 11. Let be given by (40) with Then,

Proof. Since , satisfies (41).
Therefore, for , (41) gives us

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work here is supported by MOHE Grant FRGS/1/2016/STG06/UKM/01/1.

References

P. L. Duren, Univalent Functions, vol. 259, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.

A. W. Goodman, Geometric Theory of Functions, vol. I and II, Mariner, Tampa, Fla, USA, 1983.

M. I. S. Robertson, “On the theory of univalent functions,” Annals of Mathematics. Second Series, vol. 37, no. 2, pp. 374–408, 1936.

M. Darus and R. W. Ibrahim, “Partial sums of analytic functions of bounded turning with applications,” Computational & Applied Mathematics, vol. 29, no. 1, pp. 81–88, 2010.

T. Hayami, K. Kuroki, E. Y. Duman, and S. Owa, “Partial sums of certain univalent functions,” Applied Mathematical Sciences, vol. 6, no. 13-16, pp. 779–805, 2012.