Chinese Journal of Mathematics

Volume 2017 (2017), Article ID 4674782, 4 pages

https://doi.org/10.1155/2017/4674782

## New Subclasses concerning Some Analytic and Univalent Functions

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

Correspondence should be addressed to Maslina Darus; ym.ude.mku@anilsam

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 thatand if and only if

The function given byis in the class and the function given byis in the class .

If we consider the function given byfor 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 thatfor some real , where we take the principal value for If satisfiesfor some real , then we say that

Also, if satisfiesfor 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 andfor This shows that for Further, we have that for andfor Lettingwe have thatThus, we see thatfor . This completes the proof of the theorem.

Corollary 2. *A functionbelongs to the class and *

Next, we discuss some properties of functions for

Theorem 3. *If given by (5) satisfiesfor some , then **The equality holds true for given by*

*Proof. *Let the function be given by (5); then, we have thatif satisfies (14). This shows that Further, if we consider a function given by (15), then we see that

Theorem 4. *If given by (5) satisfiesfor 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 iffor 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 thatThen, we obtain thatThis 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 iffor some The equality holds true for*

#### 3. Radius Problems

In this section, we considerfor 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 thatfor . This gives us Lettingwe see that This gives us

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

*Proof. *If we considerthen

*Remark 9. *If in (35), thenand if , then

#### 4. Partial Sums

Finally, we consider the partial sums of given by (5). In view of (5), we writefor 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 thatwhere and . This gives us Defining bywe 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. View at MathSciNet - M. I. S. Robertson, “On the theory of univalent functions,”
*Annals of Mathematics. Second Series*, vol. 37, no. 2, pp. 374–408, 1936. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Google Scholar · View at MathSciNet · View at Scopus