/ / Article

Research Article | Open Access

Volume 2017 |Article ID 4674782 | 4 pages | https://doi.org/10.1155/2017/4674782

# New Subclasses concerning Some Analytic and Univalent Functions

Accepted17 Jul 2017
Published20 Aug 2017

#### 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. ).

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

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 , thenfor

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  and Hayami et al.  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.

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