#### 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.