#### Abstract

We obtain some properties related to the coefficient bounds for certain subclass of analytic functions. We also work on the differential subordination for a certain class of functions.

#### 1. Introductions

Let denote the class of functions which is analytic in the unit disc . Let Now let be the class of functions defined by The Hadamard product of two functions and is defined by where and are analytic in .

Let , , and then is analytic in the open unit disc . The function defined in (3) is equivalent to where is the Hadamard product and is analytic in the open unit disc .

We introduce a class of functions where Authors like Saitoh [1] and Owa [2, 3] had previously studied the properties of the class of functions . They obtained many interesting results and Wang et al. [4] studied the extreme points, coefficient bounds, and radius of univalency of the same class of functions. They obtained the following theorem among other results.

Theorem 1 (see [4]). *Let . A function if and only if can be expressed as
**
where is the probability measure defined on
**
For fixed , , and , the class and the probability measure defined on are one-to-one by expression (8).*

Recently, Hayami et al. [5] studied the coefficient estimates of the class of function in the open unit disc . They derived results based on properties of the class of functions , . Xu et al. [6] used the principle of differential subordination and the Dziok-Srivastava convolution operator to investigate some analytic properties of certain subclass of analytic functions. We also note that Stanciu et al. [7] used the properties of the class of functions , , to investigate the analytic and univalent properties of the following integral operator: where .

Motivated by the work in [1–7], we used the properties of the class of function , , to investigate the coefficient estimates of the class of functions in the open unit disc . We also use the principle of differential subordination to investigate some properties of the class of functions .

We state the following known results required to prove our work.

*Definition 2. *If and are analytic in , then is said to be subordinate to , written as or . If is univalent in , then and .

Theorem 3 (see [8]). *Consider if and only if there is probability measure on such that
**
and . The correspondence between and the set of probability measures on given by Hallenbeck [9] is one-to-one.*

Theorem 4 (see [10, 11]). *Let be convex in , , , and . If and
**
then
**
The function is convex and the best -dominant.*

Lemma 5 (see [10]). *Let be starlike in , with and . If satisfies
**
then
**
and is the best -dominant.*

Lemma 6 (see [12]). *Let , with in . Then, for , *(i)*,*(ii)*.*

*Remark 7. *The combination (i) and (ii) of Lemma 6 gives

*Remark 8. *For convenience, we limit our result to the principal branch and otherwise stated the constrains on , , , , , and which remain the same throughout this paper.

#### 2. Coefficient Bounds of the Class of Functions

We begin with the following result.

Theorem 9. *Let be as defined in (3). A function , if and only if can be expressed as
**
where and is the probability measure defined on .*

*Proof. *If , then
By Theorem 3,
and (19) can be written as
which yields
and so the expression (17).

If can be expressed as (17), reverse calculation shows that .

Corollary 10. *Let be defined as in (3). A function if and only if can be expressed as
**
where is the probability measure defined on .*

*Proof. *It is as in Theorem 9.

Corollary 11. *Let be as defined in (3). If , then, for and , we have
**
where
*

*Proof. *Let from (17) and
Comparing the coefficient yields the result.

Theorem 12. *Let and . Then for we have
*

*Proof. *Since , then
and then
where
From (23) and (28), we got
The application of Remark 7 to (27) gives
Since
then Theorem 12 is proved.

#### 3. Application of Differential Subordination to the Function

Here we calculate some subordinate properties of the class .

Theorem 13. *Let and let be starlike in with and . If
**
then
**
and is the best -dominant.*

*Proof. *Let ; then
Since is analytic in and , it suffices to show that
Following the same argument in [10] (pages 76 and 77), (36) is true. Application of Lemma 5 proves Theorem 13 with as the best -dominant.

*Example 14. *Let ; if
then
and is the best -dominant.

*Solution*. If and , , then simple calculation shows that and is starlike and the argument in [10] shows (37). The proof also follows from Lemma 5.

Theorem 15. *Let and , with . If
**
then
*

*Proof. *Let , and, from (6),
where
Let ; then , where , and from (39) we have
and is convex and univalent in . So, by Lemma 6,
This completes the proof of Theorem 15.

Corollary 16. *Let
**
If
**
then
*

*Proof. *The result follows from Theorem 15.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Authors’ Contribution

Both authors read and approved the final paper.

#### Acknowledgments

The work here is fully supported by LRGS/TD/2011/UKM/ICT/03/02 and GUP-2013-004. The authors also would like to thank the referee and editor in charge for the comments and suggestions given to improve their paper.