#### Abstract

Double integral operators which were considered by S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008), 591–597) are discussed. In order to show the analytic function is starlike of order in the open unit disk , the theory of differential subordinations for analytic functions is applied. The object of the present paper is to discuss some interesting conditions for to be starlike of order in concerned with second-order differential inequalities and double integral operators.

#### 1. Introduction, Definition and Preliminaries

Let denote the class of functions which are analytic in the open unit disk For a positive integer and , we define the following classes of analytic functions:

and

with .

A function is said to be starlike of order in if it satisfies for some . We denote by the subclass of consisting of all functions which are starlike of order in .

By the familiar principle of differential subordinations between analytic functions and in , we say that is subordinate to in if there exists an analytic function with and such that .

We denote this subordination by In particular, if is univalent in , then it is known that

To obtain some results of this paper, we need the following two lemmas concerning the differential subordinations.

Lemma 1.1 (see [1, Hallenbeck and Ruscheweyh]). Letbe a convex function with and let. If and then where This result is sharp.

Lemma 1.2 (see [2, Al-Amiri and Mocanu]). Let be a positive integer, and let be real with. Let with and If satisfies then and this result is sharp.

By making use of these lemmas, Miller and Mocanu [3] have investigated some second-order differential inequality that implies starlikeness and deduced the following lemma.

Lemma 1.3. Let and let . If satisfies then is starlike.

Furthermore, by using Lemma 1.3, Miller and Mocanu [3] obtained some result concerning the double integral starlike operator as follows.

Lemma 1.4. Let and let . If satisfies then the function given by is starlike.

#### 2. Some Second-Order Differential Inequalities for Starlike of Order

In this section, we deduced some conditions concerning the second-order differential inequality to show that is starlike of order in .

Theorem 2.1. Let and let and . If satisfies then is starlike of order in .

Proof. We can rewrite the inequality (2.1) in terms of subordination as If we set then the subordination (2.2) becomes Applying Lemma 1.1 to this first-order differential subordination, we obtain or equivalently If we consider and then the subordination (2.6) can be written as Since and the function satisfies using Lemma 1.2, we obtain the subordination , or It follows from the subordination (2.6) that while from the subordination (2.11) that Combining these last two inequalities, we see that which simplifies to This gives us that which proves that is starlike of order in .

We introduce the following example for Theorem 2.1.

Example 2.2. For the function , we have Furthermore, we see that

Remark 2.3. Letting in Theorem 2.1, we obtain Lemma 1.3 given by Miller and Mocanu [3].

Also, setting in Theorem 2.1, we have the following.

Corollary 2.4. Let and let. If satisfies then is starlike of order in .

For the case in the above corollary, we find the following.

Remark 2.5. For , we have that The case was first discussed by Obradović [4].

Next, by making use of Theorem 2.1, we obtain the following result concerning the double, integral operator for starlike of order .

Theorem 2.6. Let a function satisfy for some and . Then the function given by is starlike of order in .

Proof. We first consider the function satisfying the differential equation Then, it is clear that Thus, from Theorem 2.1, we see that the solution of the differential equation (2.23) must be starlike of order . The solution of (2.23) can be obtained in two integrations. If we set , then the equation (2.23) can be simplified to which has the solution given by Since , we have that is,

Remark 2.7. Taking in Theorem 2.6, we find Lemma 1.4 given by Miller and Mocanu [3]. However, (1.14) and (2.22) are double integral operators of the same form.

Moreover, making and in Theorem 2.6, we have the following.

Corollary 2.8. If and for some , then

As examples of Corollary 2.8, we get the following.

Example 2.9. For the function , we find because

Example 2.10. For the function , we have Then, we see that that is,

#### 3. Other Result for Starlikeness of Order

To obtain that is starlike of order in Theorem 2.1, we showed that In this section, to obtain that is starlike of order , we consider some second-order differential inequality concerning the order and show the following inequality: for some .

Remark 3.1. Foe some , we see that

Now, we consider the following theorem.

Theorem 3.2. Let and let. If satisfies then or is starlike of order in .

Proof. (i) For the case , the inequality (3.4) can be written as follows: If we set then the subordination (3.6) becomes Applying Lemma 1.1 as well as the proof of Theorem 2.1, we obtain that or equivalently, that Also, if the function is defined by then the subordination (3.6) becomes namely, Since , we can use Lemma 1.1 as and obtain that is, From the subordination (3.10), we find while, from the subordination (3.15), we get By combining these last two inequalities, we obtain that which simplifies to (ii) For the case , we can rewrite (3.4) in terms of the subordination as Applying Lemma 1.1 in similar to case (i), we can obtain that Furthermore, if we set then the subordination (3.20) can be written as Since and the function satisfies we can use Lemma 1.2 and obtain the following fact: This implies that From subordinations (3.21) and (3.26), we find and Therefore, we obtain which simplifies to This completes the proof of this theorem.

Making use of Theorem 3.2, we obtain the following result concerning the double integral operator for starlike of order .

Theorem 3.3. If satisfies for some , then is a starlike function of order .

This theorem can be shown as well as the proof of Theorem 2.6.

Making in Theorem 3.3, we have

Corollary 3.4. If satisfies for some , then

Let us consider an example for Corollary 3.4.

Example 3.5. If we consider , then we find because Further, if we take , then we see that