#### 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