^{1,2}

^{1}

^{2}

^{1}

^{2}

We consider that the linear differential equations

We will assume that the reader is familiar with the fundamental results and the standard notations of Nevanlinna theory of meromorphic functions (see [

In order to estimate the rate of growth of meromorphic function of infinite order more precisely, we recall the following definition.

Let

Consider the second order linear differential equation

In this paper, we will introduce the deficient value and Borel direction into the studies of the complex differential equations. In order to give the definition of the Borel direction, we need the following notation. Let

Let

The fundamental result in angular distribution, due to Valiron, says that a meromorphic function of order

It is well known that deficient values and Borel directions are very important concepts in Nevanlinna theory of meromorphic functions. These two concepts are extensively studied. There is a striking relationship between them which was found by Yang and Zhang and says that, for a meromorphic function

Suppose that

Note that Theorem

In the sequel, we will say that an entire function

The simplest entire function extremal for

Furthermore, we state the following result due to present authors (see, [

Let

In this paper, we will consider the higher order linear differential equation

Let

The paper is organized as follows. In Section

In this section, we need some auxiliary results. The following lemma is by Gundersen.

Let

(i) There exists a set

In particular, if

(ii) There exists a set

In particular, if

(iii) There exists a set

In particular, if

Let

Suppose that

In the sequel, we will say that

Let

Before stating the following lemmas, for

Let

Let

Let

Lemma

Now we prove our main result.

Since

We consider the following two cases.

We suppose that

We deduce from (

Thus,

By using similar methods as [

We suppose that

Suppose that

Without loss of generality, we assume that there is a ray

Firstly, suppose that

Note that

By Lemma

Note that

Secondly, suppose that

Thus we can get a contradiction by using similar argument for the proof of case

Lastly, suppose that

Next we prove that

By Lemma

Hence, calculating at the points

If

In this section, we will study (

Let

We first treat the case that the entire functions

Note that if

Now suppose that (a)

We next suppose that (b)

We turn to the case that

Finally, in [

Let

Moreover, suppose that

Finally we give an example satisfying the conditions of Theorem

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

The work is supported by the United Technology Foundation of Science and Technology Department of Guizhou Province and Guizhou Normal University (Grant no. LKS [2012] 12), and the National Natural Science Foundation of China (Grant nos. 11171080 and 11171277).