#### Abstract

We consider that the linear differential equations , where , are entire functions. Assume that there exists , such that is extremal for Yang's inequality; then we will give some conditions on other coefficients which can guarantee that every solution of the equation is of infinite order. More specifically, we estimate the lower bound of hyperorder of if every solution of the equation is of infinite order.

#### 1. Introduction and Main Results

We will assume that the reader is familiar with the fundamental results and the standard notations of Nevanlinna theory of meromorphic functions (see [1, 2] or [3]). In addition, for a meromorphic function in the complex plane , we will use the notations and to denote its order and lower order, respectively.

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

Definition 1 (see [4]). Let be a meromorphic function in the complex plane . Then one defines the hyperorder of by
Consider the second order linear differential equation where and are entire functions. It is well known that if is an entire function, is a transcendental entire function, and , are two linearly independent solutions of (2), then at least one of , must have infinite order. On the other hand, there are some equations of form (2) that possess a solution of finite order; for example, satisfies . Therefore, one may ask, what assumptions on and will guarantee that every solution of (2) is of infinite order? From the works of Gundersen (see [5]) and Hellerstein et al. (see [6]), we know that if and are entire functions with , or is a polynomial, and is transcendental, or , then every solution of (2) is of infinite order. More results can be found in [712]. For entire solutions of infinite order more precise estimates for their rate of growth would be an important achievement. There are many authors investigating the hyperorder of solutions of (2), such as Chen and Yang (see [8, 13]) and Kwon (see [14, 15]).

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 such that and ; set , .

Definition 2. Let be a meromorphic function in the complex plane with . Let be a finite constant. A ray from the origin is called a Borel direction of order of , if for any positive number and for any complex number , possibly with two exceptions, the following inequality holds: where denotes the number of zeros, counting the multiplicities, of in the region .

The fundamental result in angular distribution, due to Valiron, says that a meromorphic function of order must have at least one Borel direction of order ; for example, see [3].

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 of order , the number of deficient values is less or equal to the number of Borel directions of order of . In 1988, Yang extended the above - inequality to the case of entire function of finite lower order. In order to use Yang’s result to study the complex differential equations, we will use the following Theorem which can be easily derived from [16].

Theorem 3 (see [16]). Suppose that is an entire function of finite lower order . Let () denote the number of Borel directions of order and denote the number of finite deficient values of ; then .

Note that Theorem 3 is explicitly stated in [17]. To see the valid of the conclusion of the theorem, we note that, in [17, Corollary 1], Wu has proved that if is of finite lower order and the number of Borel directions of order is finite, then the order of is also finite. As each Borel direction of order is also a Borel direction of order , this implies that, for , the number of the Borel directions of order is fewer or equal to the number of the Borel directions of order . Therefore Theorem 3 follows from Theorem 6.7 in [3].

In the sequel, we will say that an entire function is extremal for Yang’s inequality if satisfies the assumptions of Theorem 3 with .

The simplest entire function extremal for Yang’s inequality is . A little bit complicated example is , . We know that (see [3]) has deficient values and Borel direction . So .

Furthermore, we state the following result due to present authors (see, [18]).

Theorem 4 (see [18]). Let be an entire function extremal for Yang’s inequality, and let be a transcendental entire function such that . Then every solution of (2) is of infinite order.

In this paper, we will consider the higher order linear differential equation where are entire functions. Many authors have also investigated the growth of solutions of (5) and obtained lots of results on order and hyperorder of the solutions of (5) (see [1923]). We will introduce the deficient value and Borel directions into the studies of (5). The main result in the paper is as follows.

Theorem 5. Let be entire functions. Suppose that there exists an integer , such that is extremal for Yang’s inequality. Suppose that is a transcendental entire function with and for . Then every solution of (5) satisfies and .

The paper is organized as follows. In Section 2, we will give some lemmas. In Section 3, we will prove Theorem 5. In Section 4, we will discuss some further results related to the two entire coefficients in (5) which are extremal for Yang’s inequality.

#### 2. Lemmas

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

Lemma 6 (see [24]). Let denote a pair that consists of a transcendental meromorphic function and a finite set of distinct pairs of integers that satisfy for . Let and be given real constants. Then the following three statements hold.
(i) There exists a set that has linear measure zero, and there exists a constant that depends only on and , such that if , then there is a constant such that, for all satisfying and and for all , we have
In particular, if has finite order , then (7) is replaced by
(ii) There exists a set that has finite logarithmic measure, and there exists a constant that depends only on and , such that, for all satisfying and for all , inequality (7) holds.
In particular, if has finite order , then inequality (8) holds.
(iii) There exists a set that has finite linear measure, and there exists a constant that depends only on and , such that, for all satisfying and for all , we have
In particular, if has finite order , then (9) is replaced by

Let be an entire function extremal for Yang’s inequality . Suppose that the rays () () are the distinct Borel directions of order of . In [17], Wu studied the entire functions which are extremal for Yang’s inequality systematically. The following results play an important role in the proof of our results.

Lemma 7 (see [17]). Suppose that is extremal for Yang’s inequality. Then . Moreover, for every deficient value () there exists a corresponding angular domain such that for every inequality holds for , where is a positive constant depending only on , , and .

In the sequel, we will say that decays exponentially to in , if (11) holds in . Note that if is extremal for Yang’s inequality, then . Thus, for these functions, we need only to consider the Borel direction of order .

Lemma 8 (see [18]). Let be extremal for Yang’s inequality. Suppose that there exists such that where () are Borel directions of . Then .

Before stating the following lemmas, for , we define the Lebesgue measure of by and the logarithmic measure of by and define the upper and lower logarithmic density of , respectively, by

Lemma 9 (see [25]). Let be an entire function with and suppose that is defined as If , then the set has a positive upper logarithmic density.

Lemma 10 (see [26]). Let and be monotone nondecreasing functions on such that for all outside some set of finite logarithmic measure. Let be a given real constant. Then there exists a constant such that for all .

Lemma 11. Let be an entire function with order , let and be a sector with . If there is Borel direction of in , then there exists at least one of the two rays , without lose of generality, says, , such that

Lemma 11 can be founded in [27, Lemma 1], which can be proved by using a result in [28, Page 119-120].

#### 3. Proof of Theorem 5

Now we prove our main result.

Since , we know that for any given constant with , there exists a constant such that holds for all .

We consider the following two cases.

Case 1. We suppose that . Now to the contrary assume that there is a solution of (5) with . We will seek a contradiction. By Lemma 6(ii), there exists a set that has finite logarithmic measure, such that the following inequality holds for all with .
We deduce from (17) and (5) that holds for all with .
Thus, holds for all with , where . By Lemma 10, we have ; this is a contradiction. Therefore, every solution of (5) is of infinite order.
By using similar methods as [14], we can easily prove that in this case. We omit the details here.

Case 2. We suppose that . Now to the contrary assume that there is a solution (0) of (5) with . We will seek a contradiction.
Suppose that are all the finite deficient values of . Thus we have angular domains (). For any , by using Lemmas 7 and 11, we can easily obtain that has the following properties: in each sector , either there exists some such that holds for , where is a positive constant depending only on , , and , or there exists such that holds. For the sake of simplicity, in the sequel we use to represent . Note that if there exists some such that (20) holds in , then there exists such that (21) holds in and . And if there exists such that (21) holds, then there are () such that (20) holds in and , respectively.

Without loss of generality, we assume that there is a ray in such that (21) holds. Therefore, there exists a ray in each sector , such that (21) holds. By using Lemma 8, we know that all the sectors have the same magnitude .

Firstly, suppose that . Since must have a Borel direction of order , by using Lemma 11, we can see that there exists a sector such that and such that for all the rays we have

Note that . It is not hard to see that there exists a sector such that there is an such that holds for all .

By Lemma 6(i), there exist () and such that holds for all .

Note that Thus there is a sequence of with such that holds for every . Therefore, we deduce from (16), (23), and (24) that holds for all sufficiently large . Therefore, combining (26) with (27), we have that holds for all sufficiently large . This is a contradiction, so every solution of (5) is infinite order in this case.

Secondly, suppose that . By Lemma 9, there is a sequence of with such that for any , we must have

Thus we can get a contradiction by using similar argument for the proof of case . So every solution of (5) is infinite order in this case.

Lastly, suppose that . Note that is a transcendental entire function. By using the results of [29] or [8], for any , we have Thus we can get a contradiction by using similar argument for the proof of case . Therefore, every solution of (5) is infinite order in this case.

Next we prove that . Firstly, suppose that . By using similar argument as above, there exists and there is a sequence of with such that (23) and (26) (or (29)) hold for all sufficiently large .

By Lemma 6(i) that there exist and constants , , such that the following inequality holds for .

Hence, calculating at the points with , from (16), (23), (26) (or (29)), and (31), we get Thus This gives .

If , obviously, . The proof of Theorem 5 is completed.

#### 4. Further Results

In this section, we will study (5) with coefficients and which are both extremal for Yang’s inequality.

Theorem 12. Let be entire functions. Suppose that there exists an integer , such that is extremal for Yang’s inequality . Suppose that is an entire function extremal for Yang’s inequality and for . Suppose that one of the following conditions holds:(1),(2), and the set of Borel directions of is different from that of .Then every solution of (5) satisfies and .

Proof. We first treat the case that the entire functions and satisfy condition (1).
Note that if , then the conclusion of Theorem 12 follows from Theorem 5. Now suppose that . We divide the proof into two cases: (a) and (b) .
Now suppose that (a) holds. It is easy to see from Lemmas 7 and 8 that there are sectors with magnitude such that while there are sectors with magnitude such that Note that . It is easy to see that there exists a sector such that, for every , must be bounded in , while satisfies (34). So, by using the same argument in the proof of Theorem 5, we can easily prove the theorem.
We next suppose that (b) . It is not hard to see that there must exist a sector such that is bounded in , while for any satisfies (34). By using similar arguments as we did before, we can prove the theorem under the condition that and satisfy (1).
We turn to the case that and satisfy (2). In this case, it is easy to see that there exists a sector such that in it is bounded, while satisfies (34). By using similar arguments as we did before, we can prove the theorem in this case. We omit the details here. The proof of Theorem 12 is completed.

Finally, in [16], we note that if an entire function is extremal for Yang’s inequality , then for any positive integer , also has some special properties. In the sectors where, for any , satisfies for any , satisfies decays to some deficient values exponentially and decays to 0 exponentially. Therefore, in the same manner as in the proofs of Theorems 5 and 12, we have the following result.

Theorem 13. Let be entire functions. Suppose that there exists an integer , such that is extremal for Yang’s inequality . Suppose that is a transcendental entire function with and for . Then every solution of satisfies and .

Moreover, suppose that is an entire function extremal for Yang’s inequality and that one of the following assumptions holds:(1),(2), and the set of Borel directions of is different from that of .Then every solution of satisfies and , where and are two nonnegative integers.

Finally we give an example satisfying the conditions of Theorem 12. Let again , . So if we let , with and all other coefficients satisfy for , then, by Theorem 12(1), every solution of (5) satisfies and . Furthermore, if we let with , then, by Theorem 12(2), every solution of (5) satisfies and .

#### Conflict of Interests

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

#### Acknowledgments

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).