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Subnormal Solutions of Second-Order Nonhomogeneous Linear Differential Equations with Periodic Coefficients
Journal of Inequalities and Applications volume 2009, Article number: 416273 (2009)
Abstract
We obtain the representations of the subnormal solutions of nonhomogeneous linear differential equation 
, where 
and 
are polynomials in 
such that 
and 
are not all constants, 
. We partly resolve the question raised by G. G. Gundersen and E. M. Steinbart in 1994.
1. Introduction
We use the standard notations from Nevanlinna theory in this paper (see [1–3]).
The study of the properties of the solutions of a linear differential equation with periodic coefficients is one of the difficult aspects in the complex oscillation theory of differential equations. However, it is also one of the important aspects since it relates to many special functions. Some important researches were done by different authors; see, for instance, [4–9].
Now, we firstly consider the second-order homogeneous linear differential equations
where 
and 
are polynomials in 
and are not both constants. It is well known that every solution 
of (1.1) is an entire function.
Let 
be an entire function. We define
to be the 
-type order of 
.
If 
is a solution of (1.1) and if 
satisfies 
, then we say that 
is a subnormal solution of (1.1). For convenience, we also say that 
is a subnormal solution of (1.1).
-
H.
Wittich has given the general forms of all subnormal solutions of (1.1) that are shown in the following theorem.
Theorem 1 A (see [9]).
If 
is a subnormal solution of (1.1), where 
and 
are polynomials in 
and are not both constants, then 
must have the form
where 
is an integer and 
are constants with 
.
-
G.
G. Gundersen and E. M. Steinbart refined Theorem A and obtained the exact forms of subnormal solutions of (1.1) as follows.
Theorem 1 B (see [6]).
In addition to the statement of Theorem A, the following statements hold with regard to the subnormal solutions 
of (1.1).
(i)If 
and 
then any subnormal solution 
of (1.1) must have the form
where 
is an integer and 
are constants with 
(ii)If 
and 
, then any subnormal solution 
of (1.1) must be a constant.
(iii)If 
, then the only subnormal solution 
of (1.1) is 
.
Whether the conclusions of Theorem A and Theorem B can be generalized or not, Gundersen and Steinbart considered the second-order nonhomogeneous linear differential equations
where 
, and 
are polynomials in 
such that 
are not both constants. They found the exact forms of all subnormal solutions of (1.5), that is, what is mentioned in [6, Theorem ?2.2, Theorem ?2.3 and Theorem ?2.4].
In [6], they also have raised the following problem, that is, what about the forms of the subnormal solutions of the equation
where 
, and 
are polynomials in 
such that 
, 
, 
, and 
are not all constants?
In [7], we have obtained the exact forms of all subnormal solutions of homogeneous equation
where 
, and 
are polynomials in 
and are not all constants.
In this paper, we obtain the forms of subnormal solutions of nonhomogeneous linear differential equation (1.6) when 
. We have the following theorem.
Theorem 1.1.
Suppose that 
is a subnormal solution of (1.6), where 
, 
, and 
are polynomials in 
such that 
and 
are not all constants.
(i)If 
and 
, then 
must have the form
where 
is a constant, 
and 
are polynomials in 
.
(ii)If 
and 
, then 
must have the form
where 
is a constant, 
and 
are constants that may or may not be equal to zero, 
may be equal to zero or may be a polynomial in 
, 
, and 
are polynomials in 
with 
.
2. Lemmas for the Proof
In order to prove Theorem 1.1, we need some lemmas.
Lemma 2.1 (see [7]).
Suppose that 
is a subnormal solution of (1.7), where 
, 
, 
and 
are polynomials in 
and are not all constants.
(i)If 
and 
then any subnormal solution 
must be a constant.
(ii)If 
and 
then 
must have the form
where 
is a polynomial in 
with 
.
Lemma 2.2 (see [10]).
Let 
be a transcendental meromorphic function, let 
be a given real constant, and let 
. Then there exists a constant 
such that the following two statements hold (where 
).
(i)There exists a set 
that has linear measure zero such that if 
, then there is a constant 
such that for all 
satisfying 
and 
one has
(ii)There exists a set 
that has finite logarithmic measure such that (2.2) holds for all 
satisfying 
.
Lemma 2.3 (see [6]).
Let 
with 
as 
and let 
Let
and set
Let 
be analytic on the set 
. Suppose that 
is unbounded on the set 
. Then there exists an infinite sequence of points 
with 
as 
such that
Lemma 2.4 (see [8]).
Consider the nth- order differential equation of the form
where 
are polynomials in 
and 
with 
. Suppose that 
is an entire and subnormal solution of (2.6) and that 
can be expressed as 
, where 
is a constant and 
is analytic on 
. Then 
has the form
where 
is a constant and 
and 
are polynomials in 
.
As an application of Lemma 2.4, one has the following lemma.
Lemma 2.5.
Suppose that 
is an entire subnormal solution of (2.6), where 
are polynomials in 
and 
with 
, and that 
and 
are linearly dependent. Then 
has the form
where 
is a constant and 
and 
are polynomials in 
.
Proof.
Since 
is entire and is linearly dependent with 
, 
can be written as 
(see [11, page 382]), where 
is a constant and 
is analytic on 
. Then we have the representation from Lemma 2.4.
Lemma 2.6.
Suppose that 
is a solution of (1.6), where 
, 
, 
, 
, 
, and 
are polynomials in 
such that 
, 
, 
and 
are not all constants. If
then there exists a polynomial 
such that
where 
is a solution of
where 
and 
are polynomials in 
with 
Proof.
Let 
and set
where 
is the constant such that
It follows from (1.6) and (2.12) that
where
So 
and 
are polynomials in 
and 
, respectively, and 
by (2.13), but 
and 
have the exact representations that depend on the relations of 
, and 
. If 
, then (2.14) is of the form (2.11), and (2.12) gives (2.10). If 
, then we repeat the above process finite times until we obtain (2.10) and (2.11). This completes the proof of Lemma 2.6.
3. Proof of Theorem
In this section, we will prove Theorem 1.1.
Proof.
-
(i)
Suppose that
is a subnormal solution of (1.6) with 
and 
. If 
is a polynomial solution of (1.6), then 
must be a constant, which is of the form (1.8). Thus we suppose that 
is transcendental. It follows from Lemma 2.2(i) that there exists a set 
that has linear measure zero such that if 
, then there is a constant 
such that for all 
satisfying 
and 
, we have 
(3.1)
is a subnormal solution of (1.6) with 
and 
. If 
is a polynomial solution of (1.6), then 
must be a constant, which is of the form (1.8). Thus we suppose that 
is transcendental. It follows from Lemma 2.2(i) that there exists a set 
that has linear measure zero such that if 
, then there is a constant 
such that for all 
satisfying 
and 
, we have 
where 
is a constant and 
. It also follows from Lemma 2.2(ii) that there exists a set 
that has finite logarithmic measure such that (3.1) holds for all 
satisfying 
.
Now let 
be an infinite sequence satisfying 
such that 
for all 
and 
as 
Let 
be a small constant such that 
and 
. Set
and set
From above, we have that (3.1) holds on the set 
We now assert that 
is bounded on the set 
. On the contrary, it follows from Lemma 2.3 that there exists a sequence of points 
with 
as 
such that
By (1.6), we have for all 
,
It follows from (3.4)–(3.6) and 
that (3.6) yields 
as 
on the set 
This is a contradiction.
By the maximum modulus principle, 
is bounded in the angular domain
However, we know
where the integral of 
is defined on the simple contour 
, extending from a point 
to a point 
in the complex domain.
So we obtain
as 
in the angular domain 
.
Thus , from the Cauchy integral formula, we obtain
as 
in the angular domain 
. By (1.6), (3.8), and (3.9), we have for some constant 
as 
in the angular domain 
. Together with (3.8) and (3.11), 
is bounded in the angular domain 
.
If 
, it follows from Lemma 2.5 that 
must have the form (1.8).
If 
, since 
is a subnormal solution of (1.6), so is 
. Thus,
will be a subnormal solution of (1.7). Since we suppose that 
, we will discuss the following two cases.
Case 1.
If 
we have, by Lemma 2.1(i),
where 
is a constant. Hence 
, that is,
From this, 
can be written as 
(see [11, page 382]), where 
is a constant and 
is analytic on 
. Thus, 
can be written as 
, where 
is a constant and 
is analytic on 
. It follows from Lemma 2.4 that
where 
is a constant, 
and 
are polynomials in 
. Thus, 
has the form of (1.8).
Case 2.
If 
we obtain from Lemma 2.1(ii) that
where 
is a polynomial in 
with 
.
However, we can assert that 
in (3.16). Otherwise, there exists 
such that
By (3.16), we have
Thus from (3.16) and (3.18), we have
By repeating this process finite times, we obtain that for any integer 
,
We have, by (3.17) and (3.20),
as 
This is a contradiction to the fact that 
is bounded in the angular domain 
. This shows that 
is not possible when 
under the hypotheses. This completes the proof of part (i).
-
(ii)
We firstly suppose that
. Since 
is a subnormal solution of (1.6), so is 
. Set 
(3.22)
. Since 
is a subnormal solution of (1.6), so is 
. Set 
Then 
is a subnormal solution of (1.7). Now if 
, this shows that 
and 
has the form of (1.9) by Lemma 2.5. Thus, we suppose that 
in the following.
Now, assume that 
.
If 
it follows from the proof of Case 1 of Theorem 1.1(i) that 
has the form of (1.9).
If 
we obtain from Lemma 2.1(ii) that
where 
is a polynomial in 
with 
.
Set
where 
is an integer and 
are constants with 
Let 
and set
where 
Now, we will discuss the following two cases.
Case A.
We consider 
in (3.24). Let 
be a constant defined by
and set
Since 
is a subnormal solution of (1.7), it follows from (3.27) that 
satisfies
We obtain from (3.23)–(3.28) that
where 
and 
are polynomials in 
with 
Set
It follows from (1.6), (3.25), (3.29) and (3.30) that 
Set
So 
, and 
satisfies
We have by (3.27) that 
is a subnormal solution of (1.7), 
is a subnormal solution of (3.29). Moreover, 
is also a subnormal solution of (3.32) by (3.30) and 
is a subnormal solution of (1.6). Thus, we deduce from Theorem 1.1(i) and (3.32) with 
that 
has the form
where 
is a constant, 
and 
are polynomials in 
. Hence (3.23), (3.24), (3.27), (3.30), and (3.33) yield
where 
and 
are constants, 
, and 
are polynomials in 
with 
. This is the form of (1.9).
Case B.
We consider 
in (3.24). Let 
be a constant defined by
where 
is a number such that 
is the first coefficient 
in (3.24) which is not equal to zero. Set
Similar to the proof of Case A of Theorem 1.1(ii), we have
where 
and 
are constants, 
and, 
are polynomials in 
with 
. Set 
. Then 
is a polynomial in 
by the hypotheses of 
in (3.36). This is the form of (1.9). We have proved Theorem 1.1(ii) when 
Now we suppose that 
. By Lemma 2.6, there exists a polynomial 
in 
, satisfies (2.10) and (2.11).
Since 
and since we have proved Theorem 1.1 holds in the cases when 
holds, we can apply this result to (2.11).
If 
, it follows from Theorem 1.1(i) that
where 
is a constant, 
and 
are polynomials in 
. By (2.10) and (3.38), we obtain that
where 
is a constant, 
and 
are polynomials in 
. This is a form of (1.9).
If 
, it follows from the proof of Theorem 1.1(ii) when 
that
where 
and 
are polynomials in 
with 
, 
and 
are constants that may or may not be equal to zero. By (2.10) and (3.40), we obtain that 
has the form of (1.9). Theorem 1.1(ii) is completed.
Now, we give some examples to show that Theorem 1.1 is correct.
Example 3.1.
Let 
, then 
satisfies
This is an example of Theorem 1.1(i).
Example 3.2.
Let 
, then 
satisfies
This is an example of Theorem 1.1(ii) with 
and 
.
Example 3.3.
Let 
, then 
satisfies
This is an example of Theorem 1.1 (ii) with 
and 
.
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The authors are very grateful to the referee for his (her) many valuable comments and suggestions which greatly improved the presentation of this paper. The project was supposed by the National Natural Science Foundation of China (no. 10871076), and also partly supposed by the School of Mathematical Sciences Foundation of SCNU, China.
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Huang, ZB., Chen, ZX. & Li, Q. Subnormal Solutions of Second-Order Nonhomogeneous Linear Differential Equations with Periodic Coefficients. J Inequal Appl 2009, 416273 (2009). https://doi.org/10.1155/2009/416273
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DOI: https://doi.org/10.1155/2009/416273
Keywords
- Entire Function
- Linear Differential Equation
- Exact Form
- Order Differential Equation
- Infinite Sequence