Lemma 2.1 (see [12]).
Let
be a transcendental meromorphic function with
be a finite set of distinct pairs of integers satisfying
. And let
be a given constant. Then,
(i)there exists a set
with linear measure zero, such that, if
, then there is a constant
, such that for all
satisfying
and
and for all
, one has
(ii)there exists a set
with finite logarithmic measure, such that for all
satisfying
and for all
, we have
(iii)there exists a set
with finite linear measure, such that for all
satisfying
and for all
, we have
Lemma 2.2 (see [8]).
Suppose that
are real numbers,
is a polynomial with degree
, that
is an entire function with
. Set
. Then for any given
, there exists a set
that has the linear measure zero, such that for any
, there is
, such that for
, we have
(i)if
, then
(ii)if
, then
where
is a finite set.
Using Lemma 2.2, we can prove Lemma 2.3.
Lemma 2.3.
Suppose that
is a positive entire number. Let
be nonconstant polynomials, where
are complex numbers and
. Set
, then there is a set
that has linear measure zero. If
, then there exists a ray
,
, such that
or
where
is a finite set, which has linear measure zero.
Proof.
According to the values of
and
, we divide our discussion into three cases.
Case 1 (
).
-
(a)
If
, let
, Then there are three cases: (i)
; (ii)
; (iii)
.
-
(i)
. By
, we know that
.
Suppose that
, then take
is any constant in
.
Since
has linear measure zero, there exists
such that
. Thus
. By
and
that is
, we have
Therefore,
When
, then
, we can prove it by using similar argument action as in the above proof.
-
(ii)
, then
. Suppose that
, then
. Let
, and take
is any constant in
.
Since
has a linear measure zero, there exists
such that
,
Therefore
Suppose that
, then
. Let
, and take
is any constant in
.
Since
has linear measure zero, there exists
such that
,
Therefore,
-
(iii)
, then
. Using similar method as in proof of (ii), we know that there exists
such that
.
-
(b)
When
, we can prove it by using the same argument action as in (a).
-
(c)
When
, we just prove the case that
(when
, we can prove it by using the same reasoning).
Let
, take
is any constant in
.
Since
has a linear measure zero, there exists
, such that
. Then
When
,
, thus,
.
When
,
, thus,
.
Therefore
Case 2.
When
, or
and
, using a proof similar to Case 1, we can get the conclusion.
Case 3 (
and
).
By
, there are only two cases:
; or
.
If
. Take
is any constant in
.
Since
has linear measure zero, there exists
such that
. Using a proof similar to Case 1(c), we can prove it.
When
, we can prove it by using the same reasoning
Remark 2.4.
Using the similar reasoning of Lemma 2.3, we can obtain that, in Lemma 2.3, if
is replaced by
, then it has the same result.
Lemma 2.5 (see [8]).
Let
be entire functions with finite order. If
is a solution of the equation
then
.
Lemma 2.6 (see [12]).
Let
be a transcendental meromorphic function, and let
be a given constant, Then there exists a set
with finite logarithmic measure and a constant
that depends only on
and
(
), such that for all
satisfying
,
Remark 2.7.
In Lemma 2.6, when
, we have
Lemma 2.8 (see [13]).
Suppose that
and
are nondecreasing functions, such that
, where
is a set with at most finite measure, then for any constant
, there exists
such that
for all
.