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 .