Abstract
In this paper, we study the unicity of entire functions and their derivatives and obtain the following result: let
1 Introduction and main results
In this paper, we use the general notations in the Nevanlinna value distribution theory, see ([1,2,3]).
Let
is the average of the positive logarithmic
is called the counting function of poles of
Let
is the average of the positive logarithmic
where
Obviously,
We denote
Nevanlinna established two fundamental theorems, called the first fundamental theorem and the second fundamental theorem.
The first fundamental theorem. Let
The second fundamental theorem. Let
Let
In [3], Nevanlinna proved the following famous five-value theorem:
Theorem A
Let
Brosch [4], Czubiak and Gundersen [5], Gundersen [6], Steinmetz [7], and Gundersen et al. [8] studied shared pairs of values. For five shared pairs of values, Gundersen et al. [8] proved the next result, which is the best possible:
Theorem B
If
Li and Qiao [9] proved that Theorem A is still valid for five distinct small functions, and they proved the following:
Theorem C
Let
Zhang and Yang [10] and Nguyen and Si [11] studied shared pairs of small functions.
Rubel and Yang [12] investigated the uniqueness of an entire function and its derivative and proved.
Theorem D
Let
Zheng and Wang [13] improved Theorem D and proved the following: if
Li and Yang [14] improved the result of Zheng and Wang [13] and proved the following result:
Theorem E
Let
In this paper, we study shared pair of small functions and extend Theorem E as follows.
Theorem 1
Let
Remark 1
If
The following example shows that the conclusion of Theorem 1 is not valid for meromorphic functions.
Example 1
[15] Let
and let
So
The following example shows that there exist a transcendental entire function
Example 2
[16] Suppose
for
Thus, we see that
In 2020, Sahoo-Halder [16] proved.
Theorem F
Let
The authors raised two questions as follows.
Question 1
The condition of finite order of Theorem F and Corollary 1 in [16] can be removed in any way?
Question 2
How far do the conclusions in Theorem F and Corollary 1 in [16] hold for a non-constant entire function?
In Theorem 1, we answer Question 1 in a setting with stronger conditions, and we answer that for any positive integers
2 Some lemmas
Lemma 2.1
[1,2,3] Let
Lemma 2.2
[2] Let
Lemma 2.3
[1,2,3] Let
Lemma 2.4
Let
If
Proof
Suppose that
Since
Hence,
Lemma 2.5
Let
and
where
Proof
Since
Similarly, we have
Obviously,
where
Thus, by Lemma 2.2 and above two formulas, we obtain (2.1).□
Lemma 2.6
Let
where
If
or
Proof
Since
where
If
If
and
Obviously,
It follows that
3 Proof of Theorem 1
We prove Theorem 1 by contradiction. Suppose that
Since
From the fact that
Thus, we have
and
Obviously,
It follows from (3.1) and (3.4) that
On the other hand, (3.1) can be rewritten as
which implies
Thus, by (3.3), (3.7), and (3.9)
Hence, by the above formula, (3.7), (3.9), we obtain
Set
and
Obviously,
that is
Let
which implies
Rewrite (3.13) as
and set
Next, we consider two cases.
Case 1.
where
Then, by Lemma 2.6, we get
which contradicts with (3.3).
Case 2.
It follows from (3.4), (3.17), and (3.20) that
On the other hand, by Lemma 2.4, (3.1), (3.8), and (3.11), we have
Thus, by (3.21) and (3.22), we obtain
If
It follows from (3.11), (3.23), and (3.24) that
If
which implies
On the other hand, it follows from Lemma 2.2 that
Thus,
Then, by Nevanlinna’s second fundamental theorem, Lemma 2.1, (3.3), and (3.27), we have
Thus,
It is easy to see that
It follows from Lemma 2.5 and (3.16) that
By (3.3), (3.10), (3.21), and (3.29), we get
Moreover, by (3.3), (3.27), and (3.30), we have
which implies
Then by (3.3), (3.30), and (3.32), we obtain
Hence, we have
which implies that
In the following, we will prove
Rewrite (3.1) as
Set
where
where
Note that
By (3.34) and Lemma 2.2, we get
Then it follows from (3.38) that
Subcase 2.1.
We claim that
It follows from (3.11) and (3.23) that
Integrating (3.42), we get
where
It follows from (3.3), (3.11), and (3.44) that
Then combining (3.33) with (3.45), and applying Nevanlinna’s second fundamental theorem to
Because
It is impossible.
Subcase 2.2
This completes the proof of Theorem 1.
Acknowledgments
The authors thank the referee for his careful reading of the paper and giving many valuable suggestions.
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Funding information: This work was supported by the NNSF of China (Grant No. 11901119) and the Natural Science Foundation of Zhejiang Province (LY21A010012).
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Conflict of interest: The authors state no conflict of interest.
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