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Some inequalities and applications on Borel direction and exceptional values of meromorphic functions
Journal of Inequalities and Applications volume 2014, Article number: 53 (2014)
Abstract
In view of the Nevanlinna theory in the angular domain, we study the exceptional values of meromorphic functions in the Borel direction and also establish some inequalities on the exceptional values of meromorphic functions in the Borel direction. Based on these inequalities, we also give two theorems and some corollaries as regards exceptional values of meromorphic functions in the Borel direction.
MSC:30D30, 30D35.
1 Introduction and main results
To begin with, we assume that the reader is familiar with the basic results and the standard notations of the Nevanlinna theory of meromorphic functions (see [1–3]). We denote by ℂ the open complex plane, by () the extended complex plane, and by Ω () an angular domain. In addition, the order of the meromorphic function f is defined by
and the exponent of convergence of distinct a-points of f is defined by
For f a meromorphic function of order ρ (), we say that a is an exceptional value in the sense of Borel (evB for short) for f for the distinct zeros if . Thus, by the second fundamental theorem in the whole complex plane, we know that a meromorphic function f of order ρ () at most has two evB for the distinct zeros.
It is well known that exceptional values of meromorphic functions are strictly relative with singular directions. For instance, Picard exceptional value relating with Julia direction and Borel exceptional value relating with Borel direction, and so on (see [4–8]). Moreover, the characteristics of meromorphic functions in the angular domain played an important role in studying on singular directions and exceptional values of meromorphic functions (see [9–12]). Now, we firstly introduce the characteristics of meromorphic functions in the angular domain as follows [13, 14].
Let f be a meromorphic function on the angular domain and . Define
where and () are the poles of f on counted according to their multiplicities. is called Nevanlinna’s angular characteristic, and is called the angular counting function of the poles of f on , and is the reduced function of . Similarly, the order of the meromorphic function f on is defined by
and the exponent of convergence of distinct a-points of f on is defined by
For f is a meromorphic function of order (), then we say that a is an exceptional value on the angular domain in the sense of Borel (evaB for short) for f for the distinct zeros if .
An interesting subject arises naturally: Does a meromorphic function f with order () on at most have two evaB for the distinct zeros? By Lemma 2.2, Lemma 2.3, and Remark 2.1, we can give a negative answer to this question since is not valid, as () and E is the set with finite linear measure. Thus, it is an interesting topic in studying the exceptional value of meromorphic functions on the angular domain.
The main purpose of this paper is to investigate the exceptional values of the meromorphic function with infinite order in its Borel direction. Valiron [15] proved that every meromorphic function of finite order has at least one Borel direction of order ρ. Chuang [16, 17] investigated the existence of Borel directions of the meromorphic function of infinite order. Before stating Chuang’s results, we will introduce the definition as follows.
Definition 1.1 [16]
Let f be a meromorphic function of infinite order, be a real function satisfying the following conditions:
-
(i)
is continuous, non-decreasing for and as ;
-
(ii)
where ();
-
(iii)
Then is said to be of infinite order for the meromorphic function f. This definition was given by Xiong (see [16]).
We will give the definition of the Borel direction of the meromorphic functions f of infinite order as follows.
Definition 1.2 [16]
Let f be a meromorphic function of infinite order . If for any ε (), the equality
holds for any complex number , at most except two exception, where is the counting function of zero of the function in the angular domain , counting multiplicities. Then the ray is called a Borel direction of order of the meromorphic function f.
Remark 1.1 Chuang [16] proved that every meromorphic function f with infinite order has as least one Borel direction of infinite order .
Now, the main theorem of this paper is listed as follows.
Theorem 1.1 Let f be a transcendental meromorphic function of infinite order on the whole complex plane, () be one Borel direction of order of the function f and for any ε (). If there exist such that are evBB for f for distinct zeros of multiplicity , , where and are positive integers or infinity, then
Definition 1.3 Let () be one Borel direction of order of function f and k be a positive integer, we say that a is
-
(i)
an exceptional value in the sense of Borel for f in the Borel direction (evBB for short) for distinct zeros of multiplicity ≤k, if ;
-
(ii)
an exceptional value in the sense of Borel for f in the Borel direction (evBB for short) for distinct zeros, if ; where
and is the counting function of distinct a-points of f on Ω whose multiplicities do not exceed k.
In particular, we say that a is an evBB for f for simple zeros if , a is an evBB for f for simple and double zeros if .
Definition 1.4 For positive integers k, μ, we define
where the counting function of a-points of f on Ω where an a-point of multiplicity μ is counted μ times if and times if . In particular, if , we denote
Theorem 1.2 Let f be a transcendental meromorphic function of infinite order on the whole complex plane, () be one Borel direction of order of the function f and for any ε (). If there exist and two positive integers k and p such that
then there exist at most p elements which are evBB for f for distinct zeros of multiplicity not exceeding k.
2 Some lemmas
To prove our results, we need the following lemmas.
Let f be a non-constant meromorphic function on . Then for arbitrary complex number a, we have
where as .
Suppose that f is a non-constant meromorphic function in one angular domain with , then for arbitrary q distinct (), we have
where the term will be replaced by when some and
Lemma 2.3 (see [[18], p.138])
Let f be a non-constant meromorphic function in the whole complex plane ℂ. Let one angular domain be given on . Then for any , we have
and
where and K is a positive constant not depending on r and R.
Remark 2.1 Nevanlinna conjectured that
when r tends to +∞ outside an exceptional set of finite linear measure, and he proved that when the function f is meromorphic in ℂ and has finite order. In 1974, Gol’dberg [13] constructed a counter-example to show that (3) is not valid.
Lemma 2.4 (see [[20], Lemma 4])
Let f be a meromorphic function in ℂ, () be a closed angular domain, then
where is stated as in (2), , is the precise order of when f is of infinite order, E is a set of finite linear measure.
Lemma 2.5 (see [[20], Lemma 5])
Let f be a meromorphic function on a closed angular domain and , then for any and for any ,
where .
Remark 2.2 For the reduced case, that is, each multiple zero of in is counted only once (ignoring multiplicities), Lemma 2.5 still holds, and its proof is similar to the case of counting multiplicities.
Lemma 2.6 (see [17])
Let f be a meromorphic function of infinite order . Then the ray is one Borel direction of order of the meromorphic function f if and only if f satisfies the equality
for any ε ().
Lemma 2.7 Let f be a transcendental meromorphic function of infinite order on the whole complex plane, () be one Borel direction of order of the function f and for any ε (). Then
Proof Suppose that are t distinct complex constants. Since () is one Borel direction of order of the function f, then we have
where . From , then we have
it follows by Lemmas 2.4-2.6 that
Since t is arbitrary, from (4) we can easily complete the proof of Lemma 2.7. □
3 Proof of Theorem 1.1
Proof Since f is a meromorphic function of infinite order and () is one Borel direction of order of the meromorphic function f, by Lemma 2.6, we can get for any ε ()
For any positive integer k or ∞ and , we have
where and if . Then, from (6) and Lemma 2.2, we have
From (5), Lemma 2.5 and the assumptions of Theorem 1.1, there exists a constant η () such that for sufficiently large r,
Hence, from (5) and for sufficiently large r, we have
Thus, for sufficiently large r and arbitrary ε (>0), we can get from (8) and the definition of
that is,
If , we can choose an arbitrary ε (>0) satisfying and . Thus, from (5), (10) and for sufficiently large r, we easily get a contradiction.
Therefore, we get the conclusion of Theorem 1.1. □
4 Proof of Theorem 1.2
Proof Without loss of generality, we assume that . Next, we use reduction to absurdity to prove the conclusion of Theorem 1.2. Suppose that there exist elements which are evBB for f for distinct zeros of multiplicity ≤k. Since () is one Borel direction of order of the meromorphic function f and for any ε (), and if is a zero of on Ω of multiplicity d (>1) for , then is a zero of on Ω of multiplicity , and it follows that
Therefore, by Lemma 2.2 we have
From (5), Lemma 2.5 and the assumptions of Theorem 1.2, there exists a number η () such that for sufficiently large r,
From (11), (12) and Lemma 2.6, for sufficiently large r, it follows that
Since , from (5) and (13) for sufficiently large r, we can get
which is a contradiction with the assumption of Theorem 1.2.
Thus, this completes the proof of Theorem 1.2. □
5 Some consequences of Theorems 1.1 and 1.2
In this section, we will give some consequences of Theorem 1.1. Before giving these results, some definitions will be introduced below.
Definition 5.1 Let () be one Borel direction of the function f and we have any ε (), for . Then
-
(i)
a is called an exceptional value in the sense of Nevanlinna in the Borel direction (evNB for short), if ;
-
(ii)
a is called a normal value in the sense of Nevanlinna in the Borel direction (nvNB for short), if .
In addition, similar to the Picard exceptional value in the whole complex plane, by definition a is called an exceptional value in the sense of Picard in the Borel direction of f (evPB for short), if f has at most a finite number of a-points in the Borel direction.
Consequence 5.1 Under the assumptions of Theorem 1.1, if , from Theorem 1.1, we get
Since and for , it follows that
-
(i)
if f has an evBB for simple zeros which is also an evNB for f, then f has at most three evBB for simple zeros;
-
(ii)
if , are two evPB for f then no other element is an evBB for f for simple zeros;
-
(iii)
there exist at most four elements which are evBB for f simple zeros since , moreover, all these four values are nvNB for f.
Consequence 5.2 Under the assumptions of Theorem 1.1, if , , , we have
Since and , it follows from (14) that . Thus, if is an evBB for f for simple zeros, that is, , then there exist at most two other elements which are evBB for f for distinct simple zeros and double zeros. Furthermore,
-
(i)
if , then two other elements evBB are also evNB for f;
-
(ii)
if any one of the two other elements , , say , satisfies , then , are also evNB for f.
Consequence 5.3 Under the assumptions of Theorem 1.1, if , , , then we have
From the above inequality and for , we see that , and , , (). Thus, if f has an evBB for simple zeros, then there exist at most two other elements which are evBB for f distinct zeros of multiplicity ≤3, moreover, all these exceptional values are nvNB for f.
Consequence 5.4 Under the assumptions of Theorem 1.1, if , we have
Since , from (15) we have
Thus, it follows that and , , . Hence, we see that f has at most three evBB for distinct simple and double zeros, moreover, all three evBB for distinct simple and double zeros are nvNB for f.
Consequence 5.5 Under the assumptions of Theorem 1.1, if , , , then we have
Thus, it follows that . So, if there exists an evBB for f for distinct and double zeros, say , then there exist at most two other evBB for f for simple zeros, say , . Furthermore, if , , it follows that
Thus, we can see that any one of , may not be an evPB for f, furthermore, if is an evPB for f, then , are nvNB for f.
Now, some consequences of Theorem 1.2 are listed.
Consequence 5.6 Under the assumptions of Theorem 1.2, if and
we have
-
(i)
if and , then there exists at most one element which is an evBB for f simple zeros; in particular, this holds if there exists an satisfying ;
-
(ii)
if and , then there exist at most two elements which are evBB for f simple zeros; in particular, this holds if there exists an satisfying ;
-
(iii)
if and , then there exists at most three elements which are evBB for f simple zeros; in particular, this holds if there exists an satisfying .
Remark 5.1 Under the assumptions of Theorem 1.2, from Consequence 5.6, we see that if there exist four distinct elements which are evBB for f for simple zeros, then and for and .
Consequence 5.7 Under the assumptions of Theorem 1.2, if and
we have
-
(i)
if and , then there exists at most one element which is an evBB for f distinct simple and double zeros; in particular, this holds if there exists an satisfying ;
-
(ii)
if and , then there exist at most two elements which are evBB for f distinct simple and double zeros; in particular, this holds if there exists an satisfying .
Remark 5.2 Under the assumptions of Theorem 1.2, from Consequence 5.7, we find that if there exist three distinct elements which are evBB for f for distinct simple and double zeros, then and for and .
6 Remarks
From Theorems 1.1 and 1.2, it is a natural question to ask: could we get the same conclusions of Theorems 1.1 and 1.2 when f is a transcendental meromorphic function with finite order ρ () on the whole complex plane? However, we cannot give a positive answer to the above question. Now we give a simple procedure to show that the conclusion of Theorem 1.1 cannot hold when f is a transcendental meromorphic function with finite order ρ () on the whole complex plane.
If f is of finite order ρ (), that is, , then we say a is an exceptional value in the sense of Borel for f in the Borel direction (evBB for short) for distinct zeros of multiplicity ≤k, if . Thus, by Lemma 2.5 and the definition of the Borel direction, (8) can be replaced by
where and r is sufficiently large, and (10) can be replaced by
However, by Lemmas 2.1-2.5, we get
Moreover, from the above inequality, we cannot be sure whether is greater than . If , then from (16) we cannot easily get a contradiction. Therefore, Theorems 1.1 and 1.2 may not be true when f is of finite order.
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Acknowledgements
The first author was supported by the NSF of China (11301233, 61202313, 11201395), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001, 20132BAB211002). The second author was supported by the Science Foundation of Educational Commission of Hubei Province (Q20132801).
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HYX, ZJW and JT completed the main part of this article, HYX corrected the main theorems. The authors read and approved the final manuscript.
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Xu, H.Y., Wu, Z.J. & Tu, J. Some inequalities and applications on Borel direction and exceptional values of meromorphic functions. J Inequal Appl 2014, 53 (2014). https://doi.org/10.1186/1029-242X-2014-53
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DOI: https://doi.org/10.1186/1029-242X-2014-53