Abstract and Applied Analysis

Volume 2012 (2012), Article ID 217280, 16 pages

doi:10.1155/2012/217280

Research Article

A Note on the Growth of Meromorphic Functions with a Radially Distributed Value

Jun-Fan Chen

Department of Mathematics, Fujian Normal University, Fujian Province, Fuzhou 350007, China

Received 8 July 2012; Accepted 17 October 2012

Academic Editor: Ferhan M. Atici

Copyright © 2012 Jun-Fan Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This note is to investigate the growth of transcendental meromorphic functions with radially distributed values. We generalize a more recent result of Chen et al. (2011). The paper is closely related to some previous results due to Fang and Zalcman (2008), and Xu et al. (2009).

1. Introduction and Results

Let : be a meromorphic function, where is the whole complex plane and . We shall use the basic results and notations of Nevanlinna's value distribution theory of meromorphic functions (see [1–3]), such as ,  , and . Meantime, the Nevanlinna's deficiency of with respect to is defined by

and is obtained by the above formula with in place of , and in place of , respectively. The lower order and order are defined in turn as follows:

For an unbounded subset of and , we denote by the number of the zeros of counting multiplicities in .

In 1992, Yang [4], cf. [5] posed the following interesting conjecture.

Yang's Conjecture

Let be a property (or a set of properties) such that any entire (or meromorphic in ) function satisfying must be a constant. Suppose that is an entire (or meromorphic in ) function of finite lower order , and that ; ; are a finite number of rays issuing from the origin. If satisfies in , then the order of has the following estimation:

Yang's conjecture implies that if meromorphic functions satisfy certain properties in the vicinities of a finite number of rays, then the growth of meromorphic functions will be restricted.

A well-known result of Clunie [6], cf. [7] is that an entire function which satisfies in must be constant. In [5], L. Yang and C.-C. Yang chose the property as and verified the above conjecture. On the other hand, Fang and Zalcman [8] considered the value distribution of in , where is a nonzero finite complex number and is a positive integer. Actually, they [8] gave an affirmative answer to a question suggested by Ye [9]. Later on, Xu et al. [10] further generalized to in for a positive integer with and investigated its value distribution.

More recently, Chen et al. [11] chose another property as to continue to study Yang's Conjecture and proved the following results.

Theorem A. Let be a transcendental meromorphic function with in and let be a finite number of rays issued from the origin such that

with . Set . If satisfies

with a positive number , finite complex numbers and , for any positive integer , then the order of has the estimation .

Theorem B. Let be a transcendental meromorphic function of finite lower order with in . For pairs of real numbers such that

with , suppose that

with a positive number , finite complex numbers and , and , for any positive integer , and that

with . Then the order of has the estimation .

Now there arises a natural question.

Question 1. What can be said if in Theorems A and B is replaced by the th derivative ?

In this paper, we will prove the following results which generalize Theorems A and B.

Theorem 1.1. Let be a transcendental meromorphic function with for a positive integer in and let be a finite number of rays issued from the origin such that

with . Set . If satisfies

with a positive number , finite complex numbers and , for any positive integer , then the order of has the estimation .

Remark 1.2. Let . Then by Theorem 1.1 we get Theorem A.

Corollary 1.3. Let be a transcendental entire function, let the notations (), , and be defined as in Theorem 1.1, and suppose that the function fulfills the same condition (1.10) as in Theorem 1.1. Then the order of has the estimation .

Theorem 1.4. Let be a transcendental meromorphic function of finite lower order with for a positive integer in . For pairs of real numbers such that

with , suppose that

with a positive number , finite complex numbers and , and , for any positive integer , and that

with . Then the order of has the estimation .

Remark 1.5. Let . Then by Theorem 1.4 we get Theorem B.

Corollary 1.6. Let be a transcendental entire function of finite lower order , let the notations , (), , , and be defined as in Theorem 1.4, and suppose that the function fulfills the same conditions (1.12) and (1.13) as in Theorem 1.4. Then the order of has the estimation .

In order to prove our results, we require the Nevanlinna theory of meromorphic functions in an angular domain. For the sake of convenience, we recall some notations and definitions. Let be a meromorphic function on the angular domain , where . Nevanlinna et al. [12, 13] introduced the following notations:

where and are the poles of in appearing according to their multiplicities. The function is called the angular counting function (counting multiplicities) of the poles of in , and is called the angular reduced counting function (ignoring multiplicities) of the poles of in . Further, Nevanlinna's angular characteristic function is defined as follows:

Throughout the paper, we denote by a quantity satisfying

where denotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context.

2. Some Lemmas

In this section we present some lemmas which will be needed in the sequel.

Lemma 2.1 (see [5, 12–14]). Let be meromorphic in . Then in for an arbitrary finite complex number , we have

and for each positive integer , we have

Lemma 2.2 (see [10]). Let and be positive integers, let be a nonzero finite complex number, and let be a polynomial. Then the solution of the differential equation must be polynomial.

Lemma 2.3 (see [1, Theorem 3.1]; [1, page 33, (2.1)]). Let be meromorphic in , let , where , be distinct complex numbers, , and suppose that for . Then, for each positive integer , we have

Next we slightly modify the proof of Lemma 2.4 in [10] to give the following key lemma, which is an important generalization of Lemma 3 in [11].

Lemma 2.4. Let be transcendental meromorphic in , let and be finite complex numbers, and let and be positive integers with . Then in ,

Proof. Put

Then , for otherwise would be a polynomial of degree at most . This and (2.5) together with Lemma 2.2 imply that must be a polynomial, a contradiction. By the Nevanlinna's basic reasoning, Lemmas 2.1, and 2.3, (2.5), and (2.6), we have

Now a simple calculation for (2.5) shows that

where is a homogeneous differential polynomial in of degree and of the form

Then by (2.5), (2.6), and (2.8), we get

From (2.5)–(2.10), and Lemma 2.1, it thus follows that

On the other hand, by the Nevanlinna's basic reasoning, Lemmas 2.1 and 2.3, we deduce that

This, together with Lemma 2.1, yields

Next we divide into two cases.

Case  1 (). Suppose that is a zero of of multiplicity . Then we can know that is a zero of of multiplicity at least . Thus, we have

Substituting this into (2.13) gives

From this and (2.11) it follows that

implying that

Thereby, noting , we get

so that

This is the desired result.

Case  2 (). Then, by (2.11) and (2.13), we have

which leads to

This yields

obtaining the desired result.

This completes the proof of Lemma 2.4.

The following auxiliary results regarding Pólya peaks and the spread relation are necessary in the proofs of our theorems.

Lemma 2.5 (see [14–16]). Let be a transcendental meromorphic function of finite lower order and order in . Then, for an arbitrary positive number satisfying and any set of finite linear measure, there exist Pólya peaks satisfying the following:

(i), ;(ii);(iii).

A sequence of satisfying (i), (ii), and (iii) in Lemma 2.5 is called a Pólya peak of order of outside . Given a positive function on with as , we define

Lemma 2.6 (see [17]). Let be a transcendental meromorphic function of finite lower order and order in . Suppose that for some . Then for an arbitrary Pólya peak of order and an arbitrary positive function with as , we have

Now a more precise estimation of in terms of is introduced as follows.

Lemma 2.7 (see [18]). Let be transcendental meromorphic in . Then, for a positive integer and a real number , we have

where is a positive number depending on only and .

At last, we state the following results due to Edrei, Hayman, and Miles, respectively.

Lemma 2.8 (see [19]). Let be a transcendental meromorphic function with in . Then, given , we have

where

and is a set of positive real numbers with finite logarithmic measure (i.e., depending on only.

Lemma 2.9 (see [20]). Let be a transcendental meromorphic function in . Then for each there exists a set of the lower logarithmic density at least , that is,

such that, for every positive integer , we have

3. Proofs of Theorems 1.1 and 1.4

In this section, we state the detailed proofs of Theorems 1.1 and 1.4 by using the method in [14]. To begin with, we give the proof of Theorem 1.4. Finally the proof of Theorem 1.1 can be derived from Theorem 1.4.

3.1. Proof of Theorem 1.4

Assume on the contrary that Theorem 1.4 does not hold. Then . Now by (1.12), we have

for arbitrarily small and sufficiently large . Let be the zeros of on appearing according to their multiplicities, and set . By the definition of , we deduce that

where is a positive number depending on only and , which is not necessarily the same for every occurrence in the context. From Lemma 2.4, we have

Thus, it follows by (3.2) and (3.3) that

where the exceptional set associated with is of at most finite linear measure.

Now we discuss two cases separately.

Case 1 (). Then by the assumption and , we have . Now from (1.13), we can find a real number such that

Applying Lemma 2.5 to gives the existence of the Pólya peak of order of outside the set . Then, noting that and , by applying Lemma 2.6 to the Pólya peak , for sufficiently large we have

Without loss of generality, we can assume that (3.7) holds for all the . Set

It then follows from (3.5) and (3.7) that

By (3.9), it is easy to see that there exists a such that, for infinitely many , we have

Without loss of generality, we can assume that (3.10) holds for all the . Set and . From the definition of , we deduce that

On the other hand, by the definition of and (3.4), it follows that

where , , and is a positive number depending on only and . Combining (3.11) with (3.12) gives

implying together with (iii) in Lemma 2.5 and Lemma 2.7 that

Thus, from (ii) in Lemma 2.5 for , we have

which is impossible.

Case 2 (). Then by the assumption and , we have . By the same argument as in Case 1 with all the replaced by , we can derive

which is also impossible.

This completes the proof of Theorem 1.4.

3.2. Proof of Theorem 1.1

By Theorem 1.4, it suffices to prove that the lower order of is finite. As in the proof of Theorem 1.4, we have, for each ,

where the exceptional set associated with is of at most finite linear measure.

For in Lemma 2.8 and in (3.17), and hence for in Lemma 2.9 when , . Applying Lemma 2.8 to gives the existence of a sequence of positive numbers such that , , and

Set

Then, from (3.18) and (3.19), it follows that

Hence, there exists a such that, for infinitely many , we have

Without loss of generality, we can assume that this holds for all the . Let . Thus, by the definition of and (3.21), it follows that

On the other hand, by the definition of and (3.17), we have

where , , and is a positive number depending on only and . Combining (3.22) with (3.23) now yields

so that, together with (3.19) and Lemma 2.9, we have

Thus and so Theorem 1.1 follows from Theorem 1.4.

This completes the proof of Theorem 1.1.

Acknowledgments

The author is extremely grateful to the referees for their many valuable suggestions to improve the presentation. The project was supported by the National Natural Science Foundation of China (Grant no. 11126351) and the Natural Science Foundation of Fujian Province, China (Grant no. 2010J05003).

References

W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964.

L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.

H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Pure and Applied Mathematics Monographs No. 32, Science Press, Beijing, China, 1995.

L. Yang, “Some recent results and problems in the theory of value-distribution,” in Proceedings Symposium on Value Distribution Theory in Several Complex Variables, vol. 12, pp. 157–171, University of Notre Dame Press, 1992.

L. Yang and C.-C. Yang, “Angular distribution of values of ,” Science in China Series A, vol. 37, no. 3, pp. 284–294, 1994.

J. Clunie, “On a result of Hayman,” Journal of the London Mathematical Society, vol. 42, pp. 389–392, 1967.

W. K. Hayman, “Picard values of meromorphic functions and their derivatives,” Annals of Mathematics, vol. 70, pp. 9–42, 1959.

M. Fang and L. Zalcman, “On the value distribution of ,” Science in China, vol. 51, no. 7, pp. 1196–1202, 2008.

Y. S. Ye, “A Picard type theorem and Bloch law,” Chinese Annals of Mathematics Series B, vol. 15, no. 1, pp. 75–80, 1994.

Y. Xu, F. Q. Wu, and L. W. Liao, “Picard values and normal families of meromorphic functions,” Proceedings of the Royal Society of Edinburgh A, vol. 139, no. 5, pp. 1091–1099, 2009.

S.-J. Chen, W.-C. Lin, and J.-F. Chen, “On the growth of meromorphic functions with a radially distributed value,” Indian Journal of Pure and Applied Mathematics, vol. 42, no. 1, pp. 53–70, 2011.

R. Nevanlinna, “Uber die Eigenschaften meromorpher Funktionen in einem Winkelraum,” Acta Societatis Scientiarum Fennicae, vol. 50, pp. 1–45, 1925.

A. A. Gol’dberg and I. V. Ostrovskiĭ, The Distribution of Values of Meromorphic Functions, Izdat-vo Nauka, Moscow, Russia, 1970.

J. Zheng, “On transcendental meromorphic functions with radially distributed values,” Science in China. Series A, vol. 47, no. 3, pp. 401–416, 2004.

A. Edrei, “Sums of deficiencies of meromorphic functions,” Journal d'Analyse Mathématique, vol. 14, pp. 79–107, 1965.

L. Yang, “Borel directions of meromorphic functions in an angular domain,” Scientia Sinica, pp. 149–164, 1979.

A. Baernstein, II, “Proof of Edrei's spread conjecture,” Proceedings of the London Mathematical Society, vol. 26, pp. 418–434, 1973.

C.-T. Chuang, “Sur la comparaison de la croissance d'une fonction méromorphe et de celle de sa dérivée,” Bulletin des Sciences Mathématiques, vol. 75, pp. 171–190, 1951.

A. Edrei, “Meromorphic functions with three radially distributed values,” Transactions of the American Mathematical Society, vol. 78, pp. 276–293, 1955.

W. K. Hayman and J. Miles, “On the growth of a meromorphic function and its derivatives,” Complex Variables, vol. 12, no. 1–4, pp. 245–260, 1989.