Value Distribution of the kth Derivatives of Meromorphic Functions

Abstract

In the paper, we take up a new method to prove a result of value distribution of meromorphic functions: let f be a meromorphic function in , and let , where P is a polynomial. Suppose that all zeros of f have multiplicity at least , except possibly finite many, and as . Then has infinitely many zeros.

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P. Yang and X. Liu, "Value Distribution of the kth Derivatives of Meromorphic Functions," Advances in Pure Mathematics, Vol. 4 No. 1, 2014, pp. 11-16. doi: 10.4236/apm.2014.41002.

1. Introduction

The value distribution theory of meromorphic functions occupies one of the central places in Complex Analysis which now has been applied to complex dynanics, complex differential and functional equations, Diophantine equations and others.

In his excellent paper [1], W.K. Hayman studied the value distribution of certain meromorphic functions and their derivatives under various conditions. Among other important results, he proves that if f(z) is a transcendental meromorphic function in the plane, then either f(z) assumes every finite value infinitely often, or every derivative of f(z) assumes every finite nonzero value infinitely often. This result is known as Hayman’s alternative. Thereafter, the value distribution of derivatives of transcendental functions continued to be studied.

In this paper, we study the value distribution of transcendental meromorphic functions, all but finitely many of whose zeros have multiplicity at least, where is a positive integer.

In 2008, Liu et al. [2] proved the following results.

Theorem A Let be an integer, let be a meromorphic function of infinite order in, and let, where is a polynomial. Suppose that 1) all zeros of have multiplicity at least, except possibly finitely many, and 2) all poles of are multiple, except possibly finitely many.

Then has infinitely many zeros.

Theorem B Let be an integer, let be a meromorphic function of finite order in, and let, where is a polynomial. Suppose that 1) all zeros of have multiplicity at least, except possibly finitely many, and 2).

Then has infinitely many zeros.

In the present paper, we prove the following result, which is a significant improvement of Theorem 1.

Theorem 1 Let be an integer, let be a meromorphic function of order in, and let, where is a polynomial. Suppose that all zeros of have multiplicity at least, except possibly finitely many. Then has infinitely many zeros.

Theorem 1 and Theorem 2 taken together imply the following result.

Theorem 2 Let be an integer, let be a meromorphic function in, and let, where is a polynomial. Suppose that 1) all zeros of have multiplicity at least, except possibly finitely many, and 2) as.

Then has infinitely many zeros.

2. Notation and Some Lemmas

We use the following notation. Let be complex plane and be a domain in. For and, and. We write in to indicate that the sequence converges to in the spherical metric uniformly on compact subsets of and in if the convergence is in the Euclidean metric.

Let be a meromorphic function in. Set

(1.1)

The Ahlfors-Shimizu characteristic is defined by

Remark Let denote the usual Nevanlinna characteristic function. Since is bounded as a function of, we can replace with in the paper.

The order of the meromorphic function is defined as

Lemma 1 [3] Let a sequence of holomorphic functions in such that locally uniformly in, where is univalent in. Let be a sequence of functions meromorphic in such that for each1) all zeros of have multiplicity at least; and 2).

Then is quasinormal of order 1 in. If, moreover, no subsequence of is normal at, then

locally uniformly in and there exists such that for all.

Remark Since Lemma 1 is not stated explicitly in [3], let us indicate how it follows from the results of that paper. The proof that is quasinormal of order 1 is essentially identical to that of Theorem of [3]. That proof also shows that condition (b) of Lemma 7 in [3] holds for. It then follows from Lemma 7 that locally uniformly on. The bound on follows from Lemma 9 of [3]. See also [4, Remark on page 484].

Lemma 2 [5, Lemma 2] Let be a family of functions meromorphic in, all of whose zeros have multiplicity at least, and suppose that there exists such that whenever. Then if is not normal at, there exist, for each1) points,;

2) functions; and 3) positive numbers

such that in, where is a nonconstant meromorphic function inall of whose zeros have multiplicity at least, such that.

Lemma 3 Let be a meromorphic function of order in, then there exist and such that

Proof We claim that there exist and such that

(1.2)

Otherwise there would exist and such that

for all. From this follows

and hence

Now we have which contradicts the hypothesis that.

Observing that hence there exists a sequence such that and as. Let. Obviously, and , and hence as.

Lemma 4 Let and. Let be a transcendental meromorphic function, all of whose zeros have multiplicity at least. Set. Suppose that. Then there exists a sequence

and such that

as.

Proof Since and, we have. By Lemma 3, there exist and such that

Set. Clearly,. Thus is not normal at 0. Obviously, all zeros of have multiplicity at least in, and hence all zeros of have multiplicity at least in for sufficiently large. Using Lemma 2 for, there exist points, and positive numbers and a subsequence of (that we continue to call) such that

in, where is a nonconstant meromorphic function in, all of whose zeros have multiplicity at least.

We claim that, where is a constant. Otherwise, , where and are constants. Then, either is a constant function, or all zeros of have multiplicity at most. A contradiction.

Let be not a zero or pole of, and let. Now we have

where. Since and is not a zero or pole of, we have, and as, where.

Set and, where. Clearly,

where satisfying as.

Now, we have and

Set. Obviously, and, and hence as.

3. Proof of Theorem

Proof We assume that has at most finitely many zeros and derive a contradiction. Let as, where and.

Set. By Lemma 4, there exists a sequence and such that

(1.3)

and

(1.4)

Set. By (1.4),

(1.5)

Hence, no subsequence of is normal at.

Since has at most finitely many zeros, we have for sufficiently large,

Observing that

in. It follows from Lemma 1 (applied to in), and there exists such that for all

(1.6)

Set. Then

and hence

(1.7)

Using the simple inequality

for, we have

(1.8)

The second term on the right of (1.7) is

(1.9)

Putting (1.7), (1.8), and (1.9) together, we have for and sufficiently large,

(1.10)

It follows from (1.1), (1.6), and (1.10),

Thus,

Acknowledgements

This work was supported by National Natural Science Foundation of China (No.11001081, No.11226095).

[1]       W. K. Hayman, “Picard Values of Meromorphic Functions and Their Derivatives,” Annals of Mathematics, Vol. 70, No. 1, 1959, pp. 9-42. http://dx.doi.org/10.2307/1969890

[2]       X. J. Liu, S. Nevo and X. C. Pang, “On the kth Derivative of Meromorphic Functions with Zeros of Multiplicity at Least k+1,” Journal of Mathematical Analysis and Applications, Vol. 348, No. 1, 2008, pp. 516-529. http://dx.doi.org/10.1016/j.jmaa.2008.07.019

[3]       S. Nevo, X. C. Pang and L. Zalcman, “Quasinormality and meromorphic functions with multiple zeros,” Journal d’Analyse Math??matique, Vol. 101, No. 1, 2007, pp. 1-23.

[4]       X. C. Pang, S. Nevo and L. Zalcman, “Derivatives of Meromorphic Functions with Multiple Zeros and Rational Functions,” Computational Methods and Function Theory, Vol. 8, No. 2, 2008, pp. 483-491. http://dx.doi.org/10.1007/BF03321700

[5]       X. C. Pang and L. Zalcman, “Normal Families and Shared Values,” Bulletin London Mathematical Society, Vol. 32, No. 3, 2000, pp. 325-331. http://dx.doi.org/10.1112/S002460939900644X

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] W. K. Hayman, “Picard Values of Meromorphic Functions and Their Derivatives,” Annals of Mathematics, Vol. 70, No. 1, 1959, pp. 9-42. http://dx.doi.org/10.2307/1969890 [2] X. J. Liu, S. Nevo and X. C. Pang, “On the kth Derivative of Meromorphic Functions with Zeros of Multiplicity at Least k+1,” Journal of Mathematical Analysis and Applications, Vol. 348, No. 1, 2008, pp. 516-529. http://dx.doi.org/10.1016/j.jmaa.2008.07.019 [3] S. Nevo, X. C. Pang and L. Zalcman, “Quasinormality and meromorphic functions with multiple zeros,” Journal d’Analyse Math??matique, Vol. 101, No. 1, 2007, pp. 1-23. [4] X. C. Pang, S. Nevo and L. Zalcman, “Derivatives of Meromorphic Functions with Multiple Zeros and Rational Functions,” Computational Methods and Function Theory, Vol. 8, No. 2, 2008, pp. 483-491. http://dx.doi.org/10.1007/BF03321700 [5] X. C. Pang and L. Zalcman, “Normal Families and Shared Values,” Bulletin London Mathematical Society, Vol. 32, No. 3, 2000, pp. 325-331. http://dx.doi.org/10.1112/S002460939900644X