Value Distribution of the k th Derivatives of Meromorphic Functions

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 ( ) ( ) = / e 0 z a z P z ≡ , where P is a polynomial. Suppose that all zeros of f have multiplicity at least 1 k + , except possibly finite many, and ( ) ( ) ( ) , , T r a o T r f = as r →∞ . Then ( ) k f R − has infinitely many zeros.


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 1 k + , where k is a positive integer.In 2008, Liu et al. [2] proved the following results.Theorem A Let 2 k ≥ be an integer, let ( ) f z be a meromorphic function of infinite order ( ) and let ( ) ( )e 0 z a z P z = ≡ / , where P is a polynomial.Suppose that 1) all zeros of f have multiplicity at least 1 k + , except possibly finitely many, and 2) all poles of f are multiple, except possibly finitely many.
k ≥ be an integer, let ( ) f z be a meromorphic function of finite order ( ) and let ( ) ( )e 0 z a z P z = ≡ / , where P is a polynomial.Suppose that 1) all zeros of f have multiplicity at least 1 k + , except possibly finitely many, and 2) z a z − has infinitely many zeros.In the present paper, we prove the following result, which is a significant improvement of Theorem 1.

Theorem 1 Let 1
k ≥ be an integer, let f be a meromorphic function of order ( ) k ≥ be an integer, let f be a meromorphic function in  , and let ( ) ( )e 0 z a z P z = ≡ / , where P is a polynomial.Suppose that 1) all zeros of f have multiplicity at least 1 k + , except possibly finitely many, and has infinitely many zeros.

Notation and Some Lemmas
We use the following notation.Let  be complex plane and D be a domain in  .For 0 ( ) Let f be a meromorphic function in  .Set The Ahlfors-Shimizu characteristic is defined by  Lemma 2 [5, Lemma 2] Let  be a family of functions meromorphic in D , all of whose zeros have multiplicity at least k , and suppose that there exists 1 Proof We claim that there exist n t → ∞ and Otherwise there would exist 0 and hence ( ) ≤ which contradicts the hypothesis that ( ) hence there exists a sequence { } is not normal at 0. Obviously, all zeros of g have multiplicity at least , and hence all zeros of ( ) n g z have multiplicity at least 1 k + in ∆ for sufficiently large n .Using Lemma 2 for ( ) , there exist points 0 in  , where G is a nonconstant meromorphic function in  , all of whose zeros have multiplicity at least ζ ≡ / , where 0 c is a constant.Otherwise, ( ) ( ) ( )

Proof of Theorem
Proof We assume that (1.7) Using the simple inequality ( ) (1.8) The second term on the right of (1.7) is ( )

,Lemma 1 [ 3 ]
T r f denote the usual Nevanlinna characteristic function.Since ( ) ( ) Let { } n h a sequence of holomorphic functions in D such that n h h H′ ⇒ = locally uniformly in D , where H is univalent in D .Let { } n f be a sequence of functions meromorphic in D such that for each n , 1) all zeros of n f have multiplicity at least 1 k + ; and 2 let us indicate how it follows from the results of that paper.The proof that { } n f is quasinormal of order 1 is essentially identical to that of Theorem 1 of [3].That proof also shows that condition (b) of Lemma 7 in [3] holds for 1 0 a z = .It then follows from Lemma 7 that 9 of [3].See also [4, Remark on page 484].

as n → ∞ . Lemma 4 1 k
Let k + ∈  and d ∈  .Let f be a transcendental meromorphic function, all of whose zeros have multiplicity at least

2 ,
, , c c  and k c are constants.Then, either G is a constant function, or all zeros of G have multiplicity OPEN ACCESS APM P. YANG, X.J. LIU 14 at most k .A contradiction.Let 0 ζ be not a zero or pole of ( ) ( ) as n → ∞ .
where g is a nonconstant meromorphic function in  , all of whose zeros have multiplicity at least k , such that