# Milloux Inequality ofE-Valued Meromorphic Function

Authors: Zhaojun Wu; Zuxing Xuan
Journal: The Scientific World Journal
 (2014)

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## Abstract

The main purpose of this paper is to establish the Milloux inequality ofE-valued meromorphic function from the complex plane ℂ to an infinite dimensional complex Banach space E with a Schauder basis. As an application, we study the Borel exceptional values of an E-valued meromorphic function and those of its derivatives; results are obtained to extend some related results for meromorphic scalar-valued function of Singh, Gopalakrishna, and Bhoosnurmath.


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## Body

## 1. Introduction

In the 1970s, the Nevanlinna theory of meromorphic function is extended to the vector-valued meromorphic function from the complex planeℂ to a finite dimensional space ℂn (see Ziegler [1]). After that, some works related to vector-valued meromorphic function in finite dimensional spaces were done by [2–5]. In 2006, C. G. Hu and Q. J. Hu [6] established Nevanlinna’s first and second fundamental theorems for an E-valued meromorphic function from the disk ℂr={|z|<r}, 0<r≤+∞, to infinite-dimensional Banach spaces E with a Schauder basis. Xuan and Wu [7] established Nevanlinna’s first and second fundamental theorems for an E-valued meromorphic function from a generic domain D⊆ℂ to E and generalized Chuang’s inequality. Motivated by [6, 7], Bhoosnurmath and Pujari [8] studied the E-valued Borel exceptional values of meromorphic functions, Wu and Xuan [9, 10] studied the characteristic functions, exceptional values, and deficiency of E-valued meromorphic function, and Hu [11] surveyed the advancements of the Nevanlinna theory of E-valued meromorphic functions and studied its related Paley problems. In this paper, we will generalize Milloux’s inequality (see [12] or [13]) to E-valued meromorphic function.


## 2. The Nevanlinna Theory in Banach Spaces

In this section, we introduce some fundamental definitions and notations ofE-valued meromorphic function which was introduced by C. G. Hu and Q. J. Hu [6]. See also [7–10].Let(E,∥•∥) be an infinite dimension complex Banach space with Schauder basis {ej} and the norm ∥•∥. Thus an E-valued meromorphic function f(z) defined in ℂr, 0<r≤+∞, can be written as
(1)f(z)=(f1(z),f2(z),…,fk(z),…)∈E,
where f1(z),f2(z),…,fk(z),… are the component functions of f(z). Let En be an n-dimensional projective space of E with a basis {ej}1n. The projective operator Pn:E→En is a realization of En associated with the basis.The elements ofE are called vectors and are usually denoted by letters from the alphabet: a,b,c,…. The symbol 0 denotes the zero vector of E. We denote vector infinity, complex number infinity, and the norm infinity by ∞^, ∞, and +∞, respectively. A vector-valued mapping is called holomorphic (meromorphic) if all component functions of f(z) are holomorphic (some of component functions of f(z) are meromorphic). The jth derivative of f(z) is defined by
(2)f(j)(z)=(f1(j)(z),f2(j)(z),…,fk(j)(z),…),
where j=1,2,…. A point z0∈ℂr is called a pole (or ∞^ point) of f(z) if z0 is a pole (or ∞ point) of at least one of the component functions of f(z). A point z0∈ℂr is called a zero of f(z) if z0 is a common zero of all the component functions of f(z). A point z0∈ℂr is called a pole or an ∞^-point of f(z) of multiplicity q∈ℕ+ which means that in such a point z0 at least one of the meromorphic component functions of f(z) has a pole of this multiplicity in the ordinary sense of function theory. A point z0∈ℂr is called a zero of f(z) of multiplicity q∈ℕ+ which means that in such a point z0 all component functions of f(z) vanish, each with at least this multiplicity.AnE-valued meromorphic function f(z) in ℂ is said to be of compact projection, if for any given ɛ>0, ∥Pn(f(z))-f(z)∥<ɛ as sufficiently large n in any fixed compact subset D⊂ℂ.Letn(r,f) or n(r,∞^) denote the number of poles of f(z) in |z|≤r and n(r,a,f) denote the number of a-points of f(z) in |z|≤r, counting with multiplicities. Define the volume function associated with E-valued meromorphic function f(z) by
(3)V(r,∞^,f)hh=V(r,f)=12π∫Cr‍log|rξ|Δlog∥f(ξ)∥dx∧dy,hhhhhhhhhhhhhhhhhhhhhhhhhlhhhllhξ=x+iy;V(r,a,f)=12π∫Cr‍log|rξ|Δlog∥f(ξ)-a∥dx∧dy,hhhhhhhhhhhhhhhhhhhhhhhhhhhhllhhξ=x+iy,
and the counting function of finite or infinite a-points by
(4)N(r,f)=n(0,f)logr+∫0r‍n(t,f)-n(0,f)tdt,N(r,∞^)=n(0,∞^)logr+∫0r‍n(t,∞^)-n(0,∞^)tdt,N(r,a,f)=n(0,a,f)logr+∫0r‍n(t,a,f)-n(0,a,f)tdt,
respectively. Next, we define
(5)m(r,f)=m(r,∞^,f)=12π∫02π‍log+∥f(reiθ)∥dθ;m(r,a,f)=12π∫02π‍log+1∥f(reiθ)-a∥dθ;T(r,f)=m(r,f)+N(r,f).
Let n¯(r,f) or n¯(r,∞^) denote the number of poles of f(z) in |z|≤r and n¯(r,a,f) denote the number of a-points of f(z) in |z|≤r, ignoring multiplicities. Similarly, we can define the counting functions N¯(r,f), N¯(r,∞^), and N¯(r,a,f) of n¯(r,f), n¯(r,∞^), and n¯(r,a,f).Letf(z)(z∈ℂr) be an E-valued meromorphic function and a∈E; if k is a positive integer, let n¯k(r,f) or n¯k(r,∞^) denote the number of distinct poles of f(z) of order ≤k in |z|≤r and n¯k(r,a,f) denote the number of distinct a-points of f(z) of order ≤k in |z|≤r. Similarly, we can define the counting functions N¯k(r,f), N¯k(r,∞^), and N¯k(r,a,f) of n¯k(r,f), n¯k(r,∞^), and n¯k(r,a,f).Iff(z) is an E-valued meromorphic function in the whole complex plane, then the order and the lower order of f(z) are defined by
(6)λ(f)=limsupr→∞log+T(r,f)logr;μ(f)=liminfr→∞log+T(r,f)logr.
We call the E-valued meromorphic function f admissible if
(7)limsupr→+∞T(r,f)logr=+∞.Definition 1.
Letf(z) be an admissible E-valued meromorphic function in ℂ. One denotes by S(r,f) any quantity such that
(8)S(r,f)=O(logT(r,f)+logr),r⟶+∞,
without restriction if f(z) is of finite order and otherwise except possibly for a set of values of r of finite linear measure.In 2006, C. G. Hu and Q. J. Hu [6] proved the following theorems.Theorem A (theE-valued Nevanlinna’s first fundamental theorem).
Letf(z) be a nonconstant E-valued meromorphic function in ℂR={|z|<R}, 0<R≤+∞. Then, for 0<r<R, a∈E, and f(z)≢a,
(9)T(r,f)=V(r,a)+N(r,a)+m(r,a)+log+∥cq(a)∥+ɛ(r,a).
Here ɛ(r,a) is a function such that
(10)|ɛ(r,a)|≤log+∥a∥+log2,ɛ(r,0)≡0,
and cq(a)∈E is the coefficient of the first term in the Laurent series at the point a.Theorem B (theE-valued Nevanlinna’s second fundamental theorem).
Letf(z) be an admissible E-valued meromorphic function of compact projection in ℂR={|z|<R}, 0<R≤+∞, and a[k]∈E(k=1,2,…,q) be q≥3 distinct points. Then, for 0<r<R,
(11)∑k=1q‍m(r,a[k],f)≤2T(r,f)-N1(r)+S(r,f),
where N1(r)=2N(r,f)-N(r,f′)+N(r,0,f′).


## 3. Milloux Inequality ofE-Valued Meromorphic Function

In this section, we will establish the Milloux inequality ofE-valued meromorphic function and prove the following theorems.Theorem 2 (Milloux inequality).
Suppose thatf(z) is an admissible E-valued meromorphic function of compact projection in ℂR={|z|<R}, 0<R≤+∞. Let a,b∈E be distinct points and b≠0. Then, for 0<r<R,
(12)T(r,f)≤N¯(r,f)+(k+1){N¯(r,a,f)+V(r,a,f)}+{N¯(r,b,f(k))+V(r,b,f(k))}+S(r,f).In order to prove Theorem2, we will prove the following general form of Milloux inequality of E-valued meromorphic function when the multiple values are considered.Theorem 3 (general form of Milloux inequality).
Suppose thatf(z) is an admissible E-valued meromorphic function of compact projection in ℂR={|z|<R}, 0<R≤+∞. Let a[i],b[j]∈E(i=1,2,…,p;j=1,2,…,q) be distinct points such that b[j]≠0  (j=1,2,…,q) and let mi,nj(i=1,2,…,p;j=1,2,…,q), and l be any positive integers. Then
(13){(1+k∑j=1q‍1nj+1))pq-((∑j=1q‍)∑i=1p‍kq+1mi+1+∑j=1q‍1nj+1+1l+1(1+k∑j=1q‍1nj+1))}T(r,f)≤ll+1(1+k∑j=1q‍1nj+1)N¯l(r,f)+(kq+1)∑i=1p‍{N¯mi(r,a[i],f)+V(r,a[i],f)}+∑j=1q‍{N¯nj(r,b[j],f(k))+V(r,b[j],f(k))}+S(r,f).By lettingp=q=1 and l,mi,nj tend to infinity in (13), we can get Theorem 2. In order to prove Theorem 3, we need the following lemma.Lemma 4 (see [10]).
Letf(z) be of compact projection in ℂ; then, for a positive integer k, one has
(14)12π∫02π‍log+∥f(k)(reiθ)∥∥f(reiθ)∥dθ=S(r,f).We are now in the position to prove Theorem3.Proof.
We set(15)F(z)=∑i=1p‍1∥f(z)-a[i]∥;
then
(16)12π∫02π‍log+F(reiθ)dθ≤m(r,0,f(k))+12π∫02π‍log+{F(reiθ)∥f(k)(reiθ)∥}dθ.
By [6], we have
(17)12π∫02π‍log+F(reiθ)dθ≥∑i=1p‍m(r,a[i])-log+2qδ.
From (16) and (17), we can get
(18)∑i=1p‍m(r,a[i],f)≤m(r,0,f(k))+12π∫02π‍log+{F(reiθ)∥f(k)(reiθ)∥}dθ+log+2qδ.
Hence, we can get from the above inequality and Lemma 4 that
(19)∑i=1p‍m(r,a[i],f)≤m(r,0,f(k))+S(r,f).
It follows from Theorem A that
(20)T(r,f(k))=m(r,0,f(k))+N(r,0,f(k))+V(r,0,f(k))+O(1).
Thus from (19) and (20) we deduce
(21)∑i=1p‍m(r,a[i],f)≤T(r,f(k))-N(r,0,f(k))-V(r,0,f(k))+S(r,f).
By Theorem A, we have
(22)pT(r,f)≤T(r,f(k))+∑i=1p‍[N(r,a[i],f)+V(r,a[i],f)]-N(r,0,f(k))-V(r,0,f(k))+S(r,f).
Now it follows from Theorems A and B and Lemma 4 that
(23)qT(r,f(k))≤∑j=1q‍{N(r,b[j],f(k))+V(r,b[j],f(k))}+N(r,0,f(k))+V(r,0,f(k))+N(r,f(k))-(N(r,0,f(k+1))+2N(r,f(k))-N(r,f(k+1)))+S(r,f(k))=∑j=1q‍{N(r,b[j],f(k))+V(r,b[j],f(k))}+N(r,0,f(k))+V(r,0,f(k))+N(r,f(k+1))-N(r,f(k))+N(r,0,f(k+1))+S(r,f)≤∑j=1q‍{N(r,b[j],f(k))+V(r,b[j],f(k))}+N(r,0,f(k))+V(r,0,f(k))+N¯(r,f)-N(r,0,f(k+1))+S(r,f).
It follows from (22) and (23) that
(24)pqT(r,f)≤N¯(r,f)+(q-1){∑i=1p‍N(r,a[i],f)-N(r,0,f(k))}+{∑j=1q(r,0,f(k+1))‍∑i=1p‍N(r,a[i],f)+∑j=1q‍N(r,b[j],f(k))-N(r,0,f(k+1))}+q∑i=1p‍V(r,a[i],f)+∑j=1q‍V(r,b[j],f(k))+S(r,f).
A zero of f-a of order j>k is a zero of f(k+1) of order j-(k+1) and a zero of f(k)-b of order m is a zero of f(k+1) of order m-1. Moreover, zeros of f-a of order >k are zeros of f(k) and so are not zeros of f(k)-b since b≠0. Hence
(25)∑i=1p‍N(r,a[i],f)+∑j=1q‍N(r,b[j],f(k))-N(r,0,f(k+1))≤∑i=1p‍Nk+1(r,a[i],f)+∑j=1q‍N¯(r,b[j],f(k)),∑i=1p‍N(r,a[i],f)-N(r,0,f(k))≤∑i=1p‍Nk(r,a[i],f).
Substituting (25) to (24), we obtain
(26)pqT(r,f)≤N¯(r,f)+(q-1)∑i=1p‍Nk(r,a[i],f)+∑i=1p‍Nk+1(r,a[i],f)+∑j=1q‍N¯(r,b[j],f(k))+q∑i=1p‍V(r,a[i],f)+∑j=1q‍V(r,b[j],f(k))+S(r,f),
since
(27)Nk(r,a[i],f)≤kN¯(r,a[i],f)≤kmi+1{miN¯mi(r,a[i],f)+N(r,a[i],f)}≤kmi+1{miN¯mi(r,a[i],f)+T(r,f)}+O(1),(28)Nk+1(r,a[i],f)≤(k+1)N¯(r,a[i],f)≤k+1mi+1{miN¯mi(r,a[i],f)+N(r,a[i],f)}≤k+1mi+1{miN¯mi(r,a[i],f)+T(r,f)}+O(1).
Similarly, we can get
(29)N¯(r,b[j],f(k))≤1nj+1{njN¯nj(r,b[j],f(k))+T(r,f(k))}+O(1),N¯(r,f)≤1l+1{lN¯l(r,f)+T(r,f)}.
By Lemma 4, we can get
(30)T(r,f(k))=m(r,f(k))+N(r,f(k))≤m(r,f)+N(r,f(k))+12π∫02π‍log+∥f(k)(reiθ)∥∥f(reiθ)∥dθ≤m(r,f)+N(r,f)+kN¯(r,f)+S(r,f)≤T(r,f)+kN¯(r,f)+S(r,f).
Substituting (27)–(30) into (26), we obtain
(31)pqT(r,f)≤N¯(r,f)+(q-1)×∑i=1p‍kmi+1{miN¯mi(r,a[i],f)+T(r,f)}+∑i=1p‍k+1mi+1{miN¯mi(r,a[i],f)+T(r,f)}+∑j=1q‍1nj+1{njN¯nj(r,b[j],f(k))+T(r,f(k))}+q∑i=1p‍V(r,a[i],f)+∑j=1q‍V(r,b[j],f(k))+S(r,f)≤(1+∑j=1q‍knj+1)N¯(r,f)+(q-1)×∑i=1p‍kmimi+1N¯mi(r,a[i],f)+∑i=1p‍k+1mi+1miN¯mi(r,a[i],f)+∑j=1q‍njnj+1N¯nj(r,b[j],f(k))+(q-1)∑i=1p‍kmi+1T(r,f)+∑i=1p‍k+1mi+1T(r,f)+∑j=1q‍1nj+1njT(r,f)+q∑i=1p‍V(r,a[i],f)+∑j=1q‍V(r,b[j],f(k))+S(r,f)≤(1+∑j=1q‍knj+1)ll+1N¯l(r,f)+(kq+1)×∑i=1p‍mimi+1N¯mi(r,a[i],f)+∑j=1q‍njnj+1N¯nj(r,b[j],f(k))+q∑i=1p‍V(r,a[i],f)+∑j=1q‍V(r,b[j],f(k))+(∑i=1p‍kq+1mi+1+∑j=1q‍1nj+1+1l+1(1+k∑j=1q‍1nj+1))×T(r,f)+S(r,f).
Since mi, nj, k, and q are positive integers, it follows from (31) that
(32)pqT(r,f)≤(∑i=1p‍kq+1mi+1+∑j=1q‍1nj+1+1l+1(1+k∑j=1q‍1nj+1))×T(r,f)+(kq+1)∑i=1p‍[N¯mi(r,a[i],f)+V(r,a[i],f)]+∑j=1q‍[N¯nj(r,b[j],f(k))+V(r,b[j],f(k))]+(1+∑j=1q‍knj+1)ll+1N¯l(r,f)+S(r,f).
Hence, (13) follows from (32).


## 4.E-Valued Borel Exceptional Values of Meromorphic Function and Its Derivatives

Most recently, Bhoosnurmath and Pujari [8] studied the E-valued Borel exceptional values of meromorphic functions and gave the following definition.Definition 5.
Letf(z)(z∈ℂ) be an E-valued meromorphic function and a∈E∪{∞^}k is a positive integer. One defines
(33)ρ¯k(a,f)=limsupr→∞log+[V(a,f)+N¯k(r,a)]logr;ρ¯(a,f)=limsupr→∞log+[V(a,f)+N¯(r,a)]logr;ρ(a,f)=limsupr→∞log+[V(a,f)+N(r,a)]logr.
We say that a is an (i)
E-valued evB (exceptional value in the sense of Borel) for f for distinct zeros of order ≤k if ρ¯k(a,f)<λ(f);(ii)
E-valued evB for f for distinct zeros if ρ¯(a,f)<λ(f);(iii)
E-valued evB for f (for the whole aggregate of zeros) if ρ(a,f)<λ(f).Suppose thatf(z) is an E-valued meromorphic function with finite order ρ>0 in ℂ. Xuan and Wu [7] proved that the order of f′ is ρ. Hence for any positive integer l the order of f(l) is ρ. Therefore, we call a a vector-valued evB for f(l) for distinct zeros of order ≤k, if
(34)ρ¯k(a,f(l))=limsupr→∞log[V(r,a,f(l))+N¯k(r,a,f(l))]logr<ρ.In this section, we will prove the following theorem.Theorem 6.
Letf(z) be an admissible E-valued meromorphic function of compact projection in ℂ and the order of f(z) is ρ(0<ρ<+∞). Suppose that ∞^ is an E-valued evB for f for distinct zeros of order ≤l, a[i]∈E(i=1,2,…,p) are E-valued evB for f for distinct zeros of order ≤mi, and b[j](≠0)∈E(j=1,2,…,q) are E-valued evB for f(k) for distinct zeros of order ≤nj, where k,p,q,l and all of mi,nj are positive integers. Then
(35)∑i=1p‍kq+1mi+1+∑j=1q‍1nj+1+1l+1(1+k∑j=1q‍1nj+1)≥pq.Proof.
By Theorem3, we obtain
(36){+∑j=1q‍1nj+1+1l+1(1+k∑j=1q‍1nj+1))pq-((∑j=1q‍)∑i=1p‍kq+1mi+1+∑j=1q‍1nj+1+1l+1(1+k∑j=1q‍1nj+1))}T(r,f)≤ll+1(1+k∑j=1q‍1nj+1)N¯l(r,f)+(kq+1)∑i=1p‍{N¯mi(r,a[i],f)+V(r,a[i],f)}+∑j=1q‍{N¯nj(r,b[j],f(k))+V(r,b[j],f(k))}+S(r,f).
Since ∞^ is an E-valued evB for f for distinct zeros of order ≤l, a[i]∈E(i=1,2,…,p) is an E-valued evB for f for distinct zeros of order ≤mi and b[j](≠0)∈E(j=1,2,…,q) is an E-valued evB for f(k) for distinct zeros of order ≤nj. Thus there is a 0<μ<ρ such that for any i,j(i=1,2,…,p;j=1,2,…,q) we have
(37)N¯l(r,f)≤Rμ,N¯mi(r,a[i],f)+V(r,a[i],f)≤Rμ,N¯nj(r,b[j],f(k))+V(r,b[j],f(k))≤Rμ.
It follows from 0<μ<ρ and (36) and (37) that
(38)∑i=1p‍kq+1mi+1+∑j=1q‍1nj+1+1l+1(1+k∑j=1q‍1nj+1)≥pq.Lettingp=q=1 in Theorem 6, we can get the following corollary.Corollary 7.
Letf(z) be an admissible E-valued meromorphic function of compact projection in ℂ and the order of f(z) is ρ(0<ρ<+∞). Suppose that ∞^ is an E-valued evB for f for distinct zeros of order ≤l, where l is an integer ≥1. If there exist a, b∈E, b≠0, such that a is an E-valued evB for f for distinct zeros of order ≤p and b is a an E-valued evB for f(k) for distinct zeros of order ≤q, where p,q are positive integers, then
(39)q+1+k(q+1)(l+1)+k+1p+1+1q+1≥1.If∞^, a are E-valued evB for f for distinct zeros, that is, letting l,p tend to infinity in (39), we can get 1/(q+1)≥1. This means that, for each integer k, q≥1,ρ¯q(b,f(k))≥ρ, for all b≠0,≠∞^. Hence, we can get the following corollary.Corollary 8.
Letf(z) be an admissible E-valued meromorphic function of compact projection in ℂ and the order of f(z) is ρ(0<ρ<+∞). Suppose that ∞^, a∈E are E-valued evB for f for distinct zeros. Then, for all positive integers k and q, ρ¯q(b,f(k))=ρ for all b≠0,≠∞^.The corresponding results of Corollaries7 and 8 for the meromorphic scalar value function were obtained by Gopalakrishna and Bhoosnurmath [14] and Singh and Gopalakrishna [15]. The corresponding results of Corollaries 7 and 8 for the meromorphic scalar value function on annuli were obtained by Chen and Wu [16].



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