On normal and non-normal holomorphic functions on complex Banach manifolds

On normal and non-normal holomorphic functions on complex Banach manifolds

PETERV. DOVBUSH

Abstract. Let X be a complex Banach manifold. A holomorphic function f :X→Cis called a normal function if the familyFf = {f◦ϕ:ϕ∈O(,X)}

forms a normal family in the sense of Montel (hereO(,X)denotes the set of all holomorphic maps from the complex unit disc intoX). Characterizations of normal functions are presented. A sufficient condition for the sum of a normal function and non-normal function to be non-normal is given. Criteria for a holo- morphic function to be non-normal are obtained.

These results are used to draw one interesting conclusion on the boundary behavior of normal holomorphic functions in a convex bounded domain Din a complex Banach spaceV.Let{xn}be a sequence of points inDwhich tends to a boundary pointξ∈∂Dsuch that limn→∞ f(xn)=Lfor someL ∈C.Sufficient conditions on a sequence{x_n}of points inDand a normal holomorphic function f are given for f to have the admissible limit valueL,thus extending the result obtained by Bagemihl and Seidel.

Mathematics Subject Classification (2000):32A18 (primary).

1.

Introduction

The idea of associating to a meromorphic function f on the complex unit disc = {z ∈

C

^{: |z| < 1}a family}

F

^{= {f ◦g : g ∈ Aut()}, where Aut()is} the group of biholomorphic automorphisms of, and of ascribing to the function f properties of the family

F

apparently arose in work of Yosida [16] in 1934 and was considered by Noshiro [14] in 1937. In 1957, Lehto and Virtanen [12] defined

“normal functions” to be those meromorphic functions f whose associated families

F

were normal in the sense of Montel.

Since that time, the subject of normal functions has been studied extensively, resulting in substantial development in the single complex variable context and in generalizations to the setting of several complex variables (see lists of references in [11, 13], and [17]).

Received January 30, 2008; accepted in revised form June 29, 2008.

In this paper, we investigate normal holomorphic functions on complex Ba- nach manifolds. After preliminaries in Section 2, in Section 3 we obtain certain relations existing between notions of normal functions, and P-sequences. In Sec- tion 4 we give criteria for a holomorphic function to be a non-normal function. In Section 5 we give sufficient condition for the sum of a normal function and not a normal function to be a non-normal function. Finally in Section 6 we investigate the boundary behavior in a fixed boundary point of a normal function defined on a convex bounded domain in a complex Banach space.

2.

Preliminaries

We refer the reader to the paper [4] and the books [3] and [5] for background on complex analysis in infinite dimension. The notation we use is essentially the same as in [4].

Let X be a complex Banach manifold modelled on a complex Banach space of positive, possibly infinite, dimension;Xis assumed to be a connected Hausdorff space. For eachxinXthe tangent space to Xat pointxwill be denoted byT_x(X).

The tangent bundleT(X)ofX consists of the ordered pairs(y, v)such thatx ∈X andv ∈ T_x(X).We shall denote the space of all holomorphic maps from the unit diskinto Xby

O

(,X).

The infinitesimal Kobayashi pseudometric on the complex Banach manifoldX is the functionk_X onT(X)defined by the formula

k_X(x, v)=inf{|a| : ∃ϕ∈

O

(,X), ϕ(0)=x, ϕ_∗(0)a=v}

whereϕ_∗(0)is the linear map induced byϕfromT₀()toT_ϕ(0)(X).

We say thatk_X is a metric ifk_X(x, v) >0 for all(x, v)∈T_x(X), v=0. The Kobayashi length of a piecewiseC¹curveγ : [0,1] → Xin Xis defined to be the upper Riemann integral

L_k(γ )= 1

0

k_X(γ (t), γ(t))dt

and the pseudometric K_X(x,y)is the infimum of the lengths of all piecewiseC¹ curves joiningxtoyinX.

Forz₁, w1inthe Poincar´e distance is expressed by

d(z₁, w1)= 1 2log

1+₁^z₋^1−w_z1w¹₁

1−₁^z₋^1−w_z1w¹₁ =tanh⁻¹

z₁−w1

1−z₁w1

.

The Kobayashi pseudometric between two pointsx,yin X is defined as follows.

Consider all finite sequences of pointsp₀=x,p₁, . . . ,p_k−1,p_k = yofXsuch that

there exist points z₁, . . . ,z_k, w1, . . . , wk ofand mapsϕ1, . . . , ϕk ∈

O

(,X) satisfyingϕj(z_j)= p_j−1andϕj(wj)= p_j, j =1, . . . ,k.

The Kobayashi pseudometricKX(x,y)is, by definition,

K_X(x,y)=inf k

j=1

d(z_j, wj)

where the infimum is taken over all possible choices of points and maps.

Thespherical arc length element dson the Riemann sphere

C

is given by ds(z,d z)= |d z|

1+ |z|². The spherical length

s(γ )=

γ ds(z,d z) of a curveγ in

C

induces a metric in the following manner.

Given distinct pointsa,bon the Riemann sphere, define s(a,b)=inf{s(γ )}

where the infimum is taken over all piecewiseC¹curves on

C

^{which joinawithb.} Thens(a,b)defines a metric on the sphere known as the spherical metric.

Denote by

O

(X)the set of all holomorphic functions on X. A family F ⊆

O

⁽⁾ is said to be normal in if every sequence{f_n} ⊂ F has a subsequence which converges uniformly (with respect to the Euclidean metric) on compacta in or diverges uniformly to∞on compacta in.

Definition 2.1. A function f in

O

^(X) is called a normal function if the family

F

^{= {f ◦ϕ : ϕ∈}

O

^{(,X)}is normal.}

A familyF⊆

O

()is said to be spherically equicontinuous at a pointz₀ ∈ if for each positive numberthere is a positive numberδsuch thats(f(z), f(z₀)) <

forK(z,z₀) < δand f inF.

Following Gauthier [7] we shall define a sequence {x_n}of points in X to be a P-sequence of f ∈

O

(X)if there is a sequence{y_n}of points in X such that K_X(x_n,y_n) → 0 asn → ∞ buts(f(x_n), f(y_n)) ≥ for some > 0 for each positive integern.

The notion of P-sequence was originally defined by Gavrilov [6] for mero- morphic functions of the unit disk.The present version of P-sequence was in- troduced by Gauthier [7] in which he proved that the two notions are equivalent for meromorphic functions.

The following lemmas are key tools of this paper.

Lemma 2.2 (Zalcman’s Lemma [15]). Let

F

be a family of analytic functions in .Then

F

is not normal inif and only if there exist(i)a number r with0<r <

1;(ii)points zn satisfying|zn| < r;(iii)functions fn ∈

F

;(iv) positive numbers ρn →0as n→ ∞;such that

f_n(z_n+ρnξ)→g(ξ) as n→ ∞, (2.1) uniformly on compact subsets of

C

^,where g is a nonconstant entire function in

C

^. The function g may be taken to satisfy the normalization g (z)<g (0)=1 (z∈

C

).

Hereg (z)denotes the spherical derivative g (z)= |g(z)|

1+ |g(z)|².

Lemma 2.3. Let f be a normal function on a complex Banach manifold X and suppose k_X is a metric. There exists a constant c>1such that

log(cµ(f,x))≤log(cµ(f,y))exp^2KX(x,y) for all x,y∈X. Hereµ(f,x):=max{1,|f(x)|}.

This result was originally proved by Zaidenberg [17] in the case of complex manifolds. The proof of this result given in [9, page 39] extends immediately to the complex Banach manifold case.

3.

Normality and

P

-sequences

In this section we obtain certain relations existing between notions of normal func- tions, and P-sequences.

Theorem 3.1. Let X be a complex Banach manifold and suppose k_X is a metric.

The following statements are equivalent for f ∈

O

^(X): (a) f is normal;

(b) there exists a constant Q >0such that Q_f(x):= sup

v∈T_x(X)\{0}

ds(f(x), f_∗(x)v)

k_X(x, v) <Q for all x ∈X; (3.1) (c) there exists a constant L>0such that

s(f(x), f(y))≤L·K_X(x,y) for all x,y∈ X; (3.2) (d) f has no P-sequence.

Proof. (a)⇒(b): Assume that the family

F

= {f ◦ϕ:ϕ∈

O

(,X)}is a normal family. By Marty’s theorem [15, page 75], there exists a constantL>0 such that

|(f ◦ϕ)(0)|

1+ |f ◦ϕ(0)|² <L for allϕ∈

O

(,X). (3.3) By the definition ofk_X there existsψ ∈

O

(,X)such thatψ(0)=x, ψ_∗(0)a=v fora>0 anda/2<k_X(x, v)≤a.Therefore, from (3.3),

ds(f(x), f_∗(x)v) <2L·k_X(x, v) for all(x, v)∈T(X).

Namely,Q_f ≤2L.

(b)⇒(c): Letxandybe distinct points of X.It follows readily from (3.1) that ds(f(x), f_∗(x)v) <Q·k_X(x, v).

By integrating both side of the above inequality along the piecewise C¹ curves joiningxto yin Xand by the definitions we have

s(f(x), f(y))≤L·K_X(x,y).

SinceK_X is the integrated form of the infinitesimal metrick_X (see [4, Corollary 3]) we have (3.2).

(c)⇒(d): This follows immediately.

(d)⇒(a): If(d)holds, then the family

F

= {f ◦ϕ : ϕ ∈

O

(,X)}is equicon- tinuous at each point of,since otherwise there is a pointz₀∈,some >0,a sequence{z_n}of points inwithz_n→z₀,and a sequence{f◦ϕn} ⊆

F

^satisfying s(f ◦ϕn(z_n), f ◦ϕn(z₀))≥, n=1,2, . . . . (3.4) Since K = d, andz_n → z₀ then an application in [3, Proposition 3.2] gives K(z_n,z₀)→0,asz_n→z₀.By the contracting property of the Kobayashi metric, K_X(ϕ(z_n), ϕ(z₀))≤K(z_n,z₀)→0, asz_n →z₀. (3.5) From (3.4), (3.5) follows that sequence{ϕn(z₀)}of points in Xis a P-sequence for

f which contradicts(d).

Therefore,

F

^{= {f ◦ϕ: ϕ∈}

O

^(,X)}is spherically equicontinuous family at each point of,and hence, by Montel’s theorem [15, page 74],

F

^{is normal.}

This proves(a).

Remark 3.2. Hahn [8] has also published a similar characterization of a normal function on a complex hyperbolic manifold Mof finite dimension. Unfortunately, Hahn’s argument is incorrect, because the assumption that

F

= {f ◦ϕ : ϕ ∈

O

(,M)}is not a equicontinuous family does not entail that “there exist an >0 such that for all n ∈N there exist sequences{z_n}and{wn}inwith K(z_n, wn) <

1/n but s(f(ψ(z_n)), f(ψ(wn)))≥forsomeψ∈

O

(,M)” (see [8, page 60]).

Theorem 3.3. Let X and k_X be given as in Theorem 3.1and let f in

O

(X). If Q_f(x_m) → ∞as m → ∞, then {x_m} contains a subsequence which is a P- sequence of f.

Proof. Since Q_f(x_m) → ∞as m → ∞, then by (3.1) there exist a sequences {vm}, vm ∈T_xm(X),and{r_m} ⊂ R,r_m→ ∞asm→ ∞,with

ds(f(x_m), f_∗(x_m)vm) >r_m·k_X(x_m, vm). (3.6) By the definition of k_X there exists ψm ∈

O

^{(,X) such that ψm(0) = xm,} ψm∗(0)a_m = vm fora_m > 0 anda_m/2 < k_X(x_m, vm) ≤ a_m. Hence, inequality (3.6) gives,

|(f ◦ψm)(0)|

1+ |f ◦ψm(0)|² >r_m· k_X(x_m, vm) a_m > r_m

2 → ∞asm→ ∞.

By Marty’s theorem [15, page 75], the family{f◦ψm} ⊆

O

⁽⁾is not normal in any disc¹

n = {z ∈

C

^{: |z| < 1}_n^}.Then by a local adaptation of the Zalcman Lemma [15, page 152], there is a subsequence{f ◦ϕn}of{f ◦ψm}, z_n → 0, ρn → 0⁺ and a nonconstant entire functiongwith f ◦ϕn(z_n+ρnξ)→g(ξ)normally in

C

^. Passing to a subsequence if necessary, we can assume that {f ◦ϕn(0)}converges to a some pointβ ∈

C

^{.Setgn(ξ)= f ◦ϕn(zn+ρnξ).}The sequence of functions {g_n}converges locally uniformly tog.Letαbe any complex number,α = β,for which the equation g(ξ) = α has a solutionξ0 which is not a multiple solution, that is, g (ξ0) = 0.By a theorem of Hurwitz [15, page 9], in each neighborhood ofξ0 all but a finite number of the functions{g_n}assume the valueα.Thus there exists a sequence of points {ξn} ⊆

C

^{such thatξn → ξ0 andgn(ξn) = α for n} sufficiently large. It follows s(f ◦ϕn(z_n +ρnξn), f ◦ϕn(0)) ≥ s(α, β)/2 > 0 for n ≥ N₀. The Poincar´e metricd is very closed to the Euclidian metric near 0 so d(0,z_n + ρnξn) → 0 as n → ∞. Hence K_X(ϕn(z_n +ρnξn), ϕn(0)) ≤ d(z_n+ρnξn,0)→0 asn → ∞.Which proves that a subsequence{ϕn(0)}^∞_n=N0 of{x_m}is aP-sequence of f.The proof is complete.

Remark 3.4. The converse to Theorem 3.3 is not true in general, as the follow- ing example [2, Example (4.3)] shows. Let X = , f(z) = exp₁₋ⁱ_z ∈

O

^(), z_n = ₁₊ⁿ²_n2 − _n+ⁱ_n3, wn = ₁₊ⁿ²_n2. Here Q_f(wn) → ∞, and hence by Theorem 3.3 sequence{wn}contains a subsequence{wm}which is a P-sequence of f.Since K(z_m, wm)→ 0 a subsequence{z_m} ⊆ {z_n}is a P-sequence of f too, while the sequenceQ_f(z_m)→0.

4.

Criteria for non-normality

The Zalcman Lemma [18] characterizing normal families of holomorphic functions on plane domains is the main tool used in proofs of theorems in this and next chap- ter.

A relationship between non-normal holomorphic functions and the existence of a P-sequence is established in the following theorem.

Theorem 4.1. Let X and k_X be given as in Theorem3.1. A function f ∈

O

(X)is not a normal function on X iff f has a P-sequence.

Proof. If f is not normal then the family

F

= {f◦ϕ:ϕ∈

O

(,X)}is not normal in .By Zalcman’s Lemma, we can find f ◦ϕn ∈

H

,z_n ⊆ r, z_n → z₀,and ρn →0⁺such that f ◦ϕn(z_n+ρnξ)→g(ξ)uniformly on compacta, wheregis a nonconstant entire function on

C

.Sincegis nonconstant in

C

^{there is}ζ ∈

C

^such thatg(0)=g(ζ).One see at once that

nlim→∞s(f ◦ϕn(z_n), f ◦ϕn(z_n+ρnζ ))=s(g(0),g(ζ ))≥for some >0, and K_X(ϕn(z_n), ϕn(z_n +ρnζ )) ≤ d(z_n,z_n +ρnζ ) → 0 asn → ∞. Hence a sequence{ϕn(z_n)}of points in Xis a P-sequence of f.

Conversely, if f have a P-sequence {x_n}, then there is a sequence {y_n} of points in Xsuch that lim_n→∞K_X(x_n,y_n)=0 but

s(f(x_n), f(y_n))≥for some >0,n=1,2, . . . . (4.1) Suppose that f is a normal function. By (3.2)

s(f(x_n), f(y_n))≤L·K_X(x_n,y_n) for alln≥1.

It followss(f(x_n), f(y_n))→0 asn→ ∞which contradicts (4.1). Therefore f is not a normal function. This proves the theorem.

A sufficient condition for the non-normality of a holomorphic function inis given by Lappan [10, Lemma 3]. For holomorphic functions on a complex Banach manifold we have the following theorem.

Theorem 4.2. Let X and k_X be given as in Theorem3.1. The following statements are equivalent for f ∈

O

(X):

(a) f is not a normal function;

(b) There exist sequences {y_m}, {x_m} in X, and a constant M > 0 such that K_X(y_m,x_m)<M for all m≥1,lim_m→∞ f(x_m)=∞, and lim_m→∞ f(y_m)= a∈

C

^.

Proof. (a)⇒ (b): Sinceh is not normal it follows that the family

F

= {f ◦ϕ : ϕ∈

O

(,X)}is not normal in.Apply the Zalcman Lemma to

F

^withr,z_n, ρn, f◦ϕn,andgas given therein. Sincegis nonconstant entire function in

C

^{it follows} that there exist a sequence{ξm} ⊂

C

^{such that|g(ξm)|>m.}In fact, if this were not the case,gis bounded in

C

^.By Liouville’s theoremg≡constant, a contradictions

with the assumption that gis nonconstant entire function. For fixedξm chosen_m large enough so that

(i) |z_nm+ρn_mξm|< (1+r)/2; (ii) |f ◦ϕn_m(z_nm+ρn_mξm)|>m/2. Putwn_m =z_nm+ρn_mξm,x_m =ϕn_m(wn_m),andy_m=ϕn_m(z_nm).

By the contracting property of the Kobayashi metric,

K_X(y_m,x_m)=K_X(ϕn_m(wn_m), ϕn_m(z_nm))≤d(wn_m,z_nm).

By the triangle inequalityd(wn_m,z_nm)≤d(0,z_nm)+d(0, wn_m).For anyz ∈ the Poincar´e distance

d(0,z)= 1

2log1+ |z| 1− |z|. Since|z_nm|<r,and|wn_m|< (1+r)/2,we have

d(wnm,z_nm)≤ 1 2

log((1+r)/(1−r))+log((3+r)/(1−r)) .

Putting the above together we getK_X(y_m,x_m)≤ Mwhere

2M =log((1+r)/(1−r))+log((3+r)/(1−r)) <∞.

By (ii), we have that

mlim→∞ f(x_m)= lim

m→∞ f ◦ϕn_m(z_nm +ρn_mξm)= ∞.

By (2.1), we have that

mlim→∞ f(y_m)= lim

m→∞ f ◦ϕnm(z_nm +ρnm0)=g(0)∈

C

^. (b)⇒(a): Assume, to get a contradiction, that f is normal. By Lemma 2.3,

log(cµ(f,x_m))≤log(cµ(f,y_m))exp^2KX(xm,ym) for allm≥1.

The left-hand side of the above inequality tends to infinity as m → ∞while the right-hand side tends to a number which is less than log(cmax{1,|a|})exp^2r and we have a contradiction. This contradiction proves the theorem.

5.

Further results

Theorem 5.1. Let X and k_X be given as in Theorem3.1. If f ∈

O

(X)is a nor- mal function and if h is a non-normal holomorphic function on X such that each sequence{x_m}of points in X contains a subsequence{x_mk}on which at most one of f or h is unbounded, then f +h is a non-normal function.

Proof. Sinceh is not normal on X it follows that the family

H

^{= {h◦ϕ : ϕ ∈}

O

^(,X)}is not normal in.By the Zalcman Lemma, we can findh◦ϕn ∈

H

^, z_n ⊆r,andρn→0⁺such thath◦ϕn(z_n+ρnξ)→g(ξ)uniformly on compacta, wheregis a nonconstant entire function on

C

^{.But then}

ρn|h_∗(ϕn(z_n+ρnξ))ϕ_n(z_n+ρnξ)|

1+ |h(ϕn(z_n+ρnξ))|² →g (ξ).

We claim that there exist a sequence{ξm}⊂

C

^{such that}|g(ξm)|>mandg(ξm)=0. Let us suppose not.

1. If g(ξ) = 0 and g is bounded in

C

by Liouville’s theorem g ≡ constant, a contradictions with the assumption thatgis nonconstant entire function.

2. Let{a_m}be the zeroes ofg(ξ).Suppose that|g(ξ)| < L < ∞on

C

^{\ {ξ ∈}

C

^:g(ξ)=0}.The zeros ofg(ξ)are isolated. Since nonconstant holomorphic function have no local maxima, we conclude that|g(a_m)| < L.Thereforegis bounded in

C

^,hence constant by Liouville’s theorem. But this contradicts with the assumption thatgis nonconstant entire function, and the assertion is proved.

For fixedξmchosen_mlarge enough so that (i) |λn_m|< (1+r)/2; (ii) g (ξm)/ρn_m >m/2;

(iii) ρnm|h_∗(x_m)vm|/(1+ |h(x_m)|²) >g (ξm)/2; (iv) |h(x_m)|>m/2.

Hereλn_m =z_nm+ρn_mξm,x_m =ϕn_m(λn_m),andvm=ϕ_nm(λn_m).If (i), (ii), and (iii) hold, it is easy to see that

|h_∗(x_m)vm|

1+ |h(x_m)|² > g (ξm)

2ρn_m > (4−(1+r)²)m

16 · 1

1− |λn_m|². By the distance decreasing property of the Kobayashi metric

1

1− |λnm|² =k(λn_m,1)≥k_X(ϕn_m(λn_m), ϕ_nm(λn_m)·1).

Hence

|h_∗(x_m)vm|

1+ |h(x_m)|² > (4−(1+r)²)m

16 ·k_X(x_m, vm).

Since|h(x_m)| → ∞,asm → ∞,by considering a subsequence if necessary, we may assume from the hypothesis of the theorem that f is bounded on{x_m},namely

|f(x_m)|< M <∞for allm ≥1,and hence, formsufficiently large, we have¹

|(f +h)_∗(x_m)vm|

1+ |f(x_m)+h(x_m)|² > |h_∗(x_m)vm| − |f_∗(x_m)vm| 2(1+ |f(x_m)|²)(1+ |h(x_m)|²) ≥ 1

2 1

1+ |M|² · |h_∗(x_m)vm|

1+ |h(x_m)|² − |f_∗(x_m)vm| 1+ |f(x_m)|²

.

By hypothesis, f is a normal function on X hence, by Theorem 3.1, there exists Q >0 such thatQ_f(x_m) < Qfor allm≥1.Thus

Q_f+h(x_m)≥ 1 2

(4−(1+r)²)m 16(1+ |M|²) −Q

→ ∞asm→ ∞.

By Theorem 3.1, f+his not a normal function onX. Thus the proof is complete.

Lehto and Virtanen [12, page 53] remark that the sum of a normal function and a bounded function (which is necessary normal) is a normal function. The sum of two normal holomorphic functions, each omitting the values 0 and 1,need not be normal (see [10, page 191]).

In the infinite dimensional case we have the following theorem.

Theorem 5.2. If under the conditions of Theorem3.1, f₁, . . . , f_l are a finite num- ber of normal holomorphic function on X such that each sequence{x_n}of points in X contains a subsequence {x_nm} on which at most one of f_j (1 ≤ j ≤ l)is unbounded, then h :=^l_j=1f_j is a normal function.

Proof. Suppose, on the contrary, that h is not a normal function. By Theorem 4.2, we can find two sequences {x_n} and {y_n} of points in X, and a positive constant Msuch thatKX(xm,ym) < Mfor allm ≥1,limm→∞h(xm)= ∞, and lim_m→∞h(y_m)=a∈

C

^{.Since limm}→∞h(x_m)= ∞then{x_m}contains a subse- quence again denoted by{x_m}such that at most one of f_j,say f₁, is unbounded

1IfaandbinCthen

1+ |a+b|²≤1+(|a| + |b|)²<2(1+ |a|²)(1+ |b|²).

on {x_m}.Since lim_m→∞h(y_m)= a ∈

C

^then{ym}contains a subsequence{y_mk} such that either:

(i) at least two of f_j (1≤ j ≤l)is unbounded on{y_mk}; (ii) or lim_k→∞ f_j(y_mk)=αj ∈

C

^{(1≤ j ≤l).}

The case (i) is excluded by the assumption of the corollary.

Hence lim_k→∞ f₁(x_mk)= ∞,lim_k→∞ f₁(y_mk)= α1 ∈

C

^,andKX(xmk,y_mk) <

Mfor allk≥1.By Theorem 4.2 f₁is not a normal function, a contradiction which proves the corollary.

Theorem 5.3. Under the assumption of Theorem 3.1let{x_n}and{y_n}be two se- quences of points in X,and let M be a positive constant such that K_X(x_m,y_m) < M for all m ≥ 1. If f ∈

O

(X) is a normal function which omits l ∈

C

^{in X but} lim_m→∞s(f(x_m),l)=0thenlim_m→∞s(f(y_m),l)=0.

Proof. Assume first thatl ∈

C

^.Since 1+ |f(x)−l|²

1+ |f(x)|² <2(1+ |l|²)

(see footnote 1) it follows from the hypothesis of the theorem and Theorem 3.1 that g(x)=1/(f(x)−l)is a normal holomorphic function on X.It is easy to see that lim_m→∞g(x_m)= ∞.Therefore, we have lim_m→∞g(y_m) = ∞by Theorem 4.2.

Hence lim_m→∞s(f(y_m),l)=0 as desired.

Ifl = ∞the corollary is an immediate consequence of Theorem 4.2. Thus the proof is complete.

6.

Boundary behavior of normal functions in a fixed boundary point

In [1] F. Bagemihl and W. Seidel posed the following question: Given a sequence {z_j} ⊂ converging to someς ∈∂and a holomorphic mapping f ∈

O

(,

C

⁾ such thatlim_j→∞s(f(z_j),l)=0for some l∈

C

^,under what conditions on f and {z_j}can f have the limit l along some continuum inwhich is asymptotic atς?

In this section, we give the infinite dimensional analogue of their result. First we introduce some definitions and prove two lemmas.

In the rest of the paper let Dbe a bounded and convex domain in a complex Banach space(V, · ),andK_Dbe the Kobayashi metric forD.

A domain D⊆V is calledconvex (in the real sense)if(1−t)x+t y⊂ Dfor allx,y∈Dand 0≤t ≤1.

Lemma 6.1. Let D be a bounded and convex (in the real sense) domain in a complex Banach space (V, · ) and let ξ ∈ ∂D. Then for all a ∈ D the set l_ξ(a)= {y=ξ+t(a−ξ),0<t <1}is contained in D.

Proof. We haveB(a,r) :=a+ B(0,r)⊂ D,for somer > 0,where B(0,r) :=

{y∈V : y<r}.PutU = [t/(1−t)]B(0,r)+ξ,t =1.There is someb∈ D∩U. We haveb = [t/(1−t)]u+ξ where u ∈ B(0,r).Sincea−u ∈ B(a,r) ⊂ D and Dis convex y = (1−t)b+t(a−u)= ξ+t(a−ξ)∈ D.This proves that l_ξ(a)⊂ D.

Definition 6.2. The approach region of apertureα,with vertexξ ∈∂D,and pole a∈ D is the set

A_α(ξ,a)= {x ∈ D: K_D(x,l_ξ(a)) < α}

where K_D(x,l_ξ(a))=inf{K_D(x,y),y ∈l_ξ(a)}.

Lemma 6.3. Let D be a bounded and convex (in the real sense) domain in a com- plex Banach space V and letξ ∈∂D.If b∈ D,b=a,then there exists a positive constant C such that

A_α(ξ,a)⊂ A_α+C(ξ,b)⊂ A_α+2C(ξ,a).

Proof. To prove the lemma it suffices to show that if z ∈ A_α(ξ,a) then z ∈ A_α+C(ξ,b).So letz ∈ A_α(ξ,a).From the definition of A_α(ξ,a)follows that for everyε >0 there is a pointa₁inl_ξ(a)such thatK_D(z,a₁) < α+ε.Sincea₁∈l_ξ(a) there exists aτ0∈(0,1]such thata₁=ξ+τ0(a−ξ).Setb₁=ξ+τ0(b−ξ)∈l_ξ(b).

Letϕbe a mapping in

O

(,D)whose range contains bothaandb.²Suppose thatϕ(0)=aandϕ(ζ )=bfor a suitableζ ∈.Defineψ :→V by

ψ(z)=ξ+τ0(ϕ(z)−ξ).

By Lemma 6.1ψmapsintoD;hence

K_D(a₁,b₁)=K_D(ψ(0), ψ(ζ ))≤d(0, ζ ).

By the triangle inequality of the Kobayashi distance

K_D(z,b₁)≤K_D(z,a₁)+K_D(a₁,b₁).

It is clear, that

K_D(z,l_ξ(b))≤ K_D(z,b₁)≤α+ε+d(0, ζ ).

2 If D ⊂ Cⁿ then there exists a continuous pathφ : [0,1] → Dsuch thatφ(0) = aand φ(1)= b.An application of the vector-valued Stone-Weierstrass theorem and an end-point ad- justment shows that we can takeφto be a polynomial. By compactness there exists a convex open neighborhoodOof[0,1]inCsuch thatφ(O)⊂D,whereφis the polynomial onCobtained by replacing the real variablexby the complex variablez.An application of the Riemann mapping theorem yields the existence of a mappingϕ∈O(,D)whose range contains bothaandb.

SinceDis a domain in a Banach spaceV then any pair of pointsaandbinDcan be joined by a polygonal curve.IfY is the finite dimensional subspace ofV spanned by the points ofthen the connected component of∩D,containinga,is a domain in a finite dimensional space which contains bothaandb.The finite dimensional result can now be applied to show the existence of a mappingφ∈O(,D)whose range contains bothaandb(see [3, page 49]).

Sinceεwas arbitrary this gives that

K_D(z,l_ξ(b))≤α+d(0, ζ ).

It follows

z∈ A_α+C(ξ,b) whereC =d(0, ζ) <∞.

Remark 6.4. If D is the unit ball in

C

ⁿ then the approach regions A_α(ξ,a)are comparable to the admissible approach regions of Koranyi.

Definition 6.5. A function f ∈

O

(D)has anadmissible limit l ∈

C

^atξ ∈∂D,if for everyα >0 and every sequence of points{x_n} ⊂ A_α(ξ,a)which tends toξ

nlim→∞s(f(x_n),l)=0.

Remark 6.6. From Lemma 6.3 follows that the definition of admissible limit is independent of the choice of the polea∈D.

Theorem 6.7. Let D be a bounded and convex (in the real sense) domain in a complex Banach space(V, · ).Let{x_n}be a sequence of points in l_ξ(a)which tends toξ ∈∂D,such that there exists a constantε >0such that K_D(x_n,x_n+1) <

εfor all n ≥1.Suppose that the function f ∈

O

(D)is normal on D,omits l ∈

C

in D but

nlim→∞s(f(x_n),l)=0. Then f has an admissible limit l atξ.

Proof. First of all, we show that for an arbitrary sequence of points{q_n}inl_ξ(a) which tends toξ asn → ∞we have lim_n→∞s(f(q_n),l)= 0.Indeed, for every q_n there exists an integer j_n and a constantt_n,t_n ∈ [0,1],such thatq_n = x_jn + t_n(x_jn+1−x_jn).Fort ∈ [0,1]define g_t ∈

O

((D ×D)×(D× D),D× D)by g_t((x,y), (w,z))=(t x+(1−t)y,tw+(1−t)z).Then by [5, Proposition IV.1.2, page 83] for allx,y, w,z∈Dwe have

K_D×D((x,y), (w,z))≥K_D(t x+(1−t)y,tw+(1−t)z) . (6.1) By [5, Proposition V.4.2, page 136]

K_D×D((x,y), (w,z))=max{K_D(x,y),K_D(w,z)}.

Combining this with (6.1), we see that

K_D(t x+(1−t)y,tw+(1−t)z)≤max[K_D(x, w),K_D(y,z)]

for all 0≤t ≤1 and every choice ofx,y, w,zfromD.Then we have K_D(x_jn,q_n)=K_D(x_jn,x_jn+t_n(x_jn+1−x_jn)≤K_D(x_jn,x_jn+1) < ε.

As lim_n→∞s(f(x_jn),l)=0 andK_D(x_jn,q_n) < εfor alln≥1,we get

nlim→∞s(f(q_n),l)=0 by Corollary 5.3.

Let{y_n}be any sequence in A_α(ξ,a)which converges to the pointξ ∈ ∂D. For every pointy_nthere exists a pointb_n ∈l_ξ(a)such that 2α >K_D(y_n,b_n)for all n ≥1.Everyb_ncan be write in the formb_n = ξ+τn(a−ξ)whereτn ∈ [0,1].

Hence,{b_n}has a convergent subsequence{b_nj}.Letb=ξ+τ0(a−ξ)be a limit point of{b_nj}.

Assumingτ0=0 we shall derive a contradiction. Choose a positive numberr such that B_r(b)= {y∈V : y−b<r} ⊂ D.Since sequence{b_nj}tends tobas j → ∞there exits an integer j₀such thatb_nj−b<r/2 for all j ≥ j₀.By the inequality (4.3) in [3, page 52]

K_D(b_nj,b) <tanh⁻¹ 1

2

for all j ≥ j₀. Then for j ≥ j₀we have

K_D(y_nj,b) < K_D(y_nj,b_nj)+K_D(b_nj,b) <2α+tanh⁻¹ 1

2

.

This contradicts our assumption that sequence{y_n}tends toξ ∈ ∂D,since every ball in (D,K_D)is bounded away from the boundary (see [3, page 88]). Hence τ0 = 0 and sequence {b_n} ⊂ l_ξ(a) tends to ξ as n → ∞. As proved above lim_n→∞s(f(b_n),l)=0 and sinceK_D(y_n,b_n) < αfor alln≥1

nlim→∞s(f(y_n),l)=0 by Corollary 5.3. Hence f has the admissible limitlatξ.

References

[1] F. BAGEMIHLand W. SEIDEL,Sequential and continuous limits of meromorphic functions, Ann. Acad. Sci. Fenn. Math.280(1960).

[2] D. M. CAMPBELLand G. WICKES,Characterizations of normal meromorphic functions, In: “Complex Analysis”, Laine, J.et al.(eds.), Joensuu 1978, Lect. Notes Math., Vol. 747, Springer, Berlin-Heidelberg-New York, 1979, 55–72.

[3] S. DINEEN, “The Schwarz Lemma”, Oxford Mathematical Monographs, Oxford, 1989.

[4] C. J. EARLE, L. A. HARRIS, J. H. HUBBARDand S. MITRA,Schwarz’s lemma and the Kobayashi and Carath´eodory metrics on complex Banach manifolds, In: “Kleinian Groups and Hyperbolic 3-Manifolds”, Cambridge Univ. Press, Cambridge, Lond. Math. Soc. Lec.

Notes, Vol. 299, 2003, 363–384. http://www.ms.uky.edu/ larry/paper.dir/minsky.ps.

[5] T. FRANZONIand E. VESENTINI, “Holomorphic Maps and Invariant Distances”, North- Holland Mathematical Studies 40, North-Holland Publishing, Amsterdam, 1980.