aa r X i v : . [ m a t h . C V ] A ug AN IMPROVEMENT OF ZALCMAN’S LEMMA IN C n P.V.DOVBUSH
Abstract.
The aim of this paper is to give a proof of improving of Zalcman’slemma. Introduction and main result
A family F of holomorphic functions on a domain Ω ⊂ C n is normal in Ω if everysequence of functions { f j } ⊆ F contains either a subsequence which converges toa limit function f = ∞ uniformly on each compact subset of Ω , or a subsequencewhich converges uniformly to ∞ on each compact subset. A family F is said to benormal at a point z ∈ Ω if it is normal in some neighborhood of z . A family ofanalytic functions F is normal in a domain Ω if and only if F is normal at eachpoint of Ω . For every function ϕ of class C (Ω) we define at each point z ∈ Ω anhermitian form L z ( ϕ, v ) := n X k,l =1 ∂ ϕ∂z k ∂z l ( z ) v k v l , and call it the Levi form of the function ϕ at z. For a holomorphic function f in Ω , set(1.1) f ♯ ( z ) := sup | v | =1 p L z (log(1 + | f | ) , v )This quantity is well defined since the Levi form L z (log(1 + | f | ) , v ) is nonnegativefor all z ∈ Ω . Theorem 1.1. (Marty’s Criterion, see [2] ) A family F of functions holomorphicon Ω is normal on Ω ⊂ C n if and only if for each compact subset K ⊂ Ω thereexists a constant M ( K ) such that at each point z ∈ K (1.2) f ♯ ( z ) ≤ M ( K ) for all f ∈ F . Marty’s criterion is one of the more important results in function theory widelyused for determining the normality of a family of holomorphic functions. Marty’scriterion is one of the main ingredients of the proof of Zalcman’s lemma [2]. Weprove the following improved result of Zalcman-Pang’s [3].
Theorem 1.2.
Let F be a family of functions holomorphic on Ω ⊂ C n . Then F is not normal at some point z ∈ Ω if and only if for each α ∈ ( − , ∞ ) there existsequences f j ∈ F , z j → z , r j → , such that the sequence g j ( z ) := r αj f j ( z j + r j z ) Mathematics Subject Classification.
Key words and phrases.
Zalcman’s Lemma; Zalcman-Pang’s Lemma; Normal families; Holo-morphic functions of several complex variables. converges locally uniformly in C n to a non-constant entire function g satisfying g ♯ ( z ) ≤ g ♯ (0) = 1 . In case n = 1 this theorem was proved in Hua [6, Lemma 6]. A similar resultwas proved by Chen and Gu [1, Th.2] (see also Xue and Pung [8], cf. Hua [6]). Thespecial case α = 0 of Theorem 1.2 was proved in Zalcman [9, p. 814] and is knownas Zalcman’s rescaling lemma. Zalcman’s lemma - now upgraded to the status oftheorem - was first stated at [9]; for a state-of-the-art version, see [10, Lemma 2].The plan of this paper is as follows. In Section 2, we state and prove a numberof auxiliary results, some of which are of independent interest. In Section 3, we givethe proof of main theorem. In Section 4, we give two applications of main theorem.2. Auxiliary results
In this section, we state some known results and prove a lemma that is requiredin the proofs of our results.
Theorem 2.1. (Hurwitz’s theorem, see, e.g. [7, (1.5.16) Lemma, p. 24] ) Let Ω bea domain of C n and { h j } a sequence of non-vanishing holomorphic functions h j which converges uniformly on compact subsets to a holomorphic function h on Ω . Then h vanishes either everywhere or nowhere. Note that L z (log(1 + | f ( z ) | ) , v ) = | ( Df ( z ) , v ) | (1 + | f ( z ) | ) on Ω . Appealing to the Cauchy-Schwarz inequality it is easy to show that(1 + | f ( z ) | ) f ♯ ( z ) = | Df ( z ) | . The following lemma will play a crucial role in the proof of Theorem 1.2.
Lemma 2.2.
Let f be a holomorphic function on the closed unit ball B (0 , , and α be a real number with − < α < ∞ . Suppose max | z |≤ /j (1 − j | z | ) α (1 + | f ( z ) | ) f ♯ ( z )1 + (1 − j | z | ) α | f ( z ) | > . Then there exists a point ξ ∗ , | ξ ∗ | < /j, and a real number ρ, < ρ < , such that max | z |≤ /j (1 − j | z | ) α ρ α (1 + | f ( z ) | ) f ♯ ( z )1 + (1 − j | z | ) α ρ α | f ( z ) | =(1 − j | ξ ∗ | ) α ρ α (1 + | f ( ξ ∗ ) | ) f ♯ ( ξ ∗ )1 + (1 − j | ξ ∗ | ) α ρ α | f ( ξ ∗ ) | = 1 . Proof.
Set(2.1) ϕ ( t, z ) := (1 − j | z | ) α ρ α (1 + | f ( z ) | ) f ♯ ( z )1 + (1 − j | z | ) α ρ α | f ( z ) | . Suppose that ϕ (1 , z ∗ ) := max | z |≤ /j ϕ (1 , z ) > . Since (1 + | f ( z ) | ) f ♯ ( z ) isbounded on | z | ≤ /j we have(2.2) ϕ ( t, z ) ≤ (1 − j | z | ) − α t − α M. It follows ϕ ( t, z ) is continuous on [0 , × { z ∈ C n : | z | ≤ /j } and ϕ (0 , z ) = 0 on { z ∈ C n : | z | ≤ /j } . N IMPROVEMENT OF ZALCMAN’S LEMMA IN C n Hence ϕ (0 , z ∗ ) = 0 and ϕ (1 , z ∗ ) > . By continuity of ϕ ( t, z ) on [0 , × { z ∈ C n : | z | ≤ /j } , there exists ρ , < ρ < , such that ϕ ( ρ , z ∗ ) = 1 . Repeating this procedure we can find ρ m , < ρ m < , and z ∗ m , | z ∗ m | < /j, suchthat(2.3) max | z |≤ /j ϕ ( ρ . . . ρ m , z ) = ϕ ( ρ . . . ρ m , z ∗ m ) > . ϕ ( ρ . . . ρ m ρ m +1 , z ∗ m ) = 1 . The sequence { x m := ρ . . . ρ m } is a bounded and decreasing sequence. Thenthe greatest lower bound of the set { x m : m ∈ N } , say ρ, is the limit of { x m } . The sequence { z ∗ m } contains a subsequence, again denoted by { z ∗ m } , such thatlim m →∞ z ∗ m = ξ ∗ . From (2.2) follows that 0 < ρ < | ξ ∗ | < /j. We claim that(2.4) max | z |≤ /j lim m →∞ ϕ ( ρ . . . ρ m , z ) = lim m →∞ max | z |≤ /j ϕ ( ρ . . . ρ m , z ) . Since ϕ is continuous function on [0 , × B (0 , /j ) by the Weierstrass theorem (see[4, Theorem (Weierstrass) p. 565]) we can find | η | < /j and | w m | < /j such that(2.5) max | z |≤ /j lim m →∞ ϕ ( ρ . . . ρ m , z ) = max | z |≤ /j ϕ ( ρ, z ) = ϕ ( ρ, η );(2.6) ϕ ( ρ . . . ρ m , η ) ≤ max | z |≤ /j ϕ ( ρ . . . ρ m , z ) = ϕ ( ρ . . . ρ m , w m ) , m = 1 , , . . . . By the Bolzano-Weierstrass theorem there is an infinite subsequence of { w m } , againdenoted by { w m } , and ς, | ς | ≤ /j, such that w m → ς as m → ∞ . Because w m → ς and ρ . . . ρ m → ρ as m → ∞ and ϕ is continuous function on [0 , × B (0 , /j )from (2.5) and (2.6) we see ϕ ( ρ, η ) ≤ lim m →∞ max | z |≤ /j ϕ ( ρ . . . ρ m , z ) = ϕ ( ρ, ς ) ≤ max | z |≤ /j ϕ ( ρ, z ) = max | z |≤ /j lim m →∞ ϕ ( ρ . . . ρ m , z ) = ϕ ( ρ, η ) . That is, the claim (2.4) is proved. Combining (2.4) and (2.5) we obtainmax | z |≤ /j ϕ ( ρ, z ) = ϕ ( ρ, ξ ∗ ) = 1 ( | ξ ∗ | < /j ) . The proof of the lemma is complete. (cid:3) Proof of main theorem
Proof of Theorem 1.2. ” ⇒ ” To simplify matters we assume that z = 0 andall functions under consideration are holomorphic on the closed unit ball B (0 , . By Marty’s criterion (Theorem 1.1) F contains functions f j , j ∈ N, satisfyingmax | z | < / (2 j ) f ♯j ( z ) > | α | j | α | ) . Since 1 − j | z | > / | z | < / (2 j ) there existsa ξ j with | ξ j | < /j such thatmax | z |≤ /j (1 − j | z | ) | α | f ♯j ( z ) = (1 − j | ξ j | ) | α | f ♯j ( ξ j ) ≥ max | z |≤ / j (1 − j | z | ) | α | f ♯j ( z ) ≥ j | α | ) . The power function t α , t > , is continuous, monotone (increasing when α > , decreasing when α < − j | z | ) α (1 + | f ( z ) | ) ≥ − j | z | ) α | f ( z ) | ( − < α ≤ P.V.DOVBUSH and 1 + (1 − j | z | ) α | f ( z ) | ≤ [1 + | f ( z ) | ] (0 < α < ∞ arbitrary)we have(3.1) (1 − j | ξ j | ) α (1 + | f j ( ξ j ) | ) f ♯j ( ξ j )1 + (1 − j | ξ j | ) α | f j ( ξ j ) | > (1 − j | ξ j | ) | α | f ♯j ( ξ j ) > j | α | ) . Hence max | z |≤ /j (1 − j | z | ) α (1 + | f j ( z ) | ) f ♯j ( z )1 + (1 − j | z | ) α | f j ( z ) | > . According to Lemma 2.2, there exists ξ ∗ j , | ξ ∗ j | < /j, and ρ j , < ρ j < , such thatmax | z |≤ /j (1 − j | z | ) α ρ αj (1 + | f j ( z ) | ) f ♯j ( z )1 + (1 − j | z | ) α ρ αj | f j ( z ) | =(1 − j | ξ ∗ j | ) α ρ αj (1 + | f j ( ξ ∗ j ) | ) f ♯j ( ξ ∗ j )1 + (1 − j | ξ ∗ j | ) α ρ αj | f j ( ξ ∗ j ) | = 1 . Therefore inequality (3.1) shows that1 = (1 − j | ξ ∗ j | ) α ρ αj (1 + | f j ( ξ ∗ j ) | ) f ♯j ( ξ ∗ j )1 + (1 − j | ξ ∗ j | ) α ρ αj | f j ( ξ ∗ j ) | ≥ (1 − j | ξ j | ) α ρ αj (1 + | f j ( ξ j ) | ) f ♯j ( ξ j )1 + (1 − j | ξ j | ) α ρ αj | f j ( ξ j ) | ≥ (1 − j | ξ j | ) | α | ρ | α | j f ♯j ( ξ j ) ≥ ρ | α | j j | α | ) ( | ξ j | < /j ) . It follows(3.2) (cid:16) j (cid:17) ≥ ρ j → . Put r j = (1 − j | ξ ∗ j | ) ρ j → . Set h j ( z ) = r αj f j ( ξ ∗ j + r j z ) . We claim that appropriately chosen subsequences z k = ξ j k , ρ k = r j k , and g k = h j k will do. First of all, h j ( z ) is defined on | z | < jρ j , hence on | z | < j, since | ξ ∗ j + r j z | ≤ | ξ ∗ j | + r j | z | ≤ | ξ ∗ j | + r j − j | ξ ∗ j | jr j = 1 j . By the invariance of the Levi form under biholomorphic mappings, we have L z (log(1 + | h j | ) , v ) = L ξ ∗ j + r j z (log(1 + | h j | ) , r j v )and hence h ♯j ( z ) = r j h ♯j ( ξ ∗ j + r j z ) . Since r j = (1 − j | ξ ∗ j | ) ρ j a simple computations shows that h ♯j ( z ) = r j r αj (1 + | f j ( ξ ∗ j + r j z ) | ) f ♯j ( ξ ∗ j + r j z )1 + r αj | f j ( ξ ∗ j + r j z ) | =(1 − j | ξ ∗ j | ) α ρ αj (1 + | f j ( ξ ∗ j + r j z ) | ) f ♯j ( ξ ∗ j + r j z )1 + [(1 − j | ξ ∗ j | ) / (1 − j | ξ ∗ j + r j z | )] α (1 − j | ξ ∗ j + r j z | ) α ρ αj | f j ( ξ ∗ j + r j z ) | = N IMPROVEMENT OF ZALCMAN’S LEMMA IN C n (1 − j | ξ ∗ j | ) α (1 − j | ξ ∗ j + r j z | ) α · (1 − j | ξ ∗ j + r j z | ) α ρ αj (1 + | f j ( ξ ∗ j + r j z ) | ) f ♯j ( ξ ∗ j + r j z ) h (cid:16) − j | ξ ∗ j | − j | ξ ∗ j + r j z | (cid:17) α · (1 − j | ξ ∗ j + r j z | ) α ρ αj | f j ( ξ ∗ j + r j z ) | i . Bearing in mind Lemma 2.2 it is easy to see that h ♯j (0) = 1 . Since11 + 1 /j ≤ − j | ξ ∗ j | − j | ξ ∗ j + r j z | ≤ − /j we have 1 + (cid:16) − j | ξ ∗ j | − j | ξ ∗ j + r j z | (cid:17) α · (1 − j | ξ ∗ j + r j z | ) α ρ αj | f j ( ξ ∗ j + r j z ) | ≥ (cid:16) − /j (cid:17) α · h − j | ξ ∗ j + r j z | ) α ρ αj | f j ( ξ ∗ j + r j z ) | i ( − < α ≤ (cid:16) − j | ξ ∗ j | − j | ξ ∗ j + r j z | (cid:17) α · (1 − j | ξ ∗ j + r j z | ) α ρ αj | f j ( ξ ∗ j + r j z ) | ≥ (cid:16)
11 + 1 /j (cid:17) α · h − j | ξ ∗ j + r j z | ) α ρ αj | f j ( ξ ∗ j + r j z ) | i (0 < α < ∞ arbitrary) . From the above inequalities and Lemma 2.2 we infer that ) h ♯j ( z ) ≤ (cid:16) sgn ( α ) j (cid:17) α · (cid:16) − | ξ ∗ j | − j | ξ ∗ j + r j z | (cid:17) α · (1 − j | ξ ∗ j + r j z | ) α ρ αj (1 + | f j ( ξ ∗ j + r j z ) | ) f ♯j ( ξ ∗ j + r j z )1 + (1 − j | ξ ∗ j + r j z | ) α ρ αj | f j ( ξ ∗ j + r j z ) | = (cid:16) sgn ( α ) j (cid:17) α · (cid:16) − j | ξ ∗ j | − j | ξ ∗ j + r j z | (cid:17) α · ≤ (cid:16) sgn ( α ) j (cid:17) α · (cid:16) − /j (cid:17) α for all | z | < j. For every m ∈ N the sequence { h j } j>m is normal in B (0 , m ) byMarty’s criterion (Theorem 1.1). The well-known Cantor diagonal process yields asubsequence { g k = h j k } which converges uniformly on every ball B (0 , R ) . The limitfunction g satisfies g ♯ ( z ) ≤ lim sup j →∞ h ♯j ( z ) ≤ g ♯ (0) . Clearly, g is non-constantbecause g ♯ (0) = 0 . ” ⇐ ” Take α = 0 . Suppose that there exist sequences f j ∈ F , z j → , ρ j → , such that the sequence g j ( z ) = f j ( z j + ρ j z )converges locally uniformly in C n to a non-constant entire function g satisfying g ♯ ( z ) ≤ g ♯ (0) = 1 , but F is normal. By Marty’s criterion (Theorem 1.1) thereexists a constant M > | z |≤ / f ♯j ( z ) < M for all j. Since z j → , ρ j → , then for | z | < / j sufficiently large, we have | z j + ρ j z | ≤ | z j | + ρ j | z | ≤ | z j | + ρ j / < / . Thus g ♯j ( z ) = f ♯j ( z j + ρ j z ) ρ j ≤ M ρ j → | z | < / . ) sgn denotes the signum function (i.e., sgn (0) = 0 , sgn ( α ) = 1 if α > − α < P.V.DOVBUSH
This implies that g ♯ (0) = 0 , which is a contradiction to g ♯ (0) = 1 . (cid:3) Applications
Let us illustrate the use of improved Zalcman’s lemma by showing first how itcan be used to derived the following theorem.
Theorem 4.1.
Let some ε > be given and set F = { f holomorphic in Ω : f ♯ ( z ) > ε for all z ∈ Ω } . Then F is normal in Ω . Proof.
To obtain a contradiction suppose that F is not normal in point z ∈ Ω . Without restriction we may assume z = 0 . If F is not normal at 0 , it follows fromTheorem 1.2 that there exist f j ∈ F , z j → , ρ j → , such that the sequence g j ( z ) = ρ j · f j ( z j + ρ j z )converges locally uniformly in C n to a non-constant entire function g satisfying g ♯ ( z ) ≤ g ♯ (0) = 1 . Since g the non-constant entire function it follows that exists a ∈ Ω such that | g ( a ) | > . Hence | g j ( a ) | 6 = 0 for all j sufficiency large1 ≥ g ♯j ( a ) = max | v | =1 | ( Df j ( z j + ρ j a ) , v ) | · | g j ( a ) | ρ j · | f j ( z j + ρ j a ) | · (1 + | g j ( a ) | ) ≥ f ♯j ( z j + ρ j a ) ρ j · | g j ( a ) | | g j ( a ) | ≥ ερ j · | g j ( a ) | | g j ( a ) | . The right-hand side of this inequality tends to infinity as j → ∞ , a contradiction.This contradiction shows that F is normal in Ω . (cid:3) For families of holomorphic functions which do not vanish, we have the followingtheorem.
Theorem 4.2.
Let F be a family of zero-free holomorphic functions in a domain Ω ⊂ C n . The statement of Theorem 1.2 remains valid if − ≤ α < ∞ is replacedwith −∞ < α < ∞ . Proof of Theorem 4.2.
It need to consider only the case −∞ < α < . Since a family { /f, f ∈ F} conforms to the hypotheses of Theorem 4.2 the earlier argument showsthat there exist sequences 1 /f j , z j → z , r j → , such that the sequence g j ( z ) := r αj f j ( z j + r j z ) (0 ≤ α < ∞ arbitrary )converges locally uniformly in C n to a non-constant entire function g satisfying g ♯ ( z ) ≤ g ♯ (0) = 1 . By Hurwitz’s theorem either g ≡ g never vanishes. Since g ♯ (0) = 1 it is easy to see that g never vanishes then 1 /g is entire function in C n . It follows r − αj f j → /g uniformly in C n . Since Levi form vanishes for anypluriharmonic function, L z (log(1 + | /g | ) , v ) = L z (log(1 + | g | ) , v ) − L z (log | g | , v ) = L z (log(1 + | g | ) , v ) . Therefore, g ♯ ( z ) = (1 /g ) ♯ ( z ) . For every z ∈ C n we have g ♯ ( z ) ≤ g ♯ (0) = 1 , hence(1 /g ) ♯ ( z ) ≤ (1 /g ) ♯ (0) = 1 . N IMPROVEMENT OF ZALCMAN’S LEMMA IN C n The case −∞ ≤ α < (cid:3)
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