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R. Brody (1978)
Compact manifolds in hyperbolicityTransactions of the American Mathematical Society, 235
Universitätsstraße 30, D-95447 Bayreuth, Germany, E-mail address: jwinkel@member.ams.org Webpage
O Lehto (1959)
196Comment. Math. Helv., 33
R Brody (1978)
213Trans. Am. Math. Soc., 235
Alexandre Eremenko (2013)
Normal holomorphic maps from C?\documentclass[12pt]{minimal}Houston J. Math., 39
J Clunie (1966)
40Comment. Math. Helv., 117–148
J. Clunie, W. Hayman (1965)
The spherical derivative of integral and meromorphic functionsCommentarii Mathematici Helvetici, 40
K Yosida (1934)
On a class of meromorphic functionsProc. Phys-math. Soc. Jpn., 16
Olli Lehto (1959)
The spherical derivative of meromorphic functions in the neighbourhood of an isolated singularityCommentarii Mathematici Helvetici, 33
Alexandre Eremenko (2013)
1349Houston J. Math., 39
M. Tsukamoto (2006)
A Packing Problem for Holomorphic CurvesNagoya Mathematical Journal, 194
We discuss meromorphic functions on the complex plane which are Brody curves regarded as holomorphic maps to P , i.e., which have bounded spherical derivative. For some special classes we gave explicit criteria which functions are Brody. We also discuss which divisors of very slow growth may occur as zero divisor of a Brody function and show that there are transcendental entire functions of arbitrarily slow growth which are not Brody. Keywords Entire function · Brody curve · Spherical derivative Mathematics Subject Classiﬁcation 30D15 · 30D35 We discuss meromorphic functions on C which are Brody curves regarded as holomorphic curves from C to P (C). In other words, those meromorphic func- tions for which || f || calculated with respect to the euclidean metric on C and the Fubini-Study-metric on P is bounded. In concrete terms this means: | f (z)| lim sup < +∞. 1 +| f (z)| This number is also called “spherical derivative” ( [2]). We follow the established notation and denote this spherical derivative as | f (z)| f (z) = . 1 +| f (z)| Brody curves in general have been studied since the seminal paper of Brody ( [1]) illuminated their relevance for hyperbolicity questions. Jörg Winkelmann joerg.winkelmann@rub.de IB 3-111, Lehrstuhl Analysis II, Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany 123 ANNALI DELL’UNIVERSITA’ DI FERRARA The “spherical derivative” f has been studied before Brody proved his theorem, especially by Clunie, Hayman, Lehto, Virtanen and Yosida ( [2, 4, 5, 7]). Functions for which the spherical derivative is bounded (i.e. Brody functions in the notion of this article) appear also as “normal functions” or “Yosida functions” in the literature, cf. [3, 7]. Clunie and Hayman showed that a meromorphic Brody function is of order at most two and a Brody entire function is of order at most one. Examples of Fryntov (see [3], ex. 5.4) show that there exists an entire Brody function of order ρ for every ρ ∈[0, 1]. Clunie and Hayman also demonstrated the existence of Brody curves with certain divisors with high multiplicities as prescribed zero divisors. Easy examples for Brody functions are polynomials and the exponential function. As noted by Tsukamoto ( [6]), elliptic functions are also Brody: If is a lattice in C, and f is a -periodic meromorphic function, then f is continuous and -invariant. Therefore the existence of a compact fundamental region for implies that f is bounded. For certain “sparse” sets S ⊂ C one can verify that every map f : S → C extends to a Brody function f : C → C,see [3]. Brody functions can be characterized by the following condition: For every four points a ,..., a ∈ P there exists a positive number δ> 0 such that no disc of radius 1 4 1 −1 δ in C intersects more than one of the pre-images f ({a }) (see [3]). We investigate two special classes of meromorphic functions, namely R(z)e +Q(z) z λz where R, Q are rational functions and e + e with λ ∈ C. For these functions we determine completely which ones are Brody. This provides us with some surprising examples: If f , g are Brody neither f +g, nor fg need to be Brody. The Brody condition is not closed: The meromorphic function f (z) = e + is Brody if and only if t = 0. zt +1 In particular in view of the results of Clunie and Hayman on the order it might be natural to assume that every divisor of sufﬁciently slow growth can be realized as the zero divisor of an entire Brody function. We show that this is not the case, there are effective divisors of arbitrary slow growth which can not be realized as zero divisors of a Brody function. We will also see that for such a divisor its growth alone is not enough to determine whether there is a corresponding Brody entire curve (see Theorem 1). Finally we use these results on zero divisors of Brody curves to show that there are transcendental entire functions of arbitrarily slow growth. (see Theorem 2 for the exact formulation.) 1 Basic properties We start with the observation f = · h( f ) A related result has been obtained much earlier by Lehto ( [4]). 123 ANNALI DELL’UNIVERSITA’ DI FERRARA where |w| h(w) = . 1 +|w| Since h(z) ≤ for all z ∈ C, we may deduce: If f / f is bounded, then f is Brody. In fact one has the following slightly stronger statement: Lemma 1 Let f be an entire function with f (z) lim sup < +∞. f (z) z→∞ Then f is Brody. Proof There is a compact subset K ⊂ C and a constant C > 0 such that | f (z)/ f (z)| < C for all z ∈ / K . Then # # sup f (z) ≤ max C , max f (z) z∈K z∈C Corollary 1 The exponential function z → e is Brody. Concerning rational functions the Brody property follows from the observation below: Lemma 2 Let R be a non-constant rational function on C. R (z) Then lim = 0. z→∞ R(z) Proof Write R as a quotient of polynomials: R = P/Q. Then R P Q − PQ R PQ and the assertion follows from the fact that deg(P Q − PQ )< deg(PQ). Corollary 2 Rational functions have the Brody property. Proposition 1 Assume that f is a Brody entire function, that R is a rational function and that α ∈ C , β ∈ C. Then z → F (z) = R( f (αz + β)) is Brody, too. Proof Each rational function R deﬁnes a continuous self-map of P . The compactness of P implies that for each rational function R there exists a constant C such that the operator norm of the differential map DR : T P → T P is bounded by C. Then 1 1 # # sup F ≤ C |α| sup f . Corollary 3 The trigonometric functions sin, cos, tan, sinh, cosh are all Brody. 123 ANNALI DELL’UNIVERSITA’ DI FERRARA z iz Proof These functions can be expressed in the form R(e ) or R(e ) for some rational function R ∈ C(X ). In particular, there is a Brody entire function f with {z : f (z) = 0}= Z, namely f (z) = sin(π(z)). There is also a Brody function f whose zero divisor is {k : k ∈ Z}: Following a √ √ suggestion of the anonymous referee we deﬁne f (z) = cos(2π z) − 1. (Despite z being deﬁned only up to sign this is a well-deﬁned function, because cos is an even function.) 2 Products with rational functions Lemma 3 Let f : C → P be Brody and let R be a rational function with R(∞)/ ∈ {0, ∞}. Then g = R f is Brody. Proof We have g (z) g (z) = h(g(z)) g(z) |w| with h(w) = . 1+|w| We will need the following auxiliary fact: −1 Claim: Let λ> 1. Then h(λw) ≤ λh(w) and h(λ w) ≤ λ|w| for all w ∈ C. These claims are rather immediate: |λw| |λw| h(λw) = ≤ = λh(|w|) 2 2 1 +|λw| 1 +|w| and −1 |λ w| |λw| −1 h(λ w) = = ≤ λ|w|. −1 2 2 2 1 +|λ w| |λ| +|w| P(z) Now write the rational function R as quotient of two polynomials: R(z) = . Q(z) Then g f P Q = + − g f P Q and (using lim (P (z)/P(z)) = lim (Q (z)/Q(z)) = 0) we may deduce that z→∞ z→∞ there are constants R , C > 0 such that 1 1 g (z) f (z) − ≤ C g(z) f (z) for complex numbers z with |z| > R . 123 ANNALI DELL’UNIVERSITA’ DI FERRARA By assumption lim R(z) ∈ C . Therefore there are constants λ> 1 and z→∞ R > 0 such that −1 λ < |R(z)| <λ whenever |z| > R . Then h(g(z)) = h(R(z) f (z)) ≤ λh( f (z)) if |z| > R . As a consequence, we have |g (z)| | f (z)| g (z) = h(g(z) ≤ + C λh( f (z)) |g(z)| | f (z)| ≤ λ f (z) + C λh( f (z)) ≤ λ f (z) + if |z| > max{R , R }. 1 2 Hence the assertion. It is really necessary to assume R(∞) = 0, ∞. For instance, consider the functions f (z) = e + 1 and f (z) = 1/ f (z). Both are Brody functions, but Proposition 2 in 1 2 1 the next section implies that neither g (z) = zf (z) nor g (z) = f (z) is Brody. 1 1 2 2 3 Case study: R(z)e + Q(z) We will now discuss certain sums. Proposition 2 Let R, Q be rational functions. Then f (z) = R(z)e + Q(z) is a Brody function if and only if one of the following conditions is fulﬁlled: (i) R ≡ 0 or (ii) Q(∞) =∞. Proof If R ≡ 0, then f = Q is rational and therefore Brody. Assume now that Q(∞) =∞ and R ≡ 0. Then there is a rational function Q with Q (∞)/∈{0, ∞} and a complex number c with Q(z) = Q (z) + c. Then 0 0 R(z) f (z) = Q (z) e + 1 + c Q (z) R(z) and f is Brody if and only if g(z) = e is Brody by Proposition 1. Now S = R/Q Q (z) is rational and S ≡ 0. Therefore g (z) S(z) + S (z) lim = lim = 1. z→∞ z→∞ g(z) S(z) 123 ANNALI DELL’UNIVERSITA’ DI FERRARA But this implies lim sup g (z) ≤ 1 · < +∞. z→∞ Hence g is Brody and therefore f is Brody, too. It remains to show that f is not Brody whenever Q(∞) =∞ and R ≡ 0. In this case Q(z) = Q (z)z for some rational function Q with Q (∞)/∈{0, ∞} and 0 0 0 n ∈ N. Deﬁne S = R/Q . Then S is rational. Considering the equality z n f (z) = Q (z) S(z)e + z z n is Brody if and only if g(z) = S(z)e + z is Brody. Since R ≡ 0 together with Q (∞) = 0 implies S ≡ 0, the entire function g is transcendental. Therefore there is a sequence of complex numbers z with lim |z |=∞ and lim g(z ) = 0. k k→∞ k k→∞ k Now g(z) − z z n−1 n−1 g (z) = (S(z) + S (z))e + nz = S(z) + S (z) + nz S(z) which implies S(z) + S (z) g(z) n g (z) = z − 1 + . S(z) z z S+S We recall that lim = 1, lim z =∞ and lim g(z ) = 0. z→∞ k→∞ k k→∞ k Combined, these facts yield lim g (z ) =∞ k→∞ Together with lim g(z ) = 0 this implies k→∞ k lim g (z ) =∞ k→∞ Thus g is not Brody, which in turn implies that f is not Brody. z z 4 Case study: e + e z λz Proposition 3 The entire function f (z) = e + e is Brody if and only if λ ∈ R. Proof Case (1) The cases λ ∈{0, 1, −1} are trivial. z(1−λ) Case (2) Let λ/ ∈ R. Then f (z) = 0iff e =−1. Hence the set of all zeros of f is given as the set of all (2k + 1)πi a = 1 − λ 123 ANNALI DELL’UNIVERSITA’ DI FERRARA with k running through the set of integers Z.Now f (a ) = 0 implies f (a ) = k k | f (a )| and (2k+1)πi λ 1−λ f (a ) = (λ − 1)e which is unbounded because λ/ ∈ R implies that 2πi λ 1−λ e = 1. Therefore sup f (a ) =+∞ and consequently f is not Brody. Case (3) Assume λ> 0. Observe that f (z) = g(λz) for g(w) = e + e . Hence we may assume without loss of generality that 0 <λ ≤ 1. Since f (z) = 2e if λ = 1, it sufﬁces to consider the case 0 <λ< 1. We choose log 2 C = . 1 − λ Then λz z |e |≤ |e | for all z ∈ C with (z) ≥ C. It follows that z λz z z |e + λe | 3/2|e | 3/2|e | # −z −C f (z) = ≤ ≤ = 6e ≤ 6e z λz 2 z 2 z 2 1 +|e + e | 1 + 1/4|e | 1/4|e | for all z ∈ C with (z) ≥ C. On the other hand # C λC f (z) ≤| f (z)|≤ e + λe for all z with (z) ≤ C. Thus # −C C λC f (z) ≤ max{6e , e + λe }∀z ∈ C and consequently f is Brody. Case (4) We assume λ< 0. It sufﬁces to consider the case where −1 <λ< 0. As before, we deﬁne log 2 C = . 1 − λ and obtain λz z |e |≤ |e | 123 ANNALI DELL’UNIVERSITA’ DI FERRARA and therefore # −C f (z) ≤ 6e for all z ∈ C with (z) ≥ C. Next we observe that the condition (z) ≤−C implies z λz e ≤ e . As a consequence λz λz 3/2|e | 3/2|e | # −λz λC f (z) ≤ ≤ = 6e ≤ 6e λz 2 λz 2 1 + 1/4|e | 1/4|e | for all z with (z) ≤−C. Finally we observe that # z λz C −λC f (z) ≤| f (z)|≤|e |+|λe |≤ e +|λ|e for all z with −C ≤(z) ≤ C. 5 Divisors of slow growth Theorem 1 Let D be the divisor deﬁned by D ={a : k ∈ N} (reduced, i.e. all multiplicities being one). Assume that the following growth condition is fulﬁlled: ∃λ :∀k ∈ N :|a | >λ|a | > 0 (∗) k+1 k Let H denote the convex hull of the closure of the set of accumulation points of the sequence . |a | (i) If 0 is contained in the interior of H, then there does not exist any Brody function with D as zero divisor. (ii) If 0 is not contained in H, then there does exist a Brody function with D as zero divisor. Basically we show that for such a divisor D the function F (z) = 1 − a∈|D| is not a Brody function and that the only way to obtain a Brody function f with D as a zero divisor is via the approach αz αz f (z) = e F (z) = e 1 − (α ∈ C) a∈|D| 123 ANNALI DELL’UNIVERSITA’ DI FERRARA αz αz However, as we will see, the additional factor e is helpful only if e is rapidly converging to zero along the sequence a . This in turn is possible only if the divisor is concentrated in one half-plane. Proof We begin with the proof of statement (i ). Condition (∗) implies ∀k ∈ N : |a | ≥ λ |a | k+1 1 which in turn implies the absolute convergence of |a | which implies the convergence of log 1 − (k = max{k :|a |≤ 1}) 0 k |a | k>k which in turn implies the convergence of F (z) = 1 − . Thus an arbitrary entire function f with divisor D can be written as g(z) f (z) = F (z)e where F is deﬁned as above while g is an entire function. Claim (A) There is a constant C > 0 (depending only on λ) such that 1 − ≥ C k≥n for every n ∈ N and p ∈{z ∈ C :|z| λ< |a |}. To prove claim A, we observe that −(l+1/2) 1 − ≥ C = 1 − λ . k≥n l≥0 We emphasize that C is independent of n. We need a second claim. 123 ANNALI DELL’UNIVERSITA’ DI FERRARA Claim (B) For every K > 0 and m ∈ N there is a natural number N ≥ m (depending on K and m) such that 1 − ≥ K k=m for all n ≥ N and p with | p|≥ λ|a |. To prove the second claim it sufﬁces to note that n n−m 1/2+l 1 − ≥ λ − 1 k=m l=0 and 1/2+l lim λ − 1 =∞. s→∞ l=0 Due to the two above claims we know that there is a number M ∈ N such that |F (p)|= 1 − ≥ 1 k=1 for all n ≥ M and all p ∈ C with |a | n+1 λ|a | < | p| < √ . It follows that |F (p)|≥ 1 for every n ≥ M and every p ∈ C with | p|= |a a |. n n+1 Recall the notation M (r ) = max log | f (z)| |z|=r for an entire function f . g(z) With f (z) = F (z)e and |F (p)|≥ 1 for every n ≥ M and every p ∈ C with | p|= |a a | we may deduce that n n+1 M (r ) ≥ M (r ) f exp(g) for all n ≥ M and r = |a a |. n n+1 Hence, if f is Brody and therefore of order at most one, we can deduce that exp(g(z)) is likewise of order at most one, which in turn implies that g is a polynomial of degree 123 ANNALI DELL’UNIVERSITA’ DI FERRARA 1, i.e. afﬁne-linear. Thus we may assume g(z) = αz, α ∈ C. Now the condition that 0 is contained in the interior of H implies that there is a subsequence a with exp(g(a )) ≥ 1 ∀ j w (w) because |e |= e . Next we calculate | f (a )|. Since f (a ) = F (a ) = 0 for every k,wehave n k k # g(a ) f (a ) =| f (a )|=|F (a )e |∀k ∈ N k k k and therefore f (a ) ≥|F (a )|∀ j ∈ N. n n j j Thus it sufﬁces to show that lim |F (a )|=+∞ k→ in order to deduce that f is not Brody. Now 1 a |F (a )|= 1 − . |a | a n k k=n Now 1 a 1 lim 1 − = . n→∞ |a | a |a | n 1 1 By the ﬁrst claim 1 − ≥ C k>n and by the second claim for each K > 0 there is a number N ∈ N such that 1 − ≥ K 1<k<n for all n ≥ N . Combining these assertions, we show that for each K > 0 there is a number N such that |F (a )|≥ K 2|a | 123 ANNALI DELL’UNIVERSITA’ DI FERRARA for all n ≥ N . Thus lim sup | f (a )|=+∞ and f is not Brody. This proves statement (i ). Next we deal with statement (ii ). We assume that 0 is not contained in H.Bythe deﬁnition of H it follows that there are numbers M ∈ N, η ∈ S ={z ∈ C :|z|= 1} and μ ∈]0, 1[ such that (ηa ) ≥ μ|a |∀k ≥ M . k k 0 Let S = . (The convergence of this inﬁnite sum follows from the condition |a | |a |≥ λ|a |∀k.) k+1 k We choose a complex number α as α =−r η such that r is a positive real number with rμ> S. We claim that αz f (z) = e 1 − deﬁnes a Brody function. We start choosing some more constants 0 <γ < − 1, C = inf γ |a |. −1 1 + λ k Next we observe that the above choices imply (αa ) + S|a |=−r (ηa ) + S|a |≤ (−r μ + S)|a | < 0 ∀k ≥ M . k k k k k 0 It follows that there is a constant M ∈ N such that (αz) + S|z| < 0 for every complex number z for which there exists a natural number k ≥ M with |z − a | < C. Claim (C) There is a constant K > 0 such that f (z) < K f (z) for all z ∈{w ∈ C :|w − a |≥ C ∀k}. Before we prove this claim, we need another claim. Claim (D) For k = l we have {z :|z − a |≤ γ |a |} ∩ {z :|z − a |≤ γ |a |} = ∅ k k l l 123 ANNALI DELL’UNIVERSITA’ DI FERRARA Proof of claim (D). Let w be a point in the intersection. Then |w − a |≤ γ |a | and k k |w − a |≤ γ |a |, implying l l |a − a |≤|w − a |+|w − a |≤ γ (|a |+|a |) . k l k l k l We may assume that l = k + d for some d > 0. Then |a |≥ λ |a | and therefore l k −d |a − a |≥ 1 − λ |a | k l l Together: −d −d 1 − λ |a |≤ γ (|a |+|a |) ≤ γ λ + 1 |a | l k l l and therefore −d 2 1 − λ − 1 = ≤ γ. −d −d 1 + λ λ + 1 Hence 2 2 − 1 ≤ − 1 ≤ γ −1 −d 1 + λ 1 + λ which contradicts the choice of γ . Hence there can not be a point w in the intersection. This proves claim (D). Our choice of f implies f (z) 1 = α + f (z) z − a For each z ∈ C and k ∈ N with |z − a |≥ γ |a | we have k k 1 1 z − a γ |a | k k For a given z ∈ C there is at most one index k with |z − a | <γ |a | (see claim (D) k k above). Hence f (z) 1 1 1 = α + ≤|α|+ + f (z) z − a C γ |a | k k k k if |z − a |≥ C ∀k ∈ N. 123 ANNALI DELL’UNIVERSITA’ DI FERRARA This proves claim (C) with 1 1 1 S K =|α|+ + =|α|+ + . C γ |a | C γ As a next step, we deduce an upper bound for | f | on W ={z ∈ C :∃k ≥ M :|z − a |≤ C }. 2 k Fix a number k ∈ N. Then z z αz f (z) = e 1 − 1 − a a k l l=k and ⎛ ⎞ 1 z 1 αz ⎝ ⎠ f (z) =− e 1 − + α + f . a a z − a k l l l=k l=k In order to get the desired bounds we start as follows. 1 − l=k = 1 − l=k ≤ 1 + l=k ⎛ ⎞ ⎝ ⎠ = exp log 1 + l=k ⎛ ⎞ ⎝ ⎠ ≤ exp l=k ⎛ ⎞ ⎝ ⎠ = exp |z| |a | l=k S|z| ≤e . 123 ANNALI DELL’UNIVERSITA’ DI FERRARA Hence (αz) S|z| | f (z)|≤ e e |a | if |z − a |≤ C. We also observe that if |z − a |≤ C = inf γ |a | k j then |z − a |≤ γ |a | implying (due to claim (D)) that k k 1 1 z − a γ |a | l l for all l = k. Hence |z − a |≤ C implies 1 1 1 S ≤ ≤ z − a γ |a | γ l l l=k l=k Combined we obtain: 1 S C (αz) S|z| (αz) S|z| | f (z)|≤ e e + |α|+ e e (1) |a | γ |a | k k 1 CS ((αz)+S|z|) = e 1 + C |α|+ (2) |a | γ We recall that due to the choices of our constants (αz) + S|z| < 0 for all z for which there exists an index k ≥ M such that |z − a |≤ C. Therefore CS 1 + C |α|+ 1 CS γ def f (z) ≤| f (z)|≤ 1 + C |α|+ ≤ = K |a | γ |a | k M for all z ∈ W ={z :∃k ≥ M :|z − a |≤ C }. 2 k We have also seen that there is a constant K > 0 such that 1 f (z) f (z) ≤ ≤ K 2 f (z) for all z ∈ W ={z :|z − a |≥ C ∀k}. 1 k Hence f (z) ≤ max{K , K } for all z ∈ W ∪ W . Because C \ (W ∪ W ) is 1 2 1 2 1 2 contained in a ﬁnite union of bounded disks and therefore relatively compact in C,it follows that f is bounded on C, i.e., f is a Brody function. 123 ANNALI DELL’UNIVERSITA’ DI FERRARA 6 Growth conditions and characteristic function Our goal is to show that no bound on the characteristic function T (r ) forces an entire function f to be Brody except when this bound is strong enough to force f to be a polynomial. In view of the preceding section the crucial point is to verify that for every such bound there exists an entire function f fulﬁlling this condition on T (r ) and fulﬁlling simultaneously the condition (i ) of Theorem 1. Before stating the result of this section we recall some basic notions. For an entire function f with f (0) = 0 we may deﬁne the characteristic function as T (r ) = m (r ) + N (r ) with f 1/ f 1/ f 2π m (r ) = log dθ 1/ f i θ | f (re )| and N (r ) = ord ( f ) log . 1/ f a a∈C Now we can state the result: Theorem 2 Let ρ :[1, ∞[→]0, ∞[ be a continuous increasing function. ρ(t ) (i) If lim inf < ∞, then every entire function with T (r ) ≤ ρ(r ) for all t →∞ f log t r ≥ 1 must be a polynomial. ρ(t ) (ii) If lim inf =∞, then there exists an entire function which is not Brody t →∞ log t and such that T (r ) ≤ ρ(r ) ∀r ≥ 1. Proof (1) If there is a constant C > 0 such that ρ(t ) lim inf < C t →∞ log t then N (t ) f −a lim inf < C t →∞ log t for every entire function f with T (r ) ≤ ρ(r ) and for all a ∈ C. It follows that each −1 ﬁber f {a} has cardinality ≤ C and that f is a polynomial of degree bounded by C. (2) As a ﬁrst preparation, we observe that 2k+1 2 − 1 1 1 2 1 = 1 − ≥ 1 − = 1 − = . 2k+1 2k+1 2k+1 2 2 2 3 3 k k k 123 ANNALI DELL’UNIVERSITA’ DI FERRARA We will construct f as follows: We will choose a sequence c ∈ C with |c | > k k+1 4|c |≥ 1 for all k, then deﬁne P (z) = 3 − 1 . The conditions on the c ensure that P converges locally uniformly to an entire k k function f whose zeroes are precisely the points c . Let k ∈ N and let z be a complex number with |c |≤|z|≤ 2|c |. Then k+1 k − 1 ≥ 1 for all j ≤ k.For j > k then conditions |c |≥ 4|c | imply l+1 l 2( j −k)+1 ≥ 2 which in turn implies |P (z)|≥ 1 for all j and all z with |c |≤|z|≤ 2|c |. As a consequence, k+1 k m (r ) = 0 1/ f and hence T (r ) = N (r ) f 1/ f for all r ∈[ |c |, 2|c |] and all k ∈ N. k+1 k Therefore T (r ) = log ≤ k log r j ≤k for all such r. Similarily one obtains T (r ) = 0for r ≤ |c |. f 1 Let us now ﬁx k ∈ N and consider r ∈[ |c |, 2|c |]. k k Since T (r ) is increasing, we have 2c T (r ) ≤ T (2|c |) = N (2|c |) = log ≤ k log(2|c |) ≤ k log(4r ). f f k 1/ f k k j ≤k Summarizing, we have shown that T (r ) ≤ k log(4r ) for all r with r ≤ 2|c |. f k 123 ANNALI DELL’UNIVERSITA’ DI FERRARA Thus it sufﬁces to choose the c such that k log(4r ) ≤ ρ(r ) for all r ∈ [2|c |, 2|c |]. k−1 k ρ(t ) This is possible: We assumed lim inf =∞, hence for each k ∈ N there is t →∞ log t a constant R such that k log(4r ) ≤ ρ(r ) for all r ≥ R . Now it sufﬁces to choose the k k c such that 2|c |≥ R (in addition to the other conditions |c |≥ 4|c |≥ 1). k k−1 k k k−1 Thus we have established: We can choose a sequence c in C \{0} such that f (z) = 3 − 1 converges to an entire function f with T (r ) ≤ ρ(r ) for all r ≥ 1. Finally, we note that in our construction we choose the c such that |c |≥ 4|c |. k k+1 k Furthermore we may choose the c such that the set : k ∈ N |c | is dense in S ={z ∈ C :|z|= 1}. Then Theorem 1 implies such an entire function f is not Brody. Given a reduced effective divisor D on C its counting function is deﬁned as N (r ; D) = log . |a| a∈D Theorem 3 Let ρ :[1, ∞[→]0, ∞[ be a continuous increasing function. ρ(t ) (i) If lim inf < ∞, then every effective divisor D with N (r ; D) ≤ ρ(r ) for t →∞ log t all r ≥ 1 must be ﬁnite. ρ(t ) (ii) If lim inf =∞, then there exist inﬁnite reduced effective divisors D, D t →∞ log t such that • N (r ; D) ≤ ρ(r ) and N (r ; D ) ≤ ρ(r ) for all r ≥ 1. • There is an entire function f with D its zero divisor which is Brody. • There is no entire function f with D its zero divisor which is Brody. 7 Discussion Using the special cases of entire functions studied above we see that the class of entire functions which are Brody is not closed neither under addition nor under mul- z z z tiplication: the entire functions z, e + 1 and ze are all Brody, but ze + z is not, although z z z ze + z = z(e + 1) = ze + z. We see also that the Brody condition is neither closed nor open nor complex: For (s, t ) ∈ C let us consider f (z) = se + s,t tz − 1 123 ANNALI DELL’UNIVERSITA’ DI FERRARA By Proposition 2 the function f is Brody iff s,t (s, t ) ∈{(x , y) ∈ C : x = 0or y = 0} which is neither a closed nor an open set. Moreover, Proposition 3 provides an example of a family of entire functions depend- ing holomorphically on a complex parameter λ such that the function is Brody if and only if λ is real. All these properties are in stark contrast to the situation for Brody curves with values in abelian varieties. If A is an abelian variety, its universal covering is isomorphic to some C . Since every holomorphic map from C to A lifts to a holomorphic map with values in the universal covering of A, the classical theorem of Liouville implies that an entire curve with values in A is Brody if and only if it can be lifted to an afﬁne-linear map from C to C . As a consequence one obtains: • If f , g : C → A are Brody curves, so is z → f (z) + g(z). • If f : C → A is a family of entire curves depending holomorphically on a parameter t ∈ P where P is a complex manifold, then the set of all t ∈ P for which f is Brody forms a closed complex analytic subset of P. • An entire curve f : C → A is Brody if and only if log T (r ) lim sup ≤ 2. log r Acknowledgements The research for this article was done by the author partially while working at the Universität Bayreuth and partially during a stay at the Mittag-Lefﬂer Institute. The author was supported by the Mittag Lefﬂer Institute and the DFG Forschergruppe 790 “Classiﬁcation of algebraic surfaces and compact complex manifolds”. Funding Open Access funding enabled and organized by Projekt DEAL. The author has no relevant ﬁnancial or non-ﬁnancial interests to disclose. The author certiﬁes that they have no afﬁliations with or involvement in any organization or entity with any ﬁnancial interest or non-ﬁnancial interest in the subject matter or materials discussed in this manuscript. Declarations Conﬂicts of interest The author has no conﬂicts of interest to declare that are relevant to the content of this article. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 123 ANNALI DELL’UNIVERSITA’ DI FERRARA References 1. Brody, R.: Compact manifolds and hyperbolicity. Trans. Am. Math. Soc. 235, 213–219 (1978) 2. Clunie, J., Hayman, W.K.: The spherical derivative of integral and meromrophic functions. Comment. Math. Helv. 117–148, 40 (1966) 3. Eremenko, Alexandre: Normal holomorphic maps from C to a projective space. Houston J. Math. 39(4), 1349–1357 (2013) 4. Lehto, O.: The spherical derivative of meromorphic functions in the neighbourhood of an isolated singularity. Comment. Math. Helv. 33, 196–205 (1959) 5. Lehto, O., Virtanen, K.I.: On the behaviour of meromorphic functions in the neighbourhood of an isolated singularity. Ann. Acad. Sci. Fenn. Ser. A. I. 240 (1957) 6. Tsukamoto, Masaki: A packing problem for holomorphic curves. Nagoya Math. J. 194, 33–68 (2009) 7. Yosida, K.: On a class of meromorphic functions. Proc. Phys-math. Soc. Jpn. 16, 227–235 (1934) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.
ANNALI DELL UNIVERSITA DI FERRARA – Springer Journals
Published: Apr 25, 2023
Keywords: Entire function; Brody curve; Spherical derivative; 30D15; 30D35
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