On universal entire functions with zero-free derivatives

On universal entire functions with zero-free derivatives By LUIS BERNAL–GONZ´ ALEZ* Abstract. We prove in this note a generalization of a theorem due to G. Herzog on zero-free universal entire functions. Specifically, it is shown that, if a nonnegative integer qand a nonconstant entire function Φ of subexponential type are given, then there is a residual set in the class of entire functions with zero-free derivatives of orders q and q+1, such that every member of that set is universal with respect to Φ(D), where Dis the differentiation operator. 1. Introduction and notation. We denote by Cthe complex plane, by N the set of positive integers and by N0the set N∪ {0}. If r > 0, B(r) (B(r)) is the euclidean open (closed, respectively) disk with center 0 and radius r. We agree that B(+∞) = C.H(B(r)) will stand, as usual, for the space of holomorphic functions in B(r), endowed with the topology of uniform convergence on compact subsets. In H(C), this topology is induced by the metric (1) d(f, g) = ∞ ∑ j=1 1 2j ||f−g||j 1 + ||f−g||j , where ||h||r= maxB(r)|h|(∀r > 0). It is well known that H(B(r)) is a separable Fr´echet space, so it is a Polish space and also a Baire space (see, e.g., [14, pp. 213214 and 238]). In a Baire space X, a subset is residual when it contains a dense Gδ-subset of Xor, equivalently, when its complement is of first category. Such a subset is “very large” in X. *This work is supported in part by DGICYT grant PB93-0926. 1991 Mathematics Subject Classification: Primary 30E10. Secondary 47B99, 47E05. Key words and phrases: universal entire function, zero-free derivative, subexponential type, MacLane’s theorem, residual set. 1 We use a very general notion of universality, which can be found in [10], namely: Let Xand Ybe nonempty topological spaces and Lbe a family of continuous mappings from Xinto Y. Then an element x∈Xis called universal with respect to Lif the set {Lx :L∈ L} is dense in Y. As in [13], we denote the set of all universal elements by U(L). Universal elements are usually called hypercyclic in the case that X=Yis a topological vector space and Lis the sequence of iterates {Ln}∞ 1of a single linear continuous operator Lon X(see, for instance, [9]). S. Rolewicz [17] was the first to give an example of a hypercyclic element in the Banach/Hilbert setting. In 1952, G. R. MacLane [15] stated that there exist entire functions fsuch that the set of derivatives {f(n):n∈N}is dense in H(C) or, equivalently, f∈U(L) for L={Dn:n∈N}, where Dis the differentiation operator on H(C), that is, Df =f′. The result is also proved in [3] (see also [4]). S. M. Duyos Ruiz [7] has shown that, in fact, there is a residual set of such functions. Furthermore, R. M. Gethner and J. H. Shapiro [8] and K. G. Große-Erdmann [10, Satz 2.2.8] have derived the same result for every simply connected domain. For additional results about the topic, the reader is referred to [2], [11] and [12] and many others in their references. For instance, Große-Erdmann [11] provides a sharp result on growth of D-universal functions. Returning now to the general case of a family Lof continuous mappings from X into Y, G. Herzog [13] proposed recently the following interesting question: Which additional properties of elements of Xare compatible with universality? If U(L) is residual and A⊂Xis a Gδ-subset, he proves that under certain conditions on Aand L(see Theorem 1 below) the set A∩U(L) is residual in A. Then, by using this theorem, he derives the existence of zero-free universal entire functions (for D) having even a zero-free first derivative. The basic tools employed by Herzog are the theory of universality developed by Große-Erdmann [10, specially Satz 1.2.2], Alexandroff’s theorem on completeness of Gδ-subsets (see, e.g., [16, pp. 47-48]) and 2 elementary results from Complex Analysis. Herzog [13, Section 3] himself points out that there is no universal entire function fsuch that f·f′·f′′ is zero-free, since {f∈H(C) : f·f′·f′′ is zero-free}={eαz+β:α, β ∈C, α = 0}(see [6, p. 433] and [19]). If q∈N0, let us denote A(q) = {f∈H(C) : f(q)(z)f(q+1)(z)= 0 ∀z∈C}. Since exp ∈∩q∈N0A(q), every A(q) is nonempty. Our aim in this note is to furnish a strong generalization of Herzog’s theorem on zero-free derivatives. Specifically, we show in Theorem 5 that, if a nonnegative integer qand a nonconstant entire function Φ of subexponential type are given, then there is a residual subset in A(q) satisfying that every member of such a subset is universal with respect to the operator Φ(D), where Dis the differentiation operator. In terms of growth orden and type, our result is best possible. 2. Preliminary results. The technique for proving Theorem 5 will be very similar to that in [13], but we need an additional elementary result on antiderivatives together with the “good behaviour” of certain related non-linear operators relatively to convergence (this is Theorem 2; its proof is easy and left to the reader), a strong assertion due to Godefroy and Shapiro [9] (Theorem 3) and, finally, a result which asserts the continuity of subexponential differential operators on every space H(B(r)) (r > 0) (Theorem 4). Moreover, it is also employed the above mentioned Theorem 1, which is exactly Theorem 2.1 of [13]. Theorem 1. Assume that Xis a Polish space and Yis a separable metrizable space. Let dX, dYbe metrics inducing the topologies of X, Y , respectively. Let {Ak:k∈N}be a sequence of open subsets of Xwith A≡∩∞ k=1 Ak=∅. Let L={Ln:n∈N}be a sequence of continuous mappings from Xinto Y, with U(L)residual in X. Denote L|A={Ln|A:n∈N}, where Ln|Ais the restriction of Lnto A. If lim k→∞ sup n∈N inf z∈A(dX(ak, z) + dY(Lnak, Lnz)) = 0 3 for every sequence {ak}∞ 1(ak∈Ak, k ∈N), then U(L|A)is residual in A. Theorem 2. a) If q∈N0and f∈H(B(r)) then f(q)f(q+1) is zero-free if and only if there exists g∈H(B(r)) such that f(z) = {f(0) exp(∫z 0exp(g(t)) dt)if q= 0 ∑q−1 ν=0 f(ν)(0) ν!zν+f(q)(0) (q−1)! ∫z 0(z−t)q−1exp(∫t 0exp(g(u)) du)dt if q≥1 for all z∈B(r). b) Assume that q∈N0and T:f∈H(B(r)) 7→ Tf ∈H(B(r)) is the mapping given by Tf(z) = {exp(∫z 0exp(f(t)) dt)if q= 0 ∫z 0(z−t)q−1exp(∫t 0exp(f(u)) du)dt if q≥1. Then Tis a well-defined continuous operator on H(B(r)). Before stating the next two theorems, we recall that an entire function Φ(z) = ∑∞ j=0 ajzjis said to be of exponential type whenever there exist positive constants Aand Bsuch that |Φ(z)| ≤ AeB|z|for all z∈C. Cauchy’s inequalities show that this happens if and only if lim supj→∞(j!|aj|)1/j is finite (cf. [18, Chap. VII]). It is shown in [9, Section 5] that if Φ is of exponential type and L= Φ(D) (that is, L=∑∞ j=0 ajDj, where D0=I= the identity operator), then Lis a well-defined continuous linear operator on H(C). By analogy, we adopt the next terminology. We say that an entire function Φ(z) = ∑∞ j=0 ajzjis of subexponential type whenever the following property holds: Given ε > 0, there is a positive constant A=A(ε) such that |Φ(z)| ≤ Aeε|z|∀z∈C, that is, Φ is either of growth order less than one or of growth order one and minimal type. Every entire function of subexponential type is trivially of exponential type. As before, Cauchy’s inequalities show that Φ is of subexponential type if and only if limj→∞(j!|aj|)1/j = 0 (cf., e.g., [5, 2.2.9-11]). 4 Theorem 3. Suppose that Lis the continuous linear operator on H(C)given by L= Φ(D), where Φis a nonconstant entire function of exponential type. Then there is a dense, invariant submanifold of H(C)each of whose non-zero elements is hypercyclic for L. It should be pointed out here that a continuous linear operator Lon H(C) is of the form L= Φ(D), where Φ is an entire function of exponential type, if and only if Lcommutes with each of the translation operators τa(a∈C), where τaf(z) = f(z+a) (f∈H(C), z ∈C) (see [9, Theorem 5.1, Proposition 5.2] for the proof of Theorem 3 and this note; we just use de case CN=Cof [9, Section 5]). Thus the operators Lon H(C) commuting with translations are a special class of “infinite order” linear differential operators with constant coefficients. Under the hypothesis of Theorem 3, U(L) is not empty for L={Ln:n∈N}. Then U(L) is residual in H(C) (see [8, Proposition 2.1]). Theorem 4. Let Φ(z) = ∑∞ j=0 ajzjbe an entire function of subexponential type and L= Φ(D). Then Lis a well-defined continuous linear operator on H(B(r)). P r o o f. Fix t∈(0, r) and choose any s∈(t, r). Let f∈H(B(r)). Cauchy’s inequalities guarantee that ||Djf||t≤j!||f||s (s−t)jfor every j≥0. Let ε=s−t 2. By hypothesis, there is a positive constant Asuch that |aj| ≤ A·εj j!for every j≥0. Then we have that ∑∞ j=0 ||ajDjf||t=∑∞ j=0 |aj| · ||Djf||t≤∑∞ j=0 A·εj j!·j!||f||s (s−t)j= A||f||s∑∞ j=0(1/2)j= 2A||f||s<+∞. Therefore ∑∞ j=0 ajDjfconverges uniformly on every closed disk B(t) (0 < t < r) and Ldefines a mapping from H(B(r)) into itself. The linearity is trivial and, since ||Lf||t≤2A||f||s, we have also obtained that Lis continuous on H(B(r)). //// 3. The main result. We are now ready to state our theorem on universality. Herzog’s result is the special case q= 0, L=D. 5 Theorem 5. Assume that Lis the continuous linear operator on H(C)given by L= Φ(D), where Φis a nonconstant entire function of subexponential type. Fix q∈N0and set A=A(q),L={Ln:n∈N}. Then the set U(L|A)is residual in A. P r o o f. Firstly, note that A=∩∞ k=1 Akwhere Ak={f∈H(C) : minB(k)|f(q)·f(q+1)|>0}. Put X=Y=H(C). It is evident that every Ak is open in H(C), so Ais a nonempty Gδ-subset of X. From Theorem 3, U(L) is residual in X. In order to apply Theorem 1, we should demonstrate that, for every fixed sequence {fk}∞ 1(fk∈Ak, k ∈N), it holds that (2) lim k→∞ sup n∈N inf h∈A(d(fk, h) + d(Lnfk, Lnh)) = 0, dbeing defined by (1). Fix k∈Nand a function f∈Ak. There is an ε > 0 such that f(q)(z)f(q+1)(z)= 0 for all z∈B(k+2ε). From Theorem 2, there is a function g∈H(B(k+ 2ε)) such that fis given on B(k+ 2ε) by the formula given in that theorem. There exists a sequence of polynomials {Pm}∞ 1satisfying ||Pm−g||k+ε→0 (m→ ∞). With the notation of Theorem 2, we have f(z) = f(0) ·Tg(z) if q= 0 and f(z) = ∑q−1 ν=0 f(ν)(0) ν!zν+f(q)(0) (q−1)! ·Tg(z) if q≥1 for all z∈B(k+ 2ε). Let us define hm(z) = {f(0) ·TPm(z) if q= 0 ∑q−1 ν=0 f(ν)(0) ν!zν+f(q)(0) (q−1)! ·TPm(z) if q≥1 for all m∈Nand for all z∈C. Note that each hm∈A. From Theorem 2, TPm→T g (m→ ∞) uniformly on compact subsets of B(k+ε), so hm→f (m→ ∞) in the topology of H(B(k+ε)). By Theorem 4, Lnhm→Lnf(m→ ∞) uniformly on compact subsets of B(k+ε), for every n∈N. In particular, we obtain that limm→∞ ||Lnhm−Lnf||k= 0 ∀n∈N. Hence limn→∞(||hm−f||j+||Lnhm− Lnf||j) = 0 for every n∈Nand every j∈ {1,2, ..., k}. Given δ > 0, a positive integer m=m(δ, n, k) can be found in such a way that ||hm−f||j+||Lnhm−Lnf||j< δfor all j∈ {1, ..., k}, so d(f, hm) + d(Lnf, Lnhm)<∑k j=1 δ 2j+∑∞ j=k+1 1 2j+ ∑∞ j=k+1 1 2j=δ+ 21−k. Then infh∈A(d(f, h) + d(Lnf, Lnh)) < δ + 21−k∀δ > 0 and 6 ∀n∈N. Therefore we get sup n∈N inf h∈A(d(fk, h) + d(Lnfk, Lnh)) ≤21−k→0 (k→ ∞) if fk∈Ak(k∈N). Consequently, (2) is fulfilled and the proof is complete. //// Next, we show the optimality of our theorem. The result cannot be extended to all entire functions Φ of exponential type. In fact, to every type τ > 0 there exists an entire function Φ of order 1 and type τsuch that the result fails for Φ. One need only consider Φ(ζ) = eτζ. Then Φ(D)f(z) = ∑∞ n=0 f(n)(z) n!τn=f(z+τ), so that any Φ(D)-hypercyclic element is universal with respect to translates. But it follows from a classical theorem of Hurwitz (see, for instance, [1, p. 178]) that the translates of a zero-free entire function cannot approximate a non-constant entire function with zeros. Hence there cannot exist a Φ(D)-hypercyclic function. To finish, we point out that if q∈N0,{Φk:k∈N}is a sequence of nonconstant entire functions of subexponential type and Lk= Φk(D) (k∈N), then there is an entire function fwith zero-free derivatives of orders qand q+ 1 which is universal with respect to every family {Ln k:n∈N}(k∈N). This is a trivial consequence of the fact that in every Baire space the countable intersection of residual sets is residual. The author would like to thank the referee for helpful comments and suggestions. References [1] L.V. AHLFORS, Complex Variables (3rd ed.). McGraw-Hill, London 1979. [2] L. 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Apartado 1160 41080 Sevilla (S p a i n) E-mail: lb[email protected] 8