Universal functions with small derivatives and extremely fast growth

Analysis 22, 57 - 66 (2002) sis C Oldenbourg Verlag Manchen 2002 UNIVERSAL FUNCTIONS WITH SMALL DERIVATIVES AND EXTREMELY FAST GROWTH M. C. Calderón-Moreno1 Received: May 31, 2001 Abstract. We prove that if α (E (0, and Τ is an infinite order differential operator there exists a dense linear manifold M. of entire functions such that for every f e M and any plane strip S. Moreover, every non-null function in M exhibits some translation-universality property with respect to Τ and its growth index with respect to any prefixed sequence of non-constant entire functions is infinite. 2000 Mathematics Subject Classification: Primary 30D15. Secondary 30E10, 47A16, 47E05. 1 Introduction Throughout the last decades several authors have given "counterexamples" to the wellknown Liouville's theorem. For instance, there exists a non-null entire function which tends to zero on every line (see [11, 15, 16]) or such that it and even all its derivatives have also vanishing integrals on every line (see [2, 19]). In 1997 Bernal [5] (see also [4]) got many functions which not only "violated" Liouville's theorem in both senses but also possessed an extremely fast growth and "sharp" asymptotic behaviour at infinite. In order to specify exactly this result, and with it the framework of this paper, let us introduce some notation. The symbol Σ will stand for the family consisting of all strips in C (i.e., plane regions between two parallel straight lines) and all sectors 'The author has been partially supported by DGES Grant BFM2000-0514, DGES Grant BFM2001-2717 and the Junta de' Andalucía. lim exp(|z|a)T/(z) = 0 s0:={z: 0 < arg 2 < β} (/? <Ξ (0, 2ττ)) Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 1/4/17 9:12 AM 58 Calderón and L will stand for the family of all straight lines. We denote by H(C) the space of all entire functions endowed with the compact-open topology, so H{C) is a separable Fréchet space. If r > 0 and / G H(C) we denote Mf(r) := max{|/(z)| : |z| = r} and for any non-constant function h G H(C) the relative growth order of / with respect to h, (see [3]) is defined as ... log M^(Mf(r)) ph(f) = hrnsup , r—»oo log r Given any sequence Τ = {/ΐ·η}ϊ° of non-constant entire functions, the growth index of / with respect to Τ is ip(f) = min{rc G Ν : phn{î) < °°}· We set ip(f) = oo if Phn(f) = 00 f°r η. Observe that these concepts extend the older one of relative growth order [18]. With this in mind, Bemal's result [5, Theorem 3] reads as follows: THEOREM 1.1. (Bernal [5] j Assume that a e (0,1/2) and that φ : [0, +oo) (0, +oo) is a continuous function which is integrable on (l,+oo). Assum.e, in addition, that F = {Λ.η}^ is a sequence of nonconstant entire functions. Then there is a linear manifold Λ4 = Λ4(α, ψ, Τ) C H(C) satisfying the following seven properties: (a) M is dense in H{C). (b) lim exp(|z|3/V(z))/(z) = 0 for all S 6 Σ and all f e M. z€S (c) lim exp(|z|a)/ü)(2) = 0 for all S e Σ, all f e M and all j > 0. z€S (d) fW is bounded on S for all S G Σ, all f e Μ and all j > 0. (e) ftt) is integrable on S with, respect to the plane Lebesgue measure, for all S G Σ, all f e M and all j > 0. (f) f ^ integrable on S with respect to the length measure for all l G L, all f e M and all j > 0. (g) = 0 f°r al1 le L, all f £ M and all j > 1. (h.) ijr(f) = oo for all f G M\ {0}. Our aim in this paper is to show that not only the derivative operator of order j, D3 f = can be replaced by infinite order differential operators in the seven above properties, but also that we can provide an eighth property whose feature is totally different from the others, namely, a property about wild behaviour near the infinity point. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 1/4/17 9:12 AM Universal functions with fast growth 59 2 Definitions and statement of the main result An entire function φ(ζ) = akZk is said to be of exponential type if there are constants A,B>0 such that \φ(ζ)\ < AeB'2' for all ζ (E C. The function φ is of subexponential type if given ε > 0, then there is a constant A = Α(ε) > 0 such that \φ{ζ)\ < Aee^ for all ζ e C. This happens if and only if limsup^^fcllafcl)1/* = 0 (see, e.g., [8, 2.2.9- 11]). Each entire function of subexponential type is also of exponential type and every entire function φ of exponential type defines a (linear, continuous) infinite order differential operator φ{Ό) = Σ,Ζ,ο0·^ on #(c)> which is onto (see [10, 14]). Here D° = I = the identity operator. On the other hand, given a continuous selfmapping Τ on H(C), we say that an entire function / is T-universal whenever for each g € H(C) there exists a sequence (a,n) C C satisfying (Tf)(z + an)^g(z) (n —> oo) in H(C). Now, we are able to establish the main result, which will be proved in the next section. THEOREM 2.1. Let be given an α e (0, a continuous function ψ : [0, +oo) (0, +oo) which, is integrable on (1, +oo) and a sequence. Τ = {h.n}f of non-constant, entire functions. Assume, in addition, that, {ipi,m(z)}m=o fi ~ 1,2^ are two sequences of entire functions of subexponential type. Then there is a linear manifold Μ = Μ(α,φ,^,(φι:τη),(·ψ2,τη)) of entire functions satisfying the following properties: (a) M is dense in H(C). (b) lim exp(\z\3/2<p(z))f{z) = 0 for all S e Σ and all f e M. (c) lim exp(\z\a)(rphm(D)f)(z) = 0 for all S 6 Σ, all f e M and all m. > 0. (d) Ψ\,m(D)f /s bounded on S for all S 6 Σ, all f (ï M and ali m > 0. (e) iPi,m{D)f is integrable on S with, respect to the plane Lebesgue measure for all S e Σ, all f € A4 and all m > 0. (f) i>l,m(D)f is integrable on S with respect to the length. m.easure for all l G L, all f e M and all m, > 0. (9) ΙίΨιΑ0) = 0, then j^l¡m{D)fds = 0 for all I. € L and all f e M. (h) iAf) = 00 for all f e M \ {0}. Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 1/4/17 9:12 AM 60 Calderón (i) Every non-null function in M is ip2,m{D)-universal for all m > 0. Observe that any polynomial p(z) is an entire function of subexponential type. In particular, taking tpi¡m(z) = zm for all m > 0, we obtain the conditions (a)-(h) of Theorem 1.1 together with an additional universal property for a sequence of infinite order differential operators. It is noteworthy the case in which also ip2,m{z) = zm for all τη > 0. Then Theorem 2.1 provides a linear dense manifold of entire functions such that each of them and all its derivatives are universal functions in Birkhoff's sense [7] (see also [6, 12, 13] for the related concept of holomorphic monster in C) with growth conditions. Finally, we mention that in 2000 A. Bonilla [9] studied an analogous problem in the space of harmonic functions in RN, providing similar conditions (c)-(f) for any derivative operator Da and the universal condition (i) for the identity operator. 3 An auxiliary result and proof of the main result We will use the following theorem about tangential approximation due to Arakelian [1, p. 1189], From now on C^ is the extended plane. If F C C is a closed set, then A(F) is the space of all continuous functions on F which are holomorphic in the interior of F. A closed subset F C C is said to be an Arakelian set [17] whenever Coo \ F is both connected and locally connected at infinity. THEOREM 3.1. ('Arakelian [1],) Assume that F C C is an Arakelian set, and that e(t) is continuous and positive for t > 0. In addition, suppose that / oo r3/2log£(i)di > -00. (1) Then for every g € A(K) there exists an entire function f such, that, |/(*)-ί,(ζ)|<ε(|ζ|) (ζ e F). The statement does not remain valid for every F is (1) is violated. Proof, (of Theorem 2.1) Suppose that α, φ, Τ = and oo = (i = 1,2) k=0 are as in the hypotheses. Thus for each m > 0, i = 1,2, there exists a positive constant Ai¡m such that Ι&ΓΙ < aJ^- (Vfc > 0). (2) Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 1/4/17 9:12 AM Universal functions with fast growth 61 Fix a sequence {pn)S=i which is dense in H(C) and a number β e (α, For every η € Ν, the function e(t) = e„(t) := minji · γ^,βχρ(-Í3/VM - *")} is positive and continuous for t > 0 and satisfies (1). Let Ρ be the parabolic curve Ρ = {xix1'2 : a; > 0}. For each η € Ν, we define the sets En = {ze C : \z\>n+l and dist(z, Ρ) > 1 + |z|}, Bn = {z : Μ < η}· Consider a sequence of closed balls Dj = B(a,j, 1 + 2·') such that DjC{z : dist(z, Ρ) < 1 + |z|} \ Ρ and Kl + 2J+2 < |oj+i| (i>l). In particular the balls Dj are pajrwise disjoint and |z| > 2j (z € Dj). (3) Let H = {zk}kLi be a sequence of pairwise different complex numbers in Ρ with zk —» oo (k. —> oo). For each η e Ν we define Fn = Bn U En U H U ( (J Dj J , \j> jo / where jo = jo(n) is the largest index such that Dj Π Bn+\ φ 0. Then Fn is an Arakelian set. Divide {üj} into infinitely many disjoint subsequences {o.¿(m,/j)} by setting , [(m + l){m + I + 1) + 2j][(m + l)(m, + / + !) + 2(j + 1)] . i(m, i,j)— g + J for all m > 0, all / > 1 and all j > 1. Define inductively a sequence {/n}£Li of entire functions as follows. Denote rk = \zk\ for all k > 1 and for each m > 0 consider a sequence {lm,n}n °f entire functions such that 1p2,m{D)qm,n = Pn ("· > 1)· Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 1/4/17 9:12 AM 62 Calderón Recall that φ2,M(D) is an onto operator on H(C). Let g\ : F\ —» C denote the function Pi(-z) {zeB,) 0 (2 G £1) 51(2) = 1 + max^xfc Mhj(exprk) (z = zk and \zk\ > 1) qm,j(z — 1i(m,lj)) (Z e A(m,lj)) Then gj e and by Theorem 3.1 there exists an entire function fi such that \fl(z)-g1(z)\<£l(\z\) {z € Fi). Assume that rae {2,3,...} and that we have constructed 2 η — 2 functions /1,..., /„_ 1 in such a way that </¿ 6 A(FÌ), FI € #(C) and for all i 6 {1, 2,..., η — 1}. Now, we define the function gn : Fn —» C by Pn(z) (z € Bn) 0 (z e En) Sn(2) = 1 + maxi^-a Mhj{exprk) + fc^X"/ (2 = and M > "-) 0 (z G A(m,ij) I ¿ η). Trivially g„ € A(Fn) and by Theorem 3.1 there exists an entire function fn such that \fn(z)-gn(z)\<en(\z\) (zeFn). Thus, for all η e Ν, \f„(z) - pn{z)\ < - (ζ e ¿U \fn(z)\<exp(-\z\VMz)-\z\ß) (ζ e EN), \fn{z) - (1 + max Mh (exprk) + S(n,k))\ < - < 1 i<j<k η for all k such that \zk\ > n, where 5(1, k) = 0 and 5(n, k) = k ^"j/ Mfi(rk) if η > 2, I fn(z) - qm,j(z - Oi(m,nj))| < - · η 1 + \z\ for all z G A(m,nj), all j > 1 and all m > 0, and 1/nWI < Γ 1 η 1 + 1*1 (4) (5) (6) (7) (8) Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 1/4/17 9:12 AM Universal functions with fast growth 63 for all ζ e i3¿(m,¿j), all j > 1, all m > 0 and all I > 1 with Ι, φ η. Although we do not mention it explicitly, it is clear that we have (8) whenever D¿(m,í,;) ΠΒ„ = 0. By (4), the sequence is dense in //(C). Let us define M. as the linear span of {/n}n- Evidently, M. is a linear dense manifold of //(C): this proves (a). In order to verify that (b), (c) hold for every / 6 ΛΊ, it suffices to check that both properties are satisfied for every function / = fn. Prom (5), exp(|z|3^(*))l/»(*)l < exp(-M") (ζ^οο,ζε En). For any sector or strip S G Σ, we have that S\En is a bounded set, thus eM\z\3/Mz))fn(z) - 0 (z^oo,zeS). This proves (b). Now, we define the set E* as E*n = {z e C : \z\>n + 2 and dist(z, P) > 2 + |z|}. Then, one uses the Cauchy estimates and (5) to infer that l/fMl^lmaxilAHl: \w - z\ = 1} < < fc!max{exp(-M") : |ω| > \z\ - 1} < fc!exp(-(|z| - if) for all ζ e E* and all k. > 0 (remember that ψ is positive). Hence, for each m. > 0, |exp {\z\a)^m{D)fn{z)\ = < < expdzD · Alim ¿ M . fc!exp(-(M - 1)") < < 2Ai¡mexp(|z|a - (\z\ - 1)") ^ 0 (z-*oo,z€ E*n). If S e Σ, we have again that S\E* is bounded, so limejcp(|2|e^,m(£>)/n(z)) = 0, VeS which proves (c). The proofs of (d)-(f) and (h) are analogous to those we may find in [5]. In order to prove (g), we fix / 6 M. Suppose that Vi.m(0) = 0, then φ1 ¡m(D)f = £(ΣΓ=ι ^Ύ*"1'). and by the fundamental calculus theorem / OO 00 (A,m(D)f)ds = limC^b'rf^Hb)) - limC£bl'mf^(a)). b~it k= 1 aZl k=1 Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 1/4/17 9:12 AM 64 Calderón Now we have only to observe that the same way followed to obtain (c) leads us to "(c)" for T.ZibkmDik~l)· It remains only to get (i). Let be / G M and fix τη, > 0. Since every non-zero scalar multiple of a ^2,m(ö)-universal function is again ^2,m(ß)-universal, we may suppose that / = Σ]ζΐ ajfj with an = 1 and I = {ji,..., jT} finite. In order to prove that / is ip2,m(D)- universal it is enough to check that lim ((Tp2,m{D)f)(z + <lj(mji,n)) ~ Pn(*)) = 0 (9) uniformly on compact subsets. Fix η 6 N. We have \{ΦίΑϋ)!){ζ + Η^,ή,η)) ~PÁZ)\ = Σ otj • {rp2,m{D)fj){z + ai (.D)qm¡n(z) je/ < \ifo,m(D)(fh(z + ai(mJun)) - 9m,„(z))| + ^ \aj\\fj(z + ai{mJl¡n))\. iti ίΦή Now, for any ζ e ß(0,2n) C B{0,2¿(m^'n>) we have z + aiimj1¡n) e DiimJun), thus by (3) and (8), \fj(z + ai(m,jun))I < - · n p- < - (10) j (1 + |z + ai(mjlin)|) 2n for all j ξ. I with j φ . On the other hand, because of Cauchy's formula for derivatives applied to the curve 7 ξ {|ω| = 2n + ì}, we have by (7) that 1 I(/i, (2 + Ηη,ή,η)) - qmM)WI < Μ-ΐρ (k ^ °)· Therefore, by (2), life iD)(fj Áz + a (2))l < (1/2)* klJ_='^n J_ k\ ' 'il2" j! 2n < ΣΜ,πιk—0 (11) Joining (10) and (11) we obtain I(ip2,m(D)f)(z + ai{mJiìn})-pn(z)\< I ^ + / 2" ϊΦή for all ζ € B{0,2"). It is clear that this implies (9). Consequently, / is ^2,m(O)-universal and we have (i). • Brought to you by | Biblioteca de la Universidad de Sevilla Authenticated Download Date | 1/4/17 9:12 AM Universal functions with fast growth 65 References [1] N. U. Arakelian, Uniform approximation on closed sets by entire functions (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 28, 1187-1206 (1964). [2] D. H. Armitage, A non-constant continuous function on the plane whose integral on every line is zero, Amer.. Math. Monthly 101, 892-894 (1994). [3] L. Bernal-González, Orden relativo de crecimiento de funciones enteras, Collect. Math. 39, 209-229 (1988). [4] L. Bernal-González, A lot of "counterexamples" to Liouville's theorem, J. Math. Anal. Appi. 201, 1002-1009 (1996). [5] L. Bernal-González, Small entire functions with extremely fast growth, J. Math. Anal. Appi. 207, 541-548 (1997). [6] L. Bernal-González and M. C. Calderón-Moreno, Holomorphic T-monsters and strongly omnipresent operators, J. Approx. Theory 104, 204-219 (2000). [7] C. D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189, 473-475 (1929). [8] R. P. Boas, Entire Functions, Academic Press, New York, 1954. [9] A. 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