Small entire functions with extremely fast growth

TITLE: SMALL ENTIRE FUNCTIONS WITH EXTREMELY FAST GROWTH. AUTHOR: LUIS BERNAL-GONZ´ ALEZ. AFFILIATION: DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. FOOTNOTES TO THE TITLE: *This work is supported in part by DGICYT grant PB93-0926. A.M.S. Subject Classification: Primary 30D15. Secondary 30E10. Key words and phrases: Liouville’s theorem, entire functions, dense linear manifold, Arakelian set, strips and sectors, generalized order, growth index. 1 ABREVIATED TITLE: SMALL ENTIRE FUNCTIONS. NAME AND MAILING ADDRESS OF THE AUTHOR TO WHOM PROOFS SHOULD BE SENT: LUIS BERNAL-GONZ´ ALEZ. DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email protected]. 2 SMALL ENTIRE FUNCTIONS WITH EXTREMELY FAST GROWTH by LUIS BERNAL–GONZ´ ALEZ* Abstract. We prove in this note that, given α∈(0,1/2), there exists a linear manifold Mof entire functions satisfying that Mis dense in the space of all entire functions such that limz→∞ exp(|z|α)f(j)(z) = 0 on any plane strip for every f∈ M and for every derivation index j. Moreover, the growth index of each nonnull function of Mis infinite with respect to any prefixed sequence of nonconstant entire functions. 1. INTRODUCTION AND NOTATION Liouville’s theorem asserts that each bounded entire function is constant. Several authors have obtained nonconstant entire functions (nonbounded, of course) which are small on large plane subsets. For instance, Mittag-Leffler [8, pp. 290-294] *This work is supported in part by DGICYT grant PB93-0926. 1991 Mathematics Subject Classification: Primary 30D15. Secondary 30E10. Key words and phrases: Liouville’s theorem, entire functions, dense linear manifold, Arakelian set, strips and sectors, generalized order, growth index. 3 had already constructed in 1920 a nonnull entire function tending to zero on every line through the origin (therefore, this function is bounded on every ray from the origin). Two easy explicit examples of this kind of functions can be found in [9] and [12, chap. 16]. In [6] it is shown the existence of a nonconstant entire function having limit zero on every line. This is achieved by using a result of A. Roth about approximation on closed sets. But much more sophisticated “anti-Liouville” functions have been made. In 1982, L. Zalcman [14] constructed a nonconstant entire function tending to zero on every line and such that ∫l f ds = 0 for every l∈L. We have denoted by Lthe set of all straight lines in the plane and by sthe length measure. For this, Zalcman used a result (Theorem 1 below) due to Arakelian concerning tangential approximation. By an elementary pole-pushing technique, D. H. Armitage [2] has recently constructed a nonconstant entire function fsuch that each derivative f(n)is integrable on every line lwith respect to sand ∫l f(n)ds = 0 4 for every n∈N0, where N0=N∪ {0}and Nis the set of positive integers. Moreover, fsatisfies lim z→∞ z∈l f(n)(z) = 0 for every n∈N0and every line l. In [4] the author improves all above assertions and provides a dense linear manifold of entire functions “violating” Liouville’s theorem. By using Phragmen–Lindelof’s theorem (see, for instance, [7, pp. 125-126]), it is proved in [4] that all nonnull functions in this manifold have infinite growth order. There it is even found an anti-Liouville function having extremely fast growth, that is, an entire function fwith infinite growth index i(f). In this note we improve in turn the results in [4] and furnish a dense linear manifold Mall of whose functions satisfy the properties stated in [4] and, in addition, the growth index of every nunnull function of the manifold is infinite. In fact, we can get for the growth of each f∈ M \ {0}to be extremely fast with respect to any prefixed sequence of nonconstant entire functions. Moreover, the asymptotical behaviour when z→ ∞ for all functions f∈ M is sharpened by considering a positive, continuous function φwhich is integrable on (1,+∞) (compare Theorem 5 2 in [4] for j= 0 with part b) of Theorem 3 in this paper). We will need a bit of notation. If r > 0, Brdenotes the closed ball {z:|z| ≤ r}. Astrip is the plane region lying between two parallel straight lines. If β∈(0,2π), the sector sβis the set sβ={z: 0 ≤arg z≤β}. Σ will stand for the family consisting of all strips in Cand all sectors sβ(β∈(0,2π)). H(C) is the space of all entire functions, endowed with the compact-open topology. H(C) is a separable Fr´echet space. The extended plane C∞is the one-point compactification of C. If F⊂Cis 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⊂Cis said to be an Arakelian set [11] whenever C∞\Fis both connected and locally connected at infinity. Finally, let fbe an entire function. If r > 0, we define exp1r= exp r, expm+1 r= exp(expmr) (m∈N), and Mf(r) = max{|f(z)|:|z|=r}. For r > 0 large enough, we denote log1r= log r, logm+1 r= log(logmr) (m∈N). The m-order growth ρm=ρm(f) of f(see [10] and [13]) is defined to be ρm= lim sup r→∞ logm+1 Mf(r) log r. 6 Note that ρ1is the ordinary growth order of f. The growth index of fis i(f) = min{m∈N:ρm(f)<∞}, and we set i(f) = ∞when ρm(f) = ∞for all m. We can generalize these definitions (see [3]) as follows. Let hbe a nonconstant entire function. The relative growth order of fwith respect to his ρh(f) = lim sup r→∞ log M−1 h(Mf(r)) log r. Let F={hm}∞ 1be any sequence of nonconstant entire functions. We define the growth index of fwith respect to Fas iF(f) = min{m∈N:ρhm(f)<∞}, and we set iF(f) = ∞if ρhm(f) = ∞for all m. Note that ρm=ρexpmand i(f) = iF(f) for F={expm}∞ 1. 1. PRELIMINARY RESULTS We will use the following theorem due to Arakelian. For the proof and comments, see [1, p. 1189]and [5, pp. 160-162]. THEOREM 1. Assume that F⊂Cis an Arakelian set and that ε(t)is con- 7 tinuous and positive for t≥0. In addition, suppose that ∫∞ 1 t−3/2log ε(t)dt > −∞.(1) Then for every g∈A(F)there exists an entire function fsuch that |f(z)−g(z)|< ε(|z|)∀z∈F. The statement does not remain valid for every Fif (1) is violated. Note that, for instance, ε(t) = exp(−t1/2) does not satisfy (1), but ε(t) = exp(−tα) does for α < 1/2. Moreover, we will need an elementary lemma about the maximum modulus. LEMMA 2. If his a nonconstant entire function and a, b are positive constants with a > max{1, b}, then there is r0>0such that b Mh(r)< Mh(ar)∀r > r0. PROOF. We obtain |g(z)|≤|z|Mg(R)/R if |z| ≤ Rby applying Schwarz’s Lemma to g(z) = h(z)−h(0). If R=ar, then Mh(r)− |h(0)| ≤ Mg(r)≤ (r/R)(Mh(R) + |h(0)|)=(Mh(ar) + |h(0)|)/a. Choose ε=a−b 2a+2b. Then there 8 is r0>0 with |h(0)|< εMh(r0). Thus Mh(ar)≥(1−ε)a 1+εMh(r)> b Mh(r) for all r > r0. //// 3. THE MAIN THEOREM We are now ready to state our results. They are collected in the following theorem. Note that it generalizes Theorems 2,3,4 in [4]. THEOREM 3. Assume that α∈(0,1/2) and that φ: [0,+∞)→(0,+∞)is a continuous function which is integrable on (1,+∞). Assume, in addition, that F={hm}∞ 1is a sequence of nonconstant entire functions. Then there is a linear manifold M=M(α, φ, F)⊂H(C)satisfying the following seven properties: a) Mis dense in H(C). b) lim z→∞ z∈S exp(|z|3/2φ(z))f(z) = 0 ∀S∈Σand ∀f∈ M. c) lim z→∞ z∈S exp(|z|α)f(j)(z) = 0 ∀S∈Σ,∀f∈ M and ∀j∈N0. d) f(j)is bounded on S∀S∈Σ,∀f∈ M and ∀j∈N0. e) f(j)is integrable on Swith respect to plane Lebesgue measure ∀S∈Σ,∀f∈ M and ∀j∈N0. 9 3. L. BERNAL-GONZ´ ALEZ, Orden relativo de crecimiento de funciones enteras, Collect. Math. 39 (1988), 209-229. 4. L. BERNAL-GONZ´ ALEZ, A lot of “counterexamples” to Liouville’s theorem. To appear in J. Math. Anal. Appl. 5. D. GAIER, Lectures on complex approximation, Birkh¨auser, Boston, 1987. 6. P.M. GAUTHIER, Uniform approximation, 1-37, in Complex Potential Theory, Kluwer Academic Publishers, 1994. 7. A.S.B. HOLLAND, Introduction to the theory of entire functions, Academic Press, New York and London, 1973. 8. G. MITTAG-LEFFLER, Sur la repr´esentation analytique d’une branche uniforme d’une fonction monog`ene (Sixi`eme note), Acta Math. 42 (1920), 285-308. 9. D.J. NEWMAN, An entire function bounded in every direction, Amer. Math. Monthly 83 (1976), 192-193. 10. A.R. REDDY, On entire Dirichlet series of infinite order, Rev. Mat. Hisp.- Amer. 27 (1967), 120-131. 11. J.P. ROSAY and W. RUDIN, Arakelian’s approximation theorem, Amer. Math. Monthly 96 (1989), 432-434. 12. W. RUDIN, Real and Complex Analysis, Tata McGraw-Hill, 2nd. edition, Faridabad, India, 1974. 13. D. SATO, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc. 69 (1963), 410-414. 16 14. L. ZALCMAN, Uniqueness and nonuniqueness for the Radon transform, Bull. London Math. Soc. 14 (1982), 241-245. DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO FACULTAD DE MATEM´ ATICAS AVENIDA REINA MERCEDES APARTADO 1160 41080 SEVILLA, SPAIN E-MAIL: lb[email protected] 17