We prove in this note that, given α ∈ (0, 1/2), there exists a linear manifold M of entire functions satisfying that M is 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 M is infinite with respect to any prefixed sequence of nonconstant entire functions.Dirección General de Investigación Científica y Técnica (DGICYT). Españ

Bernal González, Luis

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idUS. Depósito de Investigación Universidad de Sevilla

TITLE: SMALL ENTIRE FUNCTIONS WITH EXTREMELY FAST GROWTH.AUTHOR: LUIS BERNAL-GONZALEZ.AFFILIATION: DEPARTAMENTO DE ANALISIS MATEMATICO. FACUL-TAD DE MATEMATICAS. AVENIDA REINA MERCEDES. APARTADO1160. 41080 SEVILLA, SPAIN. E-MAIL: lbernal@cica.es.FOOTNOTES TO THE TITLE: *This work is supported in part by DGICYTgrant PB93-0926.A.M.S. Subject Classication: 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.1ABREVIATED TITLE: SMALL ENTIRE FUNCTIONS.NAME AND MAILING ADDRESS OF THE AUTHOR TO WHOMPROOFS SHOULD BE SENT: LUIS BERNAL-GONZALEZ. DEPARTA-MENTO DE ANALISIS MATEMATICO. FACULTAD DE MATEMATICAS.AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN.E-MAIL: lbernal@cica.es.2SMALL ENTIRE FUNCTIONS WITH EXTREMELY FAST GROWTHbyLUIS BERNAL{GONZALEZ*Abstract. We prove in this note that, given  2 (0; 1=2), thereexists a linear manifold M of entire functions satisfying that M isdense in the space of all entire functions such thatlimz!1 exp(jzj)f (j)(z) = 0 on any plane strip for every f 2 M andfor every derivation index j. Moreover, the growth index of each non-null function of M is innite with respect to any prexed sequence ofnonconstant entire functions.1. INTRODUCTION AND NOTATIONLiouville's theorem asserts that each bounded entire function is constant. Se-veral authors have obtained nonconstant entire functions (nonbounded, of course)which are small on large plane subsets. For instance, Mittag-Leer [8, pp. 290-294]*This work is supported in part by DGICYT grant PB93-0926.1991 Mathematics Subject Classication: Primary 30D15. Secondary 30E10.Key words and phrases: Liouville's theorem, entire functions, dense linear ma-nifold, Arakelian set, strips and sectors, generalized order, growth index.3had already constructed in 1920 a nonnull entire function tending to zero on everyline through the origin (therefore, this function is bounded on every ray from theorigin). 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 functionhaving limit zero on every line. This is achieved by using a result of A. Roth aboutapproximation on closed sets.But much more sophisticated \anti-Liouville" functions have been made. In1982, L. Zalcman [14] constructed a nonconstant entire function tending to zero onevery line and such that Zlf ds = 0for every l 2 L. We have denoted by L the set of all straight lines in the plane andby s the length measure. For this, Zalcman used a result (Theorem 1 below) dueto Arakelian concerning tangential approximation. By an elementary pole-pushingtechnique, D. H. Armitage [2] has recently constructed a nonconstant entire functionf such that each derivative f (n) is integrable on every line l with respect to s andZlf (n) ds = 04for every n 2 N0, where N0 = N [ f0g and N is the set of positive integers.Moreover, f satiseslimz!1z2lf (n)(z) = 0for every n 2 N0 and every line l. In [4] the author improves all above assertions andprovides 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 isproved in [4] that all nonnull functions in this manifold have innite growth order.There it is even found an anti-Liouville function having extremely fast growth, thatis, an entire function f with innite growth index i(f).In this note we improve in turn the results in [4] and furnish a dense linearmanifold M all of whose functions satisfy the properties stated in [4] and, in addi-tion, the growth index of every nunnull function of the manifold is innite. In fact,we can get for the growth of each f 2Mn f0g to be extremely fast with respect toany prexed sequence of nonconstant entire functions. Moreover, the asymptoticalbehaviour when z ! 1 for all functions f 2 M is sharpened by considering apositive, continuous function ' which is integrable on (1;+1) (compare Theorem52 in [4] for j = 0 with part b) of Theorem 3 in this paper).We will need a bit of notation. If r > 0, Br denotes the closed ball fz : jzj  rg.A strip is the plane region lying between two parallel straight lines. If  2 (0; 2),the sector s is the set s = fz : 0  arg z  g.  will stand for the familyconsisting of all strips in C and all sectors s ( 2 (0; 2)). H(C) is the space ofall entire functions, endowed with the compact-open topology. H(C) is a separableFrechet space. The extended plane C1 is the one-point compactication of C. IfF  C is a closed set, then A(F ) is the space of all continuous functions on Fwhich are holomorphic in the interior of F . A closed subset F  C is said to bean Arakelian set [11] whenever C1 nF is both connected and locally connected atinnity.Finally, let f be an entire function. If r > 0, we dene exp1 r = exp r,expm+1 r = exp(expm r) (m 2 N), and Mf (r) = maxfjf(z)j : jzj = rg. Forr > 0 large enough, we denote log1 r = log r, logm+1 r = log(logm r) (m 2 N). Them-order growth m = m(f) of f (see [10] and [13]) is dened to bem = lim supr!1logm+1Mf (r)log r:6Note that 1 is the ordinary growth order of f . The growth index of f is i(f) =minfm 2 N : m(f) < 1g, and we set i(f) = 1 when m(f) = 1 for all m. Wecan generalize these denitions (see [3]) as follows. Let h be a nonconstant entirefunction. The relative growth order of f with respect to h ish(f) = lim supr!1logM 1h (Mf (r))log r:Let F = fhmg11 be any sequence of nonconstant entire functions. We dene thegrowth index of f with respect to F as iF (f) = minfm 2 N : hm(f) <1g, and weset iF (f) = 1 if hm(f) = 1 for all m. Note that m = expm and i(f) = iF (f)for F = fexpmg11 .1. PRELIMINARY RESULTSWe will use the following theorem due to Arakelian. For the proof and com-ments, see [1, p. 1189] and [5, pp. 160-162].THEOREM 1. Assume that F  C is an Arakelian set and that "(t) is con-7tinuous and positive for t  0. In addition, suppose thatZ 11t 3=2 log "(t) dt >  1: (1)Then for every g 2 A(F ) there exists an entire function f such thatjf(z)  g(z)j < "(jzj) 8z 2 F:The statement does not remain valid for every F if (1) is violated.Note that, for instance, "(t) = exp( t1=2) does not satisfy (1), but "(t) =exp( t) does for  < 1=2.Moreover, we will need an elementary lemma about the maximum modulus.LEMMA 2. If h is a nonconstant entire function and a; b are positive constantswith a > maxf1; bg, then there is r0 > 0 such thatbMh(r) < Mh(ar) 8r > r0:PROOF. We obtain jg(z)j  jzjMg(R)=R if jzj  R by applying Schwarz'sLemma to g(z) = h(z)   h(0). If R = ar, then Mh(r)   jh(0)j  Mg(r) (r=R)(Mh(R) + jh(0)j) = (Mh(ar) + jh(0)j)=a. Choose " = a b2a+2b . Then there8is r0 > 0 with jh(0)j < "Mh(r0). Thus Mh(ar)  (1 ")a1+" Mh(r) > bMh(r) for allr > r0. ////3. THE MAIN THEOREMWe are now ready to state our results. They are collected in the followingtheorem. Note that it generalizes Theorems 2,3,4 in [4].THEOREM 3. Assume that  2 (0; 1=2) and that ' : [0;+1) ! (0;+1) isa continuous function which is integrable on (1;+1). Assume, in addition, thatF = fhmg11 is a sequence of nonconstant entire functions. Then there is a linearmanifold M =M(; ';F)  H(C) satisfying the following seven properties:a) M is dense in H(C).b) limz!1z2Sexp(jzj3=2'(z))f(z) = 0 8S 2  and 8f 2M.c) limz!1z2Sexp(jzj)f (j)(z) = 0 8S 2 , 8f 2M and 8j 2 N0.d) f (j) is bounded on S 8S 2 , 8f 2M and 8j 2 N0.e) f (j) is integrable on S with respect to plane Lebesgue measure 8S 2 , 8f 2 Mand 8j 2 N0.9f) f (j) is integrable on l with respect to length measure 8l 2 L, 8f 2 M and8j 2 N0.g)Zlf (j) ds = 0 8l 2 L, 8f 2M and 8j 2 N.h) iF (f) =1 8f 2M n f0g.PROOF. Fix a dense sequence fPng11 in H(C),  2 (0; 1=2),  2 (; 1=2) anda function ' and a family F = fhmg11 as in the hypothesis. For every n 2 N, thefunction"(t) = "n(t)  minf1=n; exp( t3=2'(t)  t)gis positive and continuous for t  0 and satises (1). Let P be the parabolic curveP = fx  ix1=2 : x  0g:Fix any denumerable set H = fzk : k 2 Ng  fz 2 C : dist (z; P ) < 1g satisfyinglimz!1 zk =1. H does not depend upon n. Let Fn = Bn [ En [H, whereEn = fz 2 C : jzj  n+ 1 and dist(z; P )  1g:Then Fn is closed and C1 n Fn is connected and locally connected at innity, soFn is an Arakelian set. Dene inductively a sequence ffng11 of entire functions as10follows. Denote rk = jzkj for all k. Let g1 : F1 ! C denote the functiong1(z) =8<:P1(z) if z 2 B10 if z 2 E11 + max1jkMhj (exp rk) if z = zk and jzkj > 1.Since B1 \E1 = H \E1 = ;, g1 is well-dened. Trivially, g1 2 A(F1). By Theorem1, there exists an entire function f1 such thatjf1(z)  g1(z)j < "1(jzj) 8z 2 F1:Assume that n 2 f2; 3; :::g and that we have constructed 2n   2 functions g1; f1;g2; f2; :::; gn 1; fn 1 in such a way that gi 2 A(Fi), fi 2 H(C) and jfi(z)  gi(z)j <"i(jzj) 8z 2 Fi, 8i 2 f1; 2; :::; n  1g. Now, dene the function gn : Fn ! C bygn(z) =8<:Pn(z) if z 2 Bn0 if z 2 En1 + max1jkMhj (exp rk) + kPn 1i=1 Mfi(rk) if z = zk and jzkj > n.It is trivial that gn 2 A(Fn). By Theorem 1, there exists an entire function fn suchthatjfn(z)  gn(z)j < "n(jzj) 8z 2 Fn:Thus, for all n 2 N,jfn(z)  Pn(z)j < 1=n 8z 2 Bn; (2)11jfn(z)j < exp( jzj3=2'(jzj)  jzj) 8z 2 En (3)andjfn(zk)  (1 + max1jkMhj (exp rk) + S(n; k))j < 1=n  1 (4)for all k such that jzkj > n, where S(n; k) = 0 if n = 1 and S(n; k) =kPn 1i=1 Mfi(rk) if n  2.Let G 2 H(C) and K  C be a compact set. There exists a sequence of naturalnumbers n1 < n2 < ::: < nk < ::: such that Pnk ! G (k ! 1) uniformly on K.Moreover, there is k0 2 N satisfying K  Bnk whenever k > k0. Then, by using(2), we have that fnk ! G (k !1) uniformly on K. Thus the sequence ffng11 isdense in H(C).Let us dene M as the linear span of ffng11 . Evidently, M is a linear densemanifold of H(C): this is a). In order to verify that b), c) hold for every f 2 M,it suces to check that these two properties are satised for every function f = fn.We have from (3) thatexp(jzj3=2  '(z))jfn(z)j < exp( jzj)! 0 (z !1; z 2 En):12Given a sector or a strip S 2 , we have that SnEn is a bounded set. Consequently,exp(jzj3=2  '(z))jfn(z)j ! 0 (z !1; z 2 S):This proves b).Dene En asEn = fz 2 C : jzj  n+ 2 and dist(z; P )  2g:Observe that fw : jw   zj  1g  En for all z 2 En. By (3) and Cauchy'sinequalities,jf (j)n (z)j  (j!=1j)maxfjfn(w)j : jw   zj = 1g j! maxfexp( jwj ) : jwj  jzj   1g  j! exp( (jzj   1))for all z 2 En and all j 2 N0 (we have used that ' is positive). Hencej exp(jzj)f (j)n (z)j  j! exp(jzj   (jzj   1))! 0 (z !1; z 2 En):If S 2 , we have again that S nEn is bounded, so limz!1z2Sexp(jzj)f (j)n (z) = 0, whichproves c).13The proof of d), e), f), g) is very easy. Indeed, x f 2 M, S 2 , j 2 N0 andl 2 L. Property d) is inmediate from c). e) and f) are straightforward because, byc), there is R > 0 such thatjf (j)(z)j < exp( jzj) 8z 2 (S [ l) nBRand exp( jzj) is integrable on C with respect to plane Lebesgue measure andintegrable on any line with respect to length measure. In order to prove g), xj 2 N. Then, from the fundamental calculus theorem and c), we haveZlf (j) ds = limb!1b2lf (j 1)(b)  lima!1a2lf (j 1)(a) = 0  0 = 0;as required.As for h), x m 2 N and f 2 M n f0g. Then there exists p 2 N and complexconstants a1; a2; : : : ; ap such that f = a1f1+:::+apfp and ap 6= 0. Take k1 2 N withk1 > maxfm; ja1=apj; :::; jap 1=apjg such that rk > p 8k > k1. Inequality (4) im-plies that, for all k > k1,Mfp(rk)  jfp(zk)j > max1jkMhk(exp rk)+k(Mf1(rk)+::: +Mfp 1(rk)) > Mhm(exp rk) + ja1=apjMf1(rk) + ::: + jap 1=apjMfp 1(rk). Byusing the triangle inequality, we getMf (r) Mapfp(r) Ma1f1(r) Ma2f2(r)  ::: Map 1fp 1(r) 8r > 0:14But Majfj (r) = jaj jMfj (r) for all r > 0, so Mf (rk)  japj(Mfp(rk)  ja1=apjMf1(rk)   :::   jap 1=apjMfp 1(rk)) > japj  Mhm(exp rk) for all k > k1.We now apply Lemma 2 with a = 1 + 1japj , b =1japj and r =japj exp rkjapj+1 . Then thereis k2 2 N such that Mhm( japj exp rkjapj+1 ) < japjMhm(exp rk) for all k > k2. It is obviousthat there exists k3 2 N satisfying japj exp rkjapj+1 > exp(rk=2) for all k > k3. If we setk0 = maxfk1; k2; k3g, then Mf (rk) > Mhm(exp(rk=2)) for all k > k0, because themaximum modulus is increasing. Thenhm(f) = lim supr!1logM 1hm(Mf (r))log r lim supk!1logM 1hm(Mf (rk))log rk lim supk!1log exp(rk=2)log rk= lim supk!1rk2 log rk=1:Thus hm(f) =1. But this holds for all m 2 N. Thus iF (f) =1 8f 2 M n f0g,as required. ////REFERENCES1. N.U. ARAKELIAN, Uniform approximation on closed sets by entire functions,Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1187-1206.2. D.H. ARMITAGE, A non-constant continuous function on the plane whoseintegral on every line is zero, Amer. Math. Monthly 101 (1994), 892-894.153. 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, AcademicPress, New York and London, 1973.8. G. MITTAG-LEFFLER, Sur la representation analytique d'une branche uni-forme 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 innite order, Rev. Mat. Hisp.-Amer. 27 (1967), 120-131.11. J.P. ROSAY andW. 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), 410-414.1614. L. ZALCMAN, Uniqueness and nonuniqueness for the Radon transform, Bull.London Math. Soc. 14 (1982), 241-245.DEPARTAMENTO DE ANALISIS MATEMATICOFACULTAD DE MATEMATICASAVENIDA REINA MERCEDESAPARTADO 116041080 SEVILLA, SPAINE-MAIL: lbernal@cica.es17

Small entire functions with extremely fast growth

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Small entire functions with extremely fast growth

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