A question of Gol dbeg concening entie functions with pescibed zeos Walte Begweile Abstact Let (z j ) be a sequence of coplex nubes satisfying z j as j and denote by n() the nube of z j satisfying z j. Suppose that li inf log n()/ log > 0. Let φ be a positive, non-deceasing function satisfying (φ(t)t log t) <. It is poved that thee exists an entie function f whose zeos ae the z j such that log log M(, f) o((log n()) 2 φ(log n())) as outside soe exceptional set of finite logaithic easue, and that the integal condition on φ is best possible hee. These esults answe a question by A. A. Gol dbeg. Intoduction and esults Let (z j ) be a sequence of coplex nubes satisfying li z j. () j A classical theoe of Weiestaß says that thee exists an entie function f whose zeos ae the z j, with ultiplicities taken into account. Many authos have studied the poble to choose this function f such that thee is a good bound fo its axiu odulus M(, f) in tes of the nube n() of values z j satisfying z j, see [] fo efeences. A. A. Gol dbeg [6, Theoe 6] poved that if and if ε > 0, then we ay achieve 0 < li inf log n() log (2) log log M(, f) o((log n()) 2+ε ) ( F ) (3) as, whee F has finite logaithic easue. He noted that the ight side of (3) cannot be eplaced by O(log n()), but he asked whethe it is possible to eplace the nube 2 + ε in (3) by + ε. We shall show that this is not the case, although (3) can be ipoved slightly. Moe pecisely, we have the following esults. Theoe Let φ(t) be positive, non-deceasing function satisfying φ(t)t log t < and let (z j ) be a sequence of coplex nubes satisfying () and (2). Then thee exists an entie function f whose zeos ae the z j and a set F [, ) of finite logaithic easue such that log log M(, f) o((log n()) 2 φ(log n())) ( F ). (4)
Theoe 2 Let φ(t) be positive, non-deceasing function satisfying and φ(t)t log t φ(t) t β (5) fo soe β > 0 and all lage t. If 0 < α <, then thee exists a sequence (z j ) satisfying () and log n() li inf α (6) log with the following popety: if f is an entie function with zeos z j, then thee exists a set F [, ) of infinite logaithic easue such that (log n()) 2 φ(log n()) o(log log M(, f)) ( F ). (7) The condition (5) is pobably not necessay in Theoe 2. On the othe hand, since the inteesting choices fo φ(t) ae functions like φ(t) o φ(t) log log t, the hypothesis (5) sees to be a ild estiction fo the applicability of Theoe 2. The choice φ(t) shows that we cannot eplace 2 + ε by 2 in (3). Gol dbeg [6, Theoe 6] has shown that we can eplace n() by N() 0 n(t) n(0) + n(0) log t in (3). Siilaly, we can eplace n() by N() in Theoes and 2. As fa as Theoe is concened, this follows fo the inequality n() N() 2 ( F ) fo soe set F of finite logaithic easue. This inequality was also used (and poved) by Gol dbeg [6, p. 4]. Since N() n() log + O() n() 2 fo lage by (2), Theoe 2 also holds with n() eplaced by N(). Finally we note that soe hypothesis like (2) is necessay fo the validity of (3) o (4). In fact, Gol dbeg [6, Theoe 2] poved that if ψ(t) is any given inceasing function, then thee exists a sequence (z j ) satisfying () such that log M(, f) ψ(n()) ( F ) fo all entie functions f with zeos z j, whee F has uppe logaithic density one. 2 Poof of Theoe Fist we note that thee exists a positive, non-deceasing function ψ(t) satisfying ψ(t)t log t < 2
and Fo exaple, if we take ψ(t) li t φ(t) ψ(t) sup φ(s) s t then ψ(t) is non-deceasing, (8) is satisfied, and s 0. (8) dτ φ(τ)τ log τ, ψ(t)t log t φ(t)t log t t dτ φ(τ)τ log τ 2 φ(t)t log t <. We ay assue that 0 < z z 2 z 3.... We define p j [2ψ(log j) log j] and ( ) z f(z) E, p j z j j whee E(z/z j, p j ) is the usual Weiestaß piay facto. It is well-known that f is an entie function with zeos z j. Bluenthal [2, pp. 3-36] poved that log E(u, p) u p+ fo p. Clealy, this also holds if p 0. (We eak that Bluenthal s estiate can also be obtained fo the esults of Denjoy [4, pp. 8-24] and Cohn [3] if p >, and the case p is not difficult.) It follows that ( ) pj + log M(, f). z j Fo sufficiently lage, we define Then log M(, f) z j j ( ) exp. ψ(log n()) ( ) pj + + ( ) pj + + ( ) pj + S + S 2 + S 3. (9) z j < z j z j z j z > j Now log S ( ) pn() + log n() z (0) log n() + ( p n() + ) log z (2 + o())ψ(log n()) log n() log () o((log n()) 2 φ(log n())) by ou choice of p j, (2), and (8). To estiate S 2, we use a well-known gowth lea essentially due to Boel and efined by Nevanlinna [9, p. 40]. Applying it to the functions u() log n(e ) and ϕ(t) /ψ(t) and choosing ε we deduce that log n( ) (log n()) 2 ( F ) (2) 3
fo soe set F of finite logaithic easue. We eak that the book by Gol dbeg and Ostovskii contains the lea of Boel and Nevanlinna in a slightly diffeent, but equivalent fo [7, p. 20]. In fact, Nevanlinna fist poved his esult in the fo stated by Gol dbeg and Ostovskii and then ade a suitable substitution [9, p. 42-43]. To deduce (2) fo the vesion of the Boel-Nevanlinna lea given in [7], we have to choose u() log log log n(e )/ log 2. This vesion of the lea was also used by Gol dbeg [6], but he applied it to the function u() log log n(e ) and chose ϕ(u) e δu. It follows fo (2) that log S 2 log n( ) o((log n()) 2 φ(log n())) ( F ). (3) Finally, we have ( ( )) pj S 3 exp j 2 (4) ψ(log n()) jn( )+ jn( )+ fo sufficiently lage. Cobining (9), (), (3), and (4) we obtain (4). We eak that the above poof essentially follows the aguent of Gol dbeg [6], the diffeence being the way in which the lea of Boel and Nevanlinna is applied. 3 Poof of Theoe 2 Siilaly as in the poof of Theoe, thee exists a positive, non-deceasing function ψ(t) satisfying, (5) ψ(t)t log t fo soe γ > 0 and all lage t, and ψ(t) t γ (6) ψ(t) li t φ(t). (7) With α as in the stateent of the theoe, we take > so lage that the sequence ( j ) defined ecusively by log j+ (log j ) 2 ψ(α log j ) (8) is inceasing and tends to. We define l 0 0 and l j [( j+ ) α ] fo j. Clealy, l j /l j 0 as j. Next we define z k j fo l j + k l j and j. We deduce that if j < j+, then n() l j ( + o())( j+ ) α (9) as j. Thus (6) holds. Let now f be an entie function whose zeos ae the z j. By c () we denote the th Fouie coefficient of log f(e iθ ). Fist we show that c () log M(, f) (20) 4
fo and sufficiently lage. Indeed, c () c () e iϕ, so ( c () Re e iϕ 2π ) log f(e iθ ) e iθ dθ 2π log f(e iθ ) cos(θ + ϕ)dθ 2π log f(e iθ ) ( + cos(θ + ϕ))dθ 2π log M(, f) 2π ( + cos(θ + ϕ))dθ log M(, f) N() log f(0) 2π 0 2π log f(e iθ ) dθ log f(e iθ ) dθ by Jensen s foula, and (20) follows. It is well-known (see [5, p. 32], [8, p. 379], o [0, Lea ]) that if log f(z) 0 a z nea z 0, then c () 2 a + 2 fo. It follows that if j < j+, then c () 2 a + 2 We deduce that if j < < j < R < j+, then ( ) R c (R) c () (( ) R l j l j 2 We now define j by j z k (( z k ) ( ) zk ) (( ) j (l µ l µ ) µ µ ( ) ) j j R 2 log j j µ (l µ l µ ) ( (µ ( ) ) µ. ) ( ) µ R ). R 2 2ψ(log l j ). (2) Since ψ(t) as t it follows that j < j+ < j+ fo lage j. Moeove, j+/ j and j / j eains bounded as j. Hence ( ) R ( ) ( ) c (R) c () 2 + o() lj R (22) if j < < j and j+ < R < j+. Now we define j [ψ(log l j ) log l j ]. (23) Fo (9) we deduce that ( ( ) ) lj R log j log l j log + log R j ( o())ψ(log l j ) log l j log j+ (24) ( ) α o() (log n(r)) 2 ψ(log n(r)) 5
fo j+ < R < j+. Suppose now that thee exists R satisfying j+ < R < j+ and log log M(R, f) 2α (log n(r))2 ψ(log n(r)). (25) Fo (20), (22), (24), and (25) we can deduce that ( R ) ( ) ( ) c () 2 o() lj R fo j < < j, that is, ( ) ( c () 2 o() lj j fo j < < j. Cobining this with (6), (8), (9), (20), (2), and (23) we find that log log M(, f) log c () j j ) log l j log + log j j O() fo j < < j. We now define F to be the set of all satisfying log l j log ψ(log l j ) log log l j 2 log l j O() ( ) 2 o() log l j ( ) α 2 o() log j+ ( ) α 2 o() (log j ) 2 ψ(α log j ) ( ) 2α o() (log n()) 2 ψ(log n()) log log M(, f) 3α (log n())2 ψ(log n()). Fo (7) we deduce that (7) is satisfied, and the aguent given above shows that if j is lage enough, then one of the intevals ( 2j, 2j ) and ( 2j, 2j ) is contained in F. Since this iplies that log 2j 2j Fo (6) and (8) we deduce that 2ψ(log l 2j ) 2ψ(log l 2j ) log 2j F ( 2j 2, 2j ) t 2ψ(log l 2j ). log j+ (log j ) 2 (α log j ) γ (log j ) γ+3 2j, 6
fo sufficiently lage j. We define b γ + 3. Induction shows that thee exists a > 0 such that Hence so that log j+ exp(ab j ). log l j α log j+ α exp(ab j ) F ( 2j 2, 2j ) t 2ψ(α exp(ab 2j )) fo sufficiently lage j, say j j 0. Putting 0 2j0 2 we deduce that F ( 0, ) t 2 jj 0 ψ(α exp(ab 2j )) 2 j 0 ψ(α exp(ab 2t )). Using the substitution u α exp(ab 2t ) one easily deives fo (5) that the integal on the ight side diveges, that is, F has infinite logaithic easue. This copletes the poof of Theoe 2. Reak The set F in Theoe 2 depends not only on the sequence (z j ), but also on the function f. Using (20) one can also obtain a esult whee the set F does not depend on f: Suppose that φ(t) is a deceasing function which tends to zeo as t. If 0 < α <, then thee exists a sequence (z j ) satisfying () and (6) and a set F of uppe logaithic density one such that (7) holds fo all entie functions f with zeos (z j ). This esult is best possible in the following sense: Given a sequence (z j ) satisfying () and (2) and an unbounded set G [, ), thee exists an unbounded sequence ( k ) satisfying k G fo all k such that log log M( k ) o((log n( k )) 2 ) as k. This can be poved by choosing ( k ) apidly inceasing, p j [2 log j/ log z j ] fo s k < z j k, and p j vey lage fo k < z j s k+, whee (s k ) is a suitable sequence satisfying s k < k < s k+. Acknowledgents I would like to thank Pofesso A. A. Gol dbeg fo dawing y attention to the pobles consideed in this pape and fo tanslating his esults in [6] into Gean. I a also gateful to Pofesso D. F. Shea and Pofesso A. A. Gol dbeg fo soe helpful coents on a peliinay vesion of this pape. Finally, I wish to thank the efeee fo a nube of valuable suggestions. Refeences [] W. Begweile, Canonical poducts of infinite ode, J. Reine Angew. Math., to appea. [2] O. Bluenthal, Pincipes de la théoie des fonctions entièes d ode infini, Gauthies- Villas, Pais 90. [3] J. H. E. Cohn, Two piay facto inequalities, Pacific J. Math. 44 (973), 8-92. 7
[4] A. Denjoy, Su les poduits canoniques d ode infini, J. Math. Pues Appl. (6) 6 (90), -36. [5] A. Edei and W. H. J. Fuchs, Meoophic functions with seveal deficient values, Tans. Ae. Math. Soc. 93 (959), 292-328. [6] A. A. Gol dbeg, On the epesentation of a eoophic function as a quotient of entie functions (Russian), Izv. Vysš. Učebn. Zaved. Mateatika, 972, no. 0, p. 3-7. [7] A. A. Gol dbeg and I. V. Ostovskii, Distibution of values of eoophic functions (Russian), Nauka, Moscow 970. [8] J. Miles and D. F. Shea, An exteal poble in value-distibution theoy, Quat. J. Math. Oxfod (2) 24 (973), 377-383. [9] R. Nevanlinna, Reaques su les fonctions onotones, Bull. Sci. Math. 55 (93), 40-44. [0] L. A. Rubel, A Fouie seies ethod fo entie functions, Duke Math. J. 30 (963), 437-442. Hong Kong Univesity of Science & Technology Depatent of Matheatics Clea Wate Bay Kowloon Hong Kong pesent addess: Lehstuhl II fü Matheatik RWTH Aachen Teplegaben 55 W-500 Aachen Geany Eail: sf00be@dacth.bitnet 8