Taylor Problem on a Representation of Holomorphic Functions

Functional Analysis and Its Applications, Vol. 35, No. 3, pp. 237–239, 2001

On the Rubel–Taylor Problem on a Representation of Holomorphic Functions* B. N. Khabibullin

UDC 517.55

Let f be a meromorphic function on Cn (we write f ∈ Mer) and f = gf /hf a canonical representation of f as the quotient of entire functions gf and hf that are coprime in the ring of germs of holomorphic functions at each point of Cn [1]. We define a characteristic function of f ∈ Mer by the formula uf = max{ln |gf |, ln |hf |} or uf = ln(|gf |2 + |hf |2 )1/2 . Let L ( = (L1 , . . . , Lk ) be a finite set of complex vector subspaces of decomposition Cn = L1 ⊕ · · · ⊕ Lk holds. Let z = w1 + · · · + w k ,

Cn

(1)

such that the direct sum

wp ∈ Lp , p = 1, . . . , k,

(2)

Cn ,

rp = |wp | the Euclidean norm of wp , and be the (unique) representation of a vector z ∈ r = (r1 , . . . , rk ). By Sp we denote the unit sphere in Lp and by ds(p) the area element on Sp with s(p) (Sp ) = 1. Let f ∈ Mer, and let wp = rp ζp , ζp ∈ Sp , in (2). The function

TL (r ; f ) = ··· uf (r1 ζ1 , . . . , rk ζk ) ds(1) (ζ1 ) · · · ds(k) (ζk ), S1

Sk

Rk+ ,

where r ∈ R+ = {a ∈ R : a > 0}, is called the Nevanlinna L ! -characteristic of f . For k = 1 (respectively, k = n), this is the classical Nevanlinna characteristic defined via the exhaustion by balls [2–7] (respectively, by polydisks [5]). The Nevanlinna L ( -characteristic is uniquely determined up to an additive constant independently of the choice of the functions uf , gf , and hf in (1) and is continuous and convex with respect to ln r1 , . . . , ln rk everywhere in Rk+ (cf. [4–7, 9]). The following simple properties of the Nevanlinna L ( -characteristic can readily be obtained by analogy with the proofs of the corresponding facts for the above-mentioned classical Nevanlinna characteristics (see [4–6, 9]): (a) for any f1 , f2 ∈ Mer, one has TL (r ; f1 + f2 )
TL (r ; f1 ) + TL (r ; f2 ) + O(1), TL (r ; f1 f2 )
TL (r ; f1 ) + TL (r ; f2 ) + O(1),

TL (r ; 1/f1 )
TL (r ; f1 ) + O(1);

(b) for any c ∈ Cn and  > 0, there exists a constant C(, c) such that TL (r ; fc )
C(, c)TL+ ((1 + ) r + const ·1; f ),

r ∈ Rk+ ,

where 1 = (1, . . . , 1) ∈ Rk+ , fc (z) = f (z − c), and TL+ (r ; f ) = max{TL (r ; f ), 0}. Let λ(r ) > 0 be a growth function, i.e., a function continuous on Rk+ and nondecreasing in each of the variables, and let b ∈ [0, +∞]. The class Λ(b), b > 0, consists of functions f ∈ Mer such that there exist constants af < b and Af and a vector af ∈ Rk+ for which TL (r ; f )
Af λ(af r + af ),  r ∈ Rk+ . Set Λ[b] = {Λ(b
) : b < b
} for 0
b < +∞. The class Λ0 (b), b > 0, consists of functions f ∈ Mer such that lim sup|r |→∞ TL (r ; f )/λ(af r + af ) = 0, where λ(r ) is assumed to tend to +∞  as rp → +∞, the constant af is less than b, af ∈ Rk+ , and |r | = r12 + · · · + rk2 . Obviously, the *Supported by RFBR grant No. 00-01-00770. Bashkir State University, Department of Mathematics. Translated from Funktsional
nyi Analiz i Ego Prilozheniya, Vol. 35, No. 3, pp. 91–94, July–September, 2001. Original article submitted January 31, 2000. 0016–2663/01/3503–0237 $25.00

c
2001 Plenum Publishing Corporation

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characteristic TL (r ; f ) can be replaced by TL+ (r ; f ) in the definition of each of these classes. For each of these classes, we indicate the subclass containing only entire functions on Cn by a subscript E on Λ. When speaking of any one of these classes, we simply write Λ and ΛE and denote the set of quotients of functions in ΛE by ΛE /ΛE . Property (a) readily implies the following assertion. Proposition (cf. [5, Proposition] and [6]). The set Λ is a field, and ΛE is a subring of Λ. One of the main questions motivated by this proposition was stated (for k = 1) by Rubel and Taylor in the late sixties [2], [5, p. 470], [8, Problem 8]: When is Λ the field of quotients of ΛE ? ∗ For n = 1 and b = +∞, the definitive answer is “always.” It was given in the early seventies by Miles [3]. In the early nineties, the author managed to obtain a similar answer (mainly, by functional-analytic methods) for k = 1 and for arbitrary n in [10, Theorem 1.3] and [11, Theorem 5]. (The history of the problem is given in [9–11] in more detail). As to the case k = n, only Taylor’s original result [5, Theorem] is known with a growth function λ(r ), r ∈ Rn+ , of slow growth with respect to each of the variables. The following result solves the Rubel–Taylor problem under the least restrictive lower bound for the growth of λ(r ). Theorem. Let r = (r1 , . . . , rk ) ∈ Rk+ . If b ∈ [0, +∞] and ln |r |
O(λ(r )) (respectively, ln |r | = o(λ(r ))) as |r | → ∞,

(3)

then Λ(b) = ΛE (b)/ΛE (b) and Λ[b] = ΛE [b]/ΛE [b] (respectively, Λ0 (b) = Λ0E (b)/Λ0E (b)). To prove this theorem, it suffices to verify the following lemma. Lemma. Let f ∈ Mer. For each ε > 0 there exists an entire function qε ≡ 0, constants A, B, and C , and a vector a ∈ Rk+ such that (in the notation introduced after formula (2)) uf (z) + ln |qε (z)|
ATL
+ ((1 + ε)r + a ; f ) + B ln(2 + |z|) + C,

z ∈ Cn .

(4)

Indeed, by this lemma, for each function f ∈ Λ in the theorem we can choose a representation of the form f = g/h = gf /hf , where g = gf qε and h = hf qε for sufficiently small ε. Then, in view of (4) and (1), the functions ln |g| and ln |h| are majorized by the right-hand side of inequality (4). Now, by the definition of the classes Λ(b), Λ[b], and Λ0 (b) and with regard for condition (3), we obtain the desired conclusion of the theorem. Proof of the lemma. Property (b) permits us to assume without loss of generality (using shifts where necessary) that the value of f at 0 is defined, i.e., uf (0) = −∞. Consider the mean  2π values Mf (z) = (2π)−1 0 uf (eiθ z) dθ of uf over the circles. By the definition of the Nevanlinna L ( -characteristic and by the invariance of this characteristic with respect to the multiplication of the argument of uf by eiθ , we can integrate with respect to θ and obtain

TL (r ; f ) = ··· Mf (r1 ζ1 , . . . , rk ζk ) ds(1) (ζ1 ) · · · ds(k) (ζk ); (5) S1

Sk

see formula (2) and the subsequent notation concerning the relationship between r ∈ Rk+ and z ∈ Cn . It follows from well-known properties of plurisubharmonic functions [12] that Mf (z) is subharmonic on each Lp and is bounded below by the constant C = uf (0) for all z. Since Mf (z) − C is nonnegative and subharmonic on each Lp for any  > 0, we can successively apply Theorem 3.19 in [13] in each Lp to the function Mf (z) − C with some constants Cp (), p = 1, . . . , k, depending ∗ The importance of such results for the value distribution theory of meromorphic functions was also indicated by Stoll [9, p. 85], who is a classic in this field: “Unfortunately, Miles’ theorem [3] has not been proved for the case of several variables.”

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only on p and  and obtain, according to (5), Mf (z) − C = (Mf (z) − C)+

k 
Cp () ··· (Mf ((1 + )r1 ζ1 , . . . , (1 + )rk ζk ) − C)+ ds(1) (ζ1 ) · · · ds(k) (ζk ) p=1


A()

S1







S1

···

Sk

Sk

Mf ((1 + )(r1 ζ1 , . . . , rk ζk )) ds(1) (ζ1 ) · · · ds(k) (ζk ) + const

= A()TL ((1 + )r ; f ) + const,

where A() is a constant and rp  1.

It follows from the last inequality that
2π 1 uf (eiθ z) dθ
A()λ(r + 1 ) + const 2π 0 for all z ∈ Cn , where λ(r ) = TL ((1 + )r ; f ) is a growth function. Consequently [11, Theorem 3.1], there exists an entire function q ≡ 0 such that the estimate  sup λ((1 + )r
+ 1 ) + (n + 2) ln(1 + |z| + σ) − n ln σ + const, uf (z) + ln |q (z)|
A() | r
− r |
σ

 where A() is a constant, holds for any z ∈ Cn and σ > 0. By setting σ ≡ 1, (1 + )2 = 1 + ε, and qε = q , we arrive at the desired estimate (4). Conjecture. Condition (3) in the theorem can be omitted. This conjecture is corroborated by the fact that it holds for k = 1 (see [10, 11]) as well as for k = n in the case of slowly growing λ (see [5]). References 1. L. H¨ ormander, An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, 1966. 2. L. A. Rubel and B. A. Taylor, Bull. Soc. Math. France, 96, 53–96 (1968). 3. J. B. Miles, J. Analyse Math., 25, 371–388 (1972). 4. W. Stoll, In: Entire Functions and Related Parts of Analysis, Proc. Sympos. Pure Math., Vol. 11, Amer. Math. Soc., Providence, R.I., 1968, pp. 392–430. 5. B. A. Taylor, In: Entire Functions and Related Parts of Analysis. Proc. Sympos. Pure Math., Vol. 11, Amer. Math. Soc., Providence, R.I., 1968, pp. 468–474. 6. R. O. Kujala, Trans. Amer. Math. Soc., 161, 327–358 (1971). 7. H. Skoda, Ann. Inst. Fourier (Grenoble), 21, 11–23 (1971). 8. L. Ehrenpreis, In: Entire Functions and Related Parts of Analysis. Proc. Sympos. Pure Math., Vol. 11, Amer. Math. Soc., Providence, R.I., 1968, pp. 533–546. 9. W. Stoll, Value Distribution Theory for Meromorphic Maps, F. Vieweg u. Sohn, Braunschweig, 1985. 10. B. N. Khabibullin, Sib. Matem. Zh., 33, No. 3, 186–191 (1992). 11. B. N. Khabibullin, Izv. RAN, Ser. Matem., 57, No. 1, 129–146 (1993). 12. P. Lelong and L. Gruman, Entire Functions of Several Complex Variables, Springer-Verlag, Berlin–New York, 1986. 13. W. K. Hayman and P. B. Kennedy, Subharmonic Functions, Vol. I, Academic Press, London– New York, 1976. Translated by V. E. Nazaikinskii

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