Meromorphic Functions: Progress and Problems

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1 Meromorphic Functions: Progress and Problems DAVID DRASIN Mathematics Department, Purdue University West Lafayette, IN 47907, USA Zürich is a special place to workers in meromorphic function theory. Rolf Nevanlinna was Professor both at the ETH and University of Zürich. His address at the 1932 Zürich ICM centered on connections between his new theory of meromorphic functions and the Riemann surface of / _1, a perspective that continues to yield insights. Lars Ahlfors accompanied Nevanlinna to the ETH in 1928, where he developed his fundamental distortion theorem and proved Denjoy's conjecture that an entire function of order p has at most 2p distinct finite asymptotic values. Zürich has been one of the main venues of the Nevanlinna Colloquia through the years, and the home of Pólya and Pfluger. Goldberg in [20] and (with Levin and Ostrovskii) [22] has produced thorough surveys whose bibliographies contain entries. 1 Introduction We study (nonconstant) functions / meromorphic in D(R) = {\z\ < R}, 0 < r < oo, in terms of the exhaustion of D(R) by disks D(r), r < R. If a e C, 0 < r < R < oo, set ra' (r,oo) = ipog + /(re is ) <i9, 1 f 2n 1 m^a) -^J 0 l0ë+ /(re**)-a dg ( G C) ' N(r, a)= (n(t, a) - n(0, a))t~ l dt + n(0, a) log r, Jo with n(t,a) the number of roots of / = a in D(t) counted with regard to multiplicity; ÎV and n count multiple roots of f(z) a once. Define m(r, f) = m(r, oo), N(r,f) = N(r, oc), and the (Nevanlinna) characteristic by T(r)=T(r,f)=m(r,f) + N(r,f). The first fundamental theorem asserts that r(r)=t(r,-i-) + 0(l) (I) Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994 Birkhäuser Verlag, Basel, Switzerland 1995

2 Meromorphic Functions: Progress and Problems 829 for any a e C (Jensen's formula). Once (I) is known, we may define all quantities using n and N, and extend them formally to quasiregular mappings g [35]. When R = oo (as we usually assume) Nevanlinna's second fundamental theorem is the inequality Ì2N{r,a ) >(q- 2)7» + N t (r) + S(r), (II) i where Ni(r) = N(r, 1//') + 2N(r, f) N(r, /') measures the total ramification of / in D(r) and the "error term" S(r) = m(r, q J F F -) + m(r, V i ) + O(l) = o(t(r)) (r -> oo, r i E), (1) / ~ * ~ a " where E has finite measure and is empty when the order of / is finite. If we set p = lim sup T(r)j log r i»oc c, x,. r m ( r > a ) at \ v fn(r,a)-n(r,a) 6(a) = km ìnf, 0(a) = hm inf -, (2) r»oc 1 \r) i >oc i \r) so that 0(a) > 0, 6(a) > 0, 6(a) + 6(a) < 1, (II) is conveniently summarized by 5^«(o) + fl(o)<2 (II ; ) and, in particular, 5^^(a)<2. aec These give very precise conditions that a nondegenerate map / defined in C must satisfy. Further analysis shows that N(r, a) ~ T(r) as r oo for all a outside a set of capacity zero, and we call such a a normal value of /. Inequality (II ) is sharp in the sense that any sequence of nonnegative 6 U, 6 V (with 6 V -f 6 V < 1) can be associated to any sequence a u by some meromorphic function /: / solves the inverse problem. This was obtained by Drasin [5] after a long series of partial results by others (a solution to the restricted problem (II ) for entire functions was obtained in [19]). In general (see (4)) / must be of infinite order. We first construct a quasiregular formal solution g to the problem and let a; be a quasiconformal homeomorphism of C that solves the Beltrami equation wi = p(z)u z (p = gz/g z ). (If)

3 830 David Drasin where p is small near oo, so that / = gou) is meromorphic in the plane. As UJ docs not map circles exactly onto circles, it is necessary that T(r, /)/ log r approach infinity quite slowly as r oc to guarantee that / inherit the data {6 u,6 lf } from g. Thus the function / obtained in [5] has 'small' infinite order. No doubt there exist solutions to this inverse problem of arbitrarily rapid growth. A generalization of (II ), suggested by Nevanlinna, is to consider small functions a(z) with T(r,a) = o(t(r)), and define 6(a) as in (2) with a = a(z). Only rather recently has (II ) been established for small functions [32], [41]. Problem (a) Given a countable collection of functions {a y (z)} and sequence 6 V, 0 < 6 y < 1, find a meromorphic function f(z) with T(r,a v ) = o(t(r,f)) for each v and 6(a u,f) = 6 y. (b) Is something more suggestive of (II) valid for small functions: N(r, a u ) >(q- 2)T(f) + o(t(r)) (r - oc, r E)ï 1 This would have applications to the uniformization of algebraic curves. 2 Logarithmic derivative, error terms During the past decade, formal analogies have been observed between valuedistribution theory and Diophantine approximation in number theory, and this analogy inspired Osgood's proof of (II ) for small functions. In [42, p. 34] is a "dictionary" between the two subjects. This also intrigued Lang [27], who asked for precise estimates of S(r) in (1), and proposed the bound (3), based on the translation to function theory of his long-standing conjecture on rational approximation of irrational numbers. By an insightful reexamination of R. Nevanlinna's proof of (II) and the analysis of [21], Hinkkanen [25] developed a continuum of estimates for S(r) and ra(r, /'//) The simplest to state is: if t~ l (j)(t), J^ (j)(t)~ l dt < oc, then S(r)<log<j>(T(r,f)) (r i E), (3) E of finite measure; examples show that (3) and its variants are very precise. The best known estimates for S(r) in the multidimensional case have been obtained in [44], based on methods of F. Nevanlinna and Ahlfors. These require that an unbounded term be added to the right side of (3), so they may not be sharp. Ru and Wong [36] recently used ideas from multidimensional Nevanlinna theory to extend the Thue-Siegel theorem and thus give conditions that limit the number of integral points in the complement of certain hyperplanes in P n. By further analyzing /'//, Fuchs [18] (a = 1/2) and Hayman [23] (a > 1/3) proved that if p < oc, then ^6(a ) a < oo. (4) Weitsman [44], using other methods, obtained (4) when a = 1/3. According to [11], Weitsman's theorem together with (II ) give necessary and sufficient conditions that the {6 U } = {6(a u )} must fulfill for functions of finite order, except in the special situations that ^6 U = 2 or max = 1 (almost-entire functions).

4 Meromorphic Functions: Progress and Problems 831 It is far more difficult to construct entire functions (6(oc) = 1) with p < oo and infinitely many 6 V > 0. Arakelyan [1] was the first to do so, and conjectured that for entire functions, (4) should be replaced by ^(log^/ä,,))" 1 <? but this has recently been shown false by Eremenko [13], who in turn asks if (4) holds for all a > 0 or if, for all p > 1, (iog(i/m)- p < oo. The only positive result is the theorem of Lewis and Wu [29], which gives (4) with a = 1/3 2~ 260. In addition, they prove a conjecture of Littlewood about polynomials, one of whose consequences, stated informally, is that an entire function assumes most of its values on a small subset of C. Lewis and Wu used work of Eremenko and Sodin, who had obtained a weak form of the Littlewood conjecture, and one form of this application to entire functions. 3 Extremal functions If / is rational of degree ra, then J2 6 (a) + 0( a ) = 2-n~ 1, ^ *( a ) ^ L What this suggests, and what is true, is that cases of equality in (II ) are legion, whereas (II ) is usually strict. THEOREM 1. If f is meromorphic in the plane with then each 6(a) is an integral multiple of p~ x %,/) = 2 poc, (5) so that 2p 1 is a natural number. (If / is entire, p must be a positive integer [34].) F. Nevanlinna and Hille gave examples that show Theorem 1 best possible by considering meromorphic functions / = wi/u)2, where wi, w^ are linearly independent solutions to w" + Pw = 0, (6) P a polynomial; / has order (ra + 2)/2, where ra = degp. They deduced that the singularities of the Riemann surface of f~ l consist of a finite number of logarithmic branch points; even today one way to study Stokes multipliers for solutions to (6) uses the geometry of these surfaces [40, Chapter 8]. In these examples, Ni(r)=Q, (7) and only recently have Bergweiler and Eremenko [4] been able to obtain a functiontheoretic proof that (7) characterizes these surfaces: they show that whenever g is meromorphic with p < oc, the only possible singularities of the Riemann surface of g~ l are algebraic branch points, limits of algebraic branch points, and (by an extension of Ahlfors's theorem) a finite number of logarithmic branch points. Theorem 1 was conjectured by F. Nevalinna [30] in 1929, and proved by Drasin [6] who performed a quasiconformai modification of f(z 2 ): to introduce a

5 832 David Drasin quasiconformal map v with small dilatation and set g(z) =ujof(z 2 ). The role of a; is that 0 and oc become (essentially) Picard values of g. This reduces the problem to the (known) case of entire functions, and because of the specific form of UJ it is possible to directly read off all information required of /. Soon after [6] appeared, Eremenko [12] presented an alternate proof of Theorem 1, based on potential theory and convergence properties of 6-subharmonic functions. From this point of view, asymptotic equalities and inequalities become precise relations among the limit functions obtained by renormalization and normality considerations. In other papers, some joint with Sodin [16], [17], this viewpoint is systematically pursued and extended to small functions and meromorphic curves; see Section 4(C) for another application. It is very powerful but requires sophisticated potential theory when dealing with the limit functions that arise (however, the translations of these papers are inadequate). As an example of this point of view, their version of (II ) becomes THEOREM 2. Let u\,..., u q be subharmonic in a domain ficc. If for each k ^ u = maxuj = max{wfe,w4 j then Y^ Uj (q 2)w is subharmonic in Q. Eremenko [14] applies this method to obtain the deepest modern result in the theory: equality is forced in (II ) by a purely geometric hypothesis (compare with (7)). Because functions that satisfy (7) occupy a significant role in several contexts, these insights should have further application (the case p < 1/2 is in [39]). Eremenko proved THEOREM 3. Let f be meromorphic in the plane with p < oc and suppose that N x (r) = o(t(r)) (r oo). (8) Then 2p 1 is a positive integer and f satisfies (5). Problem. For a given order p < oo find the best upper bound for the left side of (II ) among all entire/meromorphic functions of order p. This is one of the oldest problems in the theory, settled for p < 1 by [8] and [43] (entire) and by [9] and [27] (meromorphic). I know of no sharp bounds for any values of p > 1 other than when (5) holds. The conjectured extremals appear in [31, p. 18] and [7]. This problem should be on our list for a long time. 4 Further topics (A) Picard properties and normal families. A long-standing principle (Bloch) is that properties P that render a function defined in the plane constant may be mated with those that yield normal families for a collection of functions in a domain Q. Examples of such P are: that / omit three fixed values; that / ^ 0, f^ ^ 1 for some fixed k > 1; that f'f m ^ 1 for some fixed ra > 1; see [37] and [46] for the history.

6 Meromorphic Functions: Progress and Problems 833 A uniform and elegant path between these two settings for a large class of P has been refined in recent years, using renormalizations and compactness arguments. The most recent ingredient is from Pang [33] (the case h=0 due to Zalcman): if T is not normal at z 0 e ft and 1 < h < 1 is given, there exist f n e T, z n -* zo, p n > 0, such that 9n{0=fkfn(Zn+PnC)->9(0 (9) with g meromorphic in the plane, nonconstant, and of finite order. The other direction goes back to the beginning of the century: if / is meromorphic and nonconstant in the plane, the family T = {ÌR(Z) = f(zo + Rz):R > 0, \z\ < 1} cannot be normal in the disk. These principles permit a uniform treatment of these P, and should have further applications. (B) Complex iteration. Baker [3] used Ahlfors's theory of covering surfaces to prove that the Julia set J (nonnormality set) of iterates of an entire function / is the closure of repelling fixed points (of all orders); for rational maps this is Julia's theorem. These theorems now have a compelling short proof due to Schwick [38], based on (II). In (9), take as T the family of iterates, h = 0, and ZQ e J such that the system {ZQ = f(wo),wo = <?(Co)} has unramified solutions; the desired fix-points arise at once. By (II), there can be at most four exceptional ZQ e J, but as J is perfect these exceptional ZQ may be ignored. (C) Quasiregular mappings. The limit function technique of Section 3 has been used by Eremenko and Lewis [15] to give a potential-theoretic proof of the THEOREM OF RICKMAN [35]. There exists q 0 = qo(n, K) < oc such that ifq > q Q and f : R n * W 1 \ {ai,...,a q ] is K-quasiregular, then f is constant. Holopainen and Rickman [26] in turn have used [15] to extend Rickman's theorem to maps / :R n > M = N\ {ai,...,a q }, N any oriented compact differentiate ra-manifold and quasiconformality defined with respect to a Riemannian metric on M; [15] allows technical matters about path families to be bypassed. (D) A return to / and /'. Nevanlinna's analysis of m(r, /'//) yields at once that lim sup : T(r,fW) / 1 (/entire) T(r, f) {l \ k + 1 (/ meromorphic), where \E\ < oc. Nevanlinna was optimistic that lower bounds might be as simple, but many counterexamples have been given to this, principally by Toppila. Hayman and Miles [24] combine estimates of 5R{z/''/'/} with a geometrical analysis of the image of { z = r} under / and the {/^} to show that given K > 1, there exists a set FK of positive lower logarithmic density, with ^TW>-{ (2eK)~ l (3eK) x (f entire) r F K independent of k. Langley suggests that perhaps lim sup T(r, f f )/T(r, p < 1/2; by [28] this would be sharp. i >oc (f meromorphic), f) = 1 if / is entire with

7 834 David Drasin References [I] N. U. Arakelyan, Entire functions of finite order with an infinite set of deficient values, Dokl. Akad. Nauk. SSSR 170 (1966), (in Russian); Engl, transi.: Soviet Math Dokl. 7 (1966), [2] A. Baernstein, Proof of Edrers spread conjecture, Proc. London Math. Soc 26 (1973), [3] I. N. Baker, Repulsive fixpoints of entire functions, Math. Z. 104 (1968), [4] W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, to appear.. [5] D. Drasin, The inverse problem of the Nevanlinna theory, Acta Math. 138 (1977), [6], Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two, Acta Math. 158 (1987), [7] D. Drasin and A. Weitsman, Meromorphic functions with large sums of deficiencies, Adv. in Math. 15 (1975), [8] E. A. Edrei, Locally Tauberian theorems for meromorphic functions of lower order less than one, Trans. Amer. Math. Soc. 140 (1969), [9], Solution of the deficiency problem for functions of small lower order, Proc. London Math. Soc. 26 (1973), [10] E. A. Edrei and W. H. J. Fuchs, Valeurs déficientes et valuers asymptotiques des fonctions méromorphes, Comm. Math. Helv. 33 (1959), [II] A. E. Eremenko, The inverse problem of the theory of meromorphic functions of finite order, Sibirsk. Mat. Zh. 27 (1986), [12], A new proof of Drasiris theorem on meromorphic functions of finite order with maximal deficiency sum, I and II, Teor. Funktsiï Funktionals Anal, i Prilozhen. (Kharkov) 51 (1989), ; 52 (1990), ; 52 (1990), [13], A counterexample to the Arakelyan conjecture, Bull. Amer. Math. Soc. 27 (1992), [14], Meromorphic functions with small ramification, Indiana Univ. Math. J. 42 (1993), [15] A. E. Eremenko and J. L. Lewis, Uniform limits of certain A-harmonic functions with applications to quasiregular mappings, Ann. Acad. Sei. Fenn. Ser. A I Math. 16 (1991), [16] A. E. Eremenko and M. Sodin, On meromorphic functions of finite order with maximal deficiency sum, Teor. Funktsiï, Funktionals Anal, i Prilozhen. (Kharkov) 59 (1992), [17], Distribution of values of meromorphic functions and meromorphic curves from the point of view of potential theory, Algebra i Analiz 3 (1991), (in Russian); Engl, trans.: St. Petersburg Math. J. 3 (1991), [18] W. H. J. Fuchs, A theorem on the Nevanlinna deficiencies of meromorphic functions of finite order, Ann. of Math. (2) 68 (1958), [19] W. H. J. and W. K. Hayman, An entire function with assigned deficiencies, Stud. Math. Analysis and Related Topics, Stanford Univ. Press, Stanford, CA, (1962), [20] A. A. Goldberg, Meromorphic functions, in v. 10 Serija Math. Analiz, Itogi Nauki i Tekhniki, Moscow (1973) (in Russian); Engl, trans.: J. Soviet Math. 4 (1975),

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