Annales Academi½ Scientiarum Fennic½ Mathematica Volumen 27, 2002, 215{220 SCHWARZIAN DERIVATIVES OF TOPOLOGICALLY FINITE MEROMORPHIC FUNCTIONS Adam Lawrence Epstein University of Warwick, Mathematics Institute Coventry CV4 7AL, United Kingdom; [email protected] Abstract. Using results from dynamics, we prove sharp upper bounds on the local vanishing order of the nonlinearity and Schwarzian derivative for topologically ¯nite meromorphic functions. The entire functions f: C C with rational nonlinearity ! f 00 Nf = f 0 and more generally the meromorphic functions f: C C with rational Schwar- zian derivative ! 1 2 S = N0 N b f f ¡ 2 f were discussed classically by the brothers Nevanlinna and a number of other re- searchers. These functions admit a purely topological characterization (for details, see [6, pp. 152{153], [7, pp. 391{393] and [11, pp. 298{303]). We show here that a topological ¯niteness property enjoyed by a considerably larger class of functions guarantees bounds on the local vanishing orders of Nf and Sf . For meromorphic f: C C, let S(f) denote the set of all singular (that is, critical or asymptotic) ! values in C. It follows by elementary covering space theory that #S(f) 2, pro- vided thatb f is not a MÄobius transformation; note further that S(f¸) for any 1 2 entire f: Cb C, provided that f is not a±ne. Our main result is the following: ! Theorem 1. A. If f: C C is meromorphic but not a MÄobius transfor- mation then ord S #S(f) !2; consequently ord N #S(f) 1, for every ³ f · ¡ ³ f · ¡ ³ C. b 2 B. If f: C C is entire but not a±ne then ord³ Nf #S(f) 2 for every ³ C; consequen!tly, if ord N 1 then ord S #S(f)· 3. ¡ 2 ³ f ¸ ³ f · ¡ 2000 Mathematics Subject Classi¯cation: Primary 32A20; Secondary 37F10. Research partially supported by NSF Grant DMS 9803242. 216 Adam Lawrence Epstein As to the converse, the mere existence of uniform bounds on ord³ Nf or ord³ Sf is not enough to guarantee the ¯niteness of S(f). For example, the function f(z) = z + 1 ez has in¯nitely many critical values 2¼ki = f(2¼ki), but N (z) = ez=(ez 1) has¡ no zeros whatsoever and f ¡ ez(ez + 2) S (z) = ¡ f 2(ez 1)2 ¡ has only simple zeros. On the other hand, the assertions in Theorem 1 are certainly sharp. Compare the following classical result discussed in the aforementioned references: Theorem 2. A. Let f: C C be meromorphic; if Sf is rational then #S(f) 2 deg S . ! ¡ · f B. Let f: C C be entire; if Nb is rational then #S(f) 2 deg N . ! f ¡ · f Combining Theorem 1 with Theorem 2 leads to the vacuous conclusion that ord ³ R deg R at every ³ C for any rational function R. In this sense, Theorem·1 may be regarded as2 a pointwise converse to Theorem 2; moreover, as ord R = deg R for R(z) = (z ³)D , the existence of meromorphic functions f ³ ¡ with Sf = R and entire functions f with Nf = R shows that Theorem 1 is sharp. For a slightly less trivial example, the entire function f(z) = sin z has critical values at 1, an asymptotic value at and no ¯nite asymptotic values. § 1 As #S(f) = 3, Theorem 1 guarantees that every zero of Nf (z) = tan z is simple. Similarly, S(g) = S(f) for the meromorphic function g(z) = csc¡(z) = 1=f(z), so Theorem 1 guarantees that every zero of G (z) = 1 3 tan2 z = G (z) is simple, f ¡ ¡ 2 g and that no such point is a zero of Nf (z) or sin2 z 2 N (z) = ¡ : g sin z cos z The proof of Theorem 1 is more interesting than the actual statement. The strategy is to reduce the assertions to well-known results in holomorphic dynamics. This idea goes back at least as far as the work [2] of Bergweiler{Eremenko (see also [8]), and our application here is quite similar but perhaps even more basic. Let F be meromorphic with an isolated ¯xed point at ³ C. The multiplicity 2 of F at ³ is the positive integer mult³ F = ord³ F (z) z . Clearly, mult³ F 2 precisely when F (³) = 1. Fatou showed that if F is¡rational then the immedi-¸ 0 ¡ ¢ ate basin of such a multiple ¯xed (or, more generally, periodic) point contains at least mult³ (F ) 1 critical values belonging to disjoint and in¯nite forward orbits. The argument ¡adapts easily to entire and meromorphic functions, provided that one also counts forward orbits of asymptotic values; for details, see [10, p. 74]. The work [2] applied Fatou's theorem to help settle a long-standing conjecture in value-distribution theory. Part of the discussion involves showing that a particular Schwarzian derivatives of topologically ¯nite meromorphic functions 217 auxiliary mapping, known by construction to have only ¯nitely many singular val- ues, necessarily has only ¯nitely many multiple ¯xed points. In fact, as Bergweiler has observed, this particular conclusion may also be established without Fatou's theorem: indeed, if S(f) is ¯nite then f 0(³ ) for any sequence of distinct k ! 1 ¯xed points ³k (see [1] and also [5]). By contrast, our application exploits the actual bound on the multiplicity of a single ¯xed point: Theorem 3. A. If F : C C is meromorphic but not a MÄobius transforma- tiom then mult (F ) #S(F )!+ 1 for every ¯xed point ³ C. ³ · 2 B. If F : C C is entire butb not a±ne then mult³ (F ) #S(F ) for every ¯xed point ³ C!. · 2 We begin by recalling some standard properties of the nonlinearity and Schwar- zian derivative. If f: U C is meromorphic and nonconstant on a connected open set U C then N! and S are de¯ned and meromorphic on U . It is f f well-known and easily veri¯edb that Nf identically vanishes if and only if f is the restriction of an a±ne transformation, while Sf identically vanishes if and only if f is the restriction of a MÄobius transformation. Moreover, NT f = Nf for any ± a±ne transformation T , while ST f = Sf for any MÄobius transformation T : these observations follow from the co± cycle relations Ng f dz = Nf dz + f ¤(Ng dz); ± (1) 2 2 2 Sg f dz = Sf dz + f ¤(Sg dz ): ± We further require two rather straightforward computations: Lemma 1. If f is meromorphic near ³ C then ord³ Nf 1 and ord S 2. 2 ¸ ¡ ³ f ¸ ¡ A. If ³ is a critical point of f then Nf has a simple pole at ³ and Sf has a double pole at ³ . B. If ³ is not a critical point of f then Sf is holomorphic at ³ , while Nf is also holomorphic at ³ if and only if f is holomorphic at ³ . Moreover, if ord N 1 then ord S = ord N 1. ³ f ¸ ³ f ³ f ¡ Proof. Let n = deg³ f be the local degree of f at ³ . If f is holomorphic at ³ then we may write f(z) = f(³) + ®(z ³)n + O (z ³)n+1 for some ® = 0, whence ¡ ¡ 6 ¡ ¢ n 2 n 1 (n 1)nan(z ³) ¡ + O (z ³) ¡ n 1 Nf (z) = ¡ ¡ ¡ = ¡ + O(1); na (z ³)n 1 + O (z ³)n z ³ n ¡ ¡ ¡¡ ¢ ¡ n2 1 1 S (z) = ¡ + O :¡ ¢ f 2(z ³)2 z ³ ¡ ¡ ¶ 1 2 In particular, if n = 1 then Nf is holomorphic at ³ , so Sf = Nf0 2 Nf is also holomorphic at ³ . Moreover, if ord N = k 1 then ¡ ³ f ¸ 2 ord N0 = k 1 < 2k = ord N ³ f ¡ ³ f 218 Adam Lawrence Epstein and consequently ord³ Sf = k 1 = ord³ Nf 1. On the other hand, if f has a pole at ³ then we may apply¡ the previous¡ observations to the function g = 1=f which is holomorphic at ³ . It follows from (1) that N = N 2g0=g , f g ¡ where Ng(z) = (n 1)=(z ³)+O(1) and g0(z)=g(z) = n=(z ³)+O(1) because ord g = n = deg ¡g ; consequen¡ tly, N (z) = (n + 1)=(z ³)¡+ O(1). Similarly, ³ ³ f ¡ ¡ Sf = Sg so n2 1 1 S (z) = ¡ + O : f 2(z ³)2 z ³ ¡ ¡ ¶ Lemma 2. Let F be holomorphic with an isolated ¯xed point at ³ C. 2 A. If mult F 2 then ord N = mult F 2. ³ ¸ ³ F ³ ¡ B. If mult F 3 then ord S = mult F 3. ³ ¸ ³ F ³ ¡ m m+1 Proof. Writing F (z) = z+®(z ³) +O (z ³) where m = mult³ F 2 and ® = 0, we have ¡ ¡ ¸ 6 ¡ ¢ m 2 m 1 (m 1)m®(z ³) ¡ + O (z ³) ¡ NF (z) = ¡ ¡ ¡ 1 + m®(z ³)m 1 + O (z ³)m ¡ ¡ ¡ ¡ ¢ m 2 m 1 = (m 1)m®(z ³) ¡ + O (z ³) ¡ ¡ ¡ ¡ ¡ ¢ ¡ ¢ whence ord ³ NF = mult³ F 2. Consequently, if mult³ F 3 then ord³ Nf 1, so ord S = mult F 3 b¡y Part B of Lemma 1. ¸ ¸ ³ F ³ ¡ Proof of Theorem 1. It su±ces to show that ord³ Nf #S(f) 2 for any entire function f: C C, and that ord S #S(f) 2 ·for any meromorphic¡ ! ³ f · ¡ function f: C C; the remaining claims then follow by Part B of Lemma 1. ! Suppose ¯rst that f is entire. If ³ is a critical point of f then Part A of Lemma 1 implies bord³ Nf = 1 < #S(f) 1. If ³ is not a critical point of f then the distinguished local inverse¡ at f(³) is ¡approximated to ¯rst-order by a unique a±ne transformation (1) z f(³) Tf;³ (z) = ³ + ¡ : f 0(³) The composition F = T (1) f is an entire function which ¯xes ³ ; indeed, we have f;³ ± F (z) = z+O (z ³)2 so ³ is a multiple ¯xed point.