Annales Academiæ Scientiarum Fennicæ 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; adame@maths.warwick.ac.uk

Abstract. Using results from dynamics, we prove sharp upper bounds on the local vanishing order of the nonlinearity and Schwarzian derivative for topologically finite meromorphic functions.

The entire functions f: C → C with rational nonlinearity

𝔑_{f} = \frac{f′′}{f′}

and more generally the meromorphic functions f: C → Ĉ with rational Schwarzian derivative

𝔖_{f} = 𝔑′_{f} − \frac{1}{2}𝔑_{f}^{2}

were discussed classically by the brothers Nevanlinna and a number of other researchers. 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 finiteness property enjoyed by a considerably larger class of functions guarantees bounds on the local vanishing orders of 𝔑_{f} and 𝔖_{f}. For meromorphic f: C → Ĉ, let S(f) denote the set of all singular (that is, critical or asymptotic) values in Ĉ. It follows by elementary covering space theory that #S(f) ≥ 2, provided that f is not a Möbius transformation; note further that ∞ ∈ S(f) for any entire f: C → C, provided that f is not affine. Our main result is the following:

Theorem 1. A. If f: C → Ĉ is meromorphic but not a Möbius transformation then ord_{ζ}𝔖_{f} ≤ #S(f) − 2; consequently ord_{ζ}𝔑_{f} ≤ #S(f) − 1, for every ζ ∈ C.

B. If f: C → C is entire but not affine then ord_{ζ}𝔑_{f} ≤ #S(f) − 2 for every ζ ∈ C; consequently, if ord_{ζ}𝔑_{f} ≥ 1 then ord_{ζ}𝔖_{f} ≤ #S(f) − 3.

2000 Mathematics Subject Classification: Primary 32A20; Secondary 37F10.

Research partially supported by NSF Grant DMS 9803242.

As to the converse, the mere existence of uniform bounds on ord_{ζ}𝔑_{f} or ord_{ζ}𝔖_{f} is not enough to guarantee the finiteness of S(f). For example, the function f(z) = z + 1 − e^{z} has infinitely many critical values 2πki = f(2πki), but 𝔑_{f}(z) = e^{z}/(e^{z} − 1) has no zeros whatsoever and

𝔖_{f}(z) = \frac{−e^{z}(e^{z} + 2)}{2(e^{z} − 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 → Ĉ be meromorphic; if 𝔖_{f} is rational then #S(f) − 2 ≤ deg 𝔖_{f}.

B. Let f: C → C be entire; if 𝔑_{f} is rational then #S(f) − 2 ≤ deg 𝔑_{f}.

Combining Theorem 1 with Theorem 2 leads to the vacuous conclusion that ord_{ζ}ℜ ≤ deg ℜ at every ζ ∈ C for any rational function ℜ. In this sense, Theorem 1 may be regarded as a pointwise converse to Theorem 2; moreover, as ord_{ζ}ℜ = deg ℜ for ℜ(z) = (z − ζ)^{D}, the existence of meromorphic functions f with 𝔖_{f} = ℜ and entire functions f with 𝔑_{f} = 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 finite asymptotic values. As #S(f) = 3, Theorem 1 guarantees that every zero of 𝔑_{f}(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_{f}(z) = −1− \frac{3}{2} tan^{2} z = 𝔊_{g}(z) is simple, and that no such point is a zero of 𝔑_{f}(z) or

𝔑_{g}(z) = \frac{sin^{2} z − 2}{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 fixed point at ζ ∈ C. The multiplicity of F at ζ is the positive integer mult_{ζ}F = ord_{ζ}(F (z) − z). Clearly, mult_{ζ}F ≥ 2 precisely when F′(ζ) = 1. Fatou showed that if F is rational then the immediate basin of such a multiple fixed (or, more generally, periodic) point contains at least mult_{ζ}(F) − 1 critical values belonging to disjoint and infinite 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 auxiliary mapping, known by construction to have only finitely many singular values, necessarily has only finitely many multiple fixed points. In fact, as Bergweiler has observed, this particular conclusion may also be established without Fatou’s theorem: indeed, if S(f) is finite then f′(ζ_{k}) → ∞ for any sequence of distinct fixed points ζ_{k} (see [1] and also [5]). By contrast, our application exploits the actual bound on the multiplicity of a single fixed point:

Theorem 3. A. If F: C → Ĉ is meromorphic but not a Möbius transformatiom then mult_{ζ}(F) ≤ #S(F) + 1 for every fixed point ζ ∈ C.

B. If F: C → C is entire but not affine then mult_{ζ}(F) ≤ #S(F) for every fixed point ζ ∈ C.

We begin by recalling some standard properties of the nonlinearity and Schwarzian derivative. If f: U → Ĉ is meromorphic and nonconstant on a connected open set U ⊆ C then 𝔑_{f} and 𝔖_{f} are defined and meromorphic on U. It is well-known and easily verified that 𝔑_{f} identically vanishes if and only if f is the restriction of an affine transformation, while 𝔖_{f} identically vanishes if and only if f is the restriction of a Möbius transformation. Moreover, 𝔑_{T ∘f} = 𝔑_{f} for any affine transformation T, while 𝔖_{T ∘f} = 𝔖_{f} for any Möbius transformation T: these observations follow from the cocycle relations

(1)  \begin{aligned}𝔑_{g∘f} dz & = 𝔑_{f} dz + f^{∗}(𝔑_{g} dz), \\ 𝔖_{g∘f} dz^{2} & = 𝔖_{f} dz^{2} + f^{∗}(𝔖_{g} dz^{2}). \end{aligned}

We further require two rather straightforward computations:

Lemma 1. If f is meromorphic near ζ ∈ C then ord_{ζ}𝔑_{f} ≥ −1 and ord_{ζ}𝔖_{f} ≥ −2.

A. If ζ is a critical point of f then 𝔑_{f} has a simple pole at ζ and 𝔖_{f} has a double pole at ζ.

B. If ζ is not a critical point of f then 𝔖_{f} is holomorphic at ζ, while 𝔑_{f} is also holomorphic at ζ if and only if f is holomorphic at ζ. Moreover, if ord_{ζ}𝔑_{f} ≥ 1 then ord_{ζ}𝔖_{f} = ord_{ζ}𝔑_{f} − 1.

Proof. Let n = deg_{ζ} f be the local degree of f at ζ. If f is holomorphic at ζ then we may write f(z) = f(ζ) + α(z − ζ)^{n} + O((z − ζ)^{n+1}) for some α ≠ 0, whence

\begin{aligned}𝔑_{f}(z) & = \frac{(n − 1)na_{n}(z − ζ)^{n−2} + O((z − ζ)^{n−1})}{na_{n}(z − ζ)^{n−1} + O((z − ζ)^{n})} = \frac{n − 1}{z − ζ} + O(1), \\ 𝔖_{f}(z) & = \frac{n^{2} − 1}{2(z − ζ)^{2}} + O(\frac{1}{z − ζ} ). \end{aligned}

In particular, if n = 1 then 𝔑_{f} is holomorphic at ζ, so 𝔖_{f} = 𝔑′_{f} − \frac{1}{2}𝔑_{f}^{2} is also holomorphic at ζ. Moreover, if ord_{ζ}𝔑_{f} = k ≥ 1 then

ord_{ζ}𝔑′_{f} = k − 1 < 2k = ord_{ζ}𝔑_{f}^{2}

and consequently ord_{ζ}𝔖_{f} = k − 1 = ord_{ζ}𝔑_{f} − 1. On the other hand, if f has a pole at ζ then we may apply the previous observations to the function g = 1/f which is holomorphic at ζ. It follows from (1) that 𝔑_{f} = 𝔑_{g} − 2g′/g, where 𝔑_{g}(z) = (n − 1)/(z − ζ)+O(1) and g′(z)/g(z) = n/(z − ζ)+O(1) because ord_{ζ}g = n = deg_{ζ} g; consequently, 𝔑_{f}(z) = −(n + 1)/(z − ζ) + O(1). Similarly, 𝔖_{f} = 𝔖_{g} so

𝔖_{f}(z) = \frac{n^{2} − 1}{2(z − ζ)^{2}} + O(\frac{1}{z − ζ} ).

Lemma 2. Let F be holomorphic with an isolated fixed point at ζ ∈ C. A. If mult_{ζ}F ≥ 2 then ord_{ζ}𝔑_{F} = mult_{ζ}F − 2. B. If mult_{ζ}F ≥ 3 then ord_{ζ}𝔖_{F} = mult_{ζ}F − 3.

Proof. Writing F (z) = z+α(z−ζ)^{m} +O((z−ζ)^{m+1}) where m = mult_{ζ}F ≥ 2 and α ≠ 0, we have

\begin{aligned}𝔑_{F} (z) & = \frac{(m − 1)mα(z − ζ)^{m−2} + O((z − ζ)^{m−1})}{1 + mα(z − ζ)^{m−1} + O((z − ζ)^{m})} \\ & = (m − 1)mα(z − ζ)^{m−2} + O((z − ζ)^{m−1}) \end{aligned}

whence ord_{ζ}𝔑_{F} = mult_{ζ}F − 2. Consequently, if mult_{ζ}F ≥ 3 then ord_{ζ}𝔑_{f} ≥ 1, so ord_{ζ}𝔖_{F} = mult_{ζ}F − 3 by Part B of Lemma 1.

Proof of Theorem 1. It suffices to show that ord_{ζ}𝔑_{f} ≤ #S(f) − 2 for any entire function f: C → C, and that ord_{ζ}𝔖_{f} ≤ #S(f) − 2 for any meromorphic function f: C → Ĉ; the remaining claims then follow by Part B of Lemma 1.

Suppose first that f is entire. If ζ is a critical point of f then Part A of Lemma 1 implies ord_{ζ}𝔑_{f} = −1 < #S(f)−1. If ζ is not a critical point of f then the distinguished local inverse at f(ζ) is approximated to first-order by a unique affine transformation

T_{f,ζ}^{(1)}(z) = ζ + \frac{z − f(ζ)}{f′(ζ)}.

The composition F = T_{f,ζ}^{(1)} ∘ f is an entire function which fixes ζ; indeed, we have F (z) = z+O((z−ζ)^{2}) so ζ is a multiple fixed point. As 𝔑_{f} = 𝔑_{F} , it follows from Part A of Lemma 2 that ord_{ζ}𝔑_{f} = mult_{ζ}F − 2; moreover, S(F) = T_{f,ζ}^{(1)}(S(f)) so

ord_{ζ}𝔑_{f} = mult_{ζ}F − 2 ≤ #S(F) − 2 = #S(f) − 2

by Part A of Theorem 3.

Suppose now that f is meromorphic. If ζ is a critical point of f then Part A of Lemma 1 implies ord_{ζ}𝔖_{f} = −2 < #S(f) − 2. If ζ is not a critical point of f then the distinguished local inverse at f(ζ) is approximated to second-order by a unique Möbius transformation T_{f,ζ}^{(2)} : in fact,

T_{f,ζ}^{(2)}(z) = ζ + \frac{2f′(ζ)(z − f(ζ))}{f′′(ζ)(z − f(ζ)) + 2f′(ζ)^{2}}

provided that ζ is not a pole. The composition F = T_{f,ζ}^{(2)} ∘ f is a meromorphic function fixing ζ, and mult_{ζ}F ≥ 3 because F (z) = z + O((z − ζ)^{3}). As we have 𝔖_{f} = 𝔖_{F} , it follows from Part B of Lemma 2 that ord_{ζ}𝔖_{f} = mult_{ζ}F − 3; moreover, S(F) = T_{f,ζ}^{(2)}(S(f)) so

ord_{ζ}𝔖_{f} = mult_{ζ}F − 3 ≤ #S(F) − 2 = #S(f) − 2

by Part B of Theorem 3.

For context, let us briefly sketch the proof of Theorem 2. The core of the argument is to show that every asymptotic value of f corresponds to a logarithmic end: a simply connected region U ⊂ C such that the restriction f|_{U} is an infinite cyclic cover of a punctured neigborhood of the asymptotic value. This assertion follows from asymptotic analysis of the solutions to the associated differential equations (see [7], [9] and [11]—for Part B one may appeal to the ex-

plicit formula f = ∫ e^{∫ 𝔑_{f}} ). Note that ord_{∞}𝔖_{f} ≥ 2 if and only the quadratic differential 𝔖_{f}(z) dz^{2} is holomorphic at ∞; in this case, it follows that f is meromorphic at ∞ and therefore rational, so that there are no logarithmic ends. On the other hand, if ord_{∞}𝔖_{f} ≤ 2 then further analysis shows that there are precisely −ord_{∞}𝔖_{f}(z) dz^{2} = 2 − ord_{∞}𝔖_{f} logarithmic ends. As ord_{ζ}𝔖_{f} = −2 at each finite critical point ζ but 𝔖_{f} is elsewhere holomorphic on C, it follows that deg 𝔖_{f} = 2k + l − 2, where k is the number of critical points and l is the number of logarithmic ends; consequently, #S(f) − 2 ≤ k + l − 2 ≤ deg 𝔖_{f} as claimed in Part A. Furthermore, under the assumption of Part B we have ord_{∞}𝔖_{f}(z) dz^{2} = 2ord_{∞}𝔑_{f}(z) dz^{2} , so that there are an even number of logarithmic ends, corresponding alternately to finite or infinite asymptotic values. As ord_{ζ}𝔑_{f} = −1 at each critical point ζ but 𝔑_{f} is elsewhere holomorphic on C, it follows that #S(f) − 2 ≤ k + \frac{1}{2}l − 1 = deg 𝔑_{f}.

Theorem 3 admits various extensions and refinements, each of which may be transformed as above into a version of Theorem 1. We conclude this note with a survey of such possible adaptations. In one direction, Fatou’s theorem on the existence of singular orbits in attracting and parabolic immediate basins has been extended to the class of finite type complex analytic maps, namely those analytic maps f: W → X where X is a finite union of compact Riemann surfaces, W ⊆ X is open, and S(f) ⊂ X is finite; details may be found in [4]. The argument proving Theorem 1 applies more generally to finite type maps f: W → Ĉ, with the conclusion that ord_{ζ}𝔖_{f}(z) dz^{2} ≤ #S(f) − 2 for every ζ ∈ W. The case when Ĉ − W is finite is of particular interest, for then 𝔖_{f}(z) dz^{2} is meromorphic except possibly at isolated essential singularities. The simplest examples of such maps would appear to be those for which 𝔖_{f} is rational, and the above sketch suggests that Theorem 2 remains valid without the assumption that f be meromorphic on all of C. In fact, it is highly plausible that Nevanlinna’s topological characterization extends, so that the following should be equivalent for analytic maps f: W → Ĉ with W ⊆ Ĉ:

(1) f is a map of finite type with finitely many critical points and finitely many logarithmic ends,

(2) 𝔖_{f} is a rational function and Ĉ − W is finite. However, we have not seen such a claim demonstrated or even explicitly stated in the literature.

One might also seek to exploit the fact that Fatou’s theorem actually bounds the parabolic multiplicity of any rationally indifferent periodic point, and sometimes furnishes a supplementary constraint on the associated holomorphic index (see [3] for precise statements). The obvious modification of the argument proving Theorem 1, namely consideration of the maps ω · T_{f,ζ}^{(j)} ∘ f for other roots of unity ω, thereby yields additional restrictions on various combinations of Taylor coefficients. However, it is not clear that the conclusions so obtained place illuminating restrictions on the nonlinearity and Schwarzian derivative.

References

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[2] Bergweiler, W., and A. Eremenko: On the singularities of the inverse to a meromorphic function of finite order. - Rev. Math. Iberoamericana 11, 1995, 355–373.

[3] Epstein, A.: Infinitesimal Thurston rigidity and the Fatou–Shishikura inequality. - Stony Brook IMS Preprint 1999/1.

[4] Epstein, A.: Towers of finite type complex analytic maps. - Ph.D. Thesis, CUNY, 1993.

[5] Epstein, A.: Dynamics of finite type complex analytic maps. - Manuscript in preparation.

[6] Eremenko, A.: Entire and meromorphic solutions of ordinary differential equations. - In: Complex Analysis I, Chapter 6, Springer-Verlag, New York, 1997.

[7] Hille, E.: Ordinary Differential Equations in the Complex Domain. - Wiley-Interscience, New York, 1976.

[8] Hinkkanen, A.: Iteration and the zeros of the second derivative of a meromorphic function. - Proc. London Math. Soc. 65, 1992, 629–650.

[9] Laine, I.: Nevanlinna Theory and Complex Differential Equations. - Walter de Gruyter, Berlin, 1993.

[10] Morosawa, S., Y. Nishimura, M. Taniguchi, and T. Ueda: Holomorphic Dynamics. - Cambridge University Press, Cambridge, 2000.

[11] Nevanlinna, R.: Analytic Functions. - Springer-Verlag, New York, 1970.

Received 24 January 2001