Some properties of Fatou and Julia sets of transcendental meromorphic functions

THEOREM 1 . 1 . Let f{z) be a transcendental meromorphic function in C with li = fi(f) < oo and 6 = 6(oo, / ) > 0. If \i = 0, then E = [0,2n); If fi > 0 and Jf has an unbounded component, then mesE ^ min< 2TT, — arcsin y - > . We make some remarks on Theorem 1.1. (1) If Jf has only bounded components, we do not know if Theorem 1.1 holds. In this case, Ff has at most an unbounded component. If Ff has no unbounded components, it is obvious that E = [0,2n). If Ff has only an unbounded component U,

Theorem 1.1 it follows that Theorem 1.1 holds. Then we are left with the case when U is invariant. In this special case, if for some a S Jf, CF{(CL) > 0 (please see the statement before Lemma 2.2 for its definition), then Theorem 1.1 still follows from Lemma 2.2 and the proof of Theorem 1.1. (2) The condition that 6(oo, f) > 0 is necessary. Observe f(z) = A tan 2, Agfi, the real axis. It is easy to get /*(/) = 1 and {(00, /) — 0. It was proved in [3] that when A > 1, Jf = 5ft, then E = {0, TT},

THEOREM 1 . 2 . Let f(z) be a transcendental meromorphic function in C satisfy- ing (I). Then E= [O,2TT). REMARK. (1) above suggests a further discussion of non-existence of the unbounded periodic components of Fj, which was investigated in Zheng [11, 12].

THEOREM 1 . 3 . Let fj (j = 1,2,.. .,N) be transcendental meromorphic functions. Assume that there exists a sequence {rn} of positive numbers which tends to infinity such that (2) l i m^«Ji) = O O ) n-»oo rn and for each j and sufficiently large n, there is a Rj>n ^ rn, such that (3) L(Rj<n,fj)>rn, j = 2,...,N. Define g(z) = fxo ••• o /AT(2). Let D be a hyperbolic domain in C such that for p > 0, gp(z) : D -» D is analytic. If for some a € D, gnp{a) —t b € 3D, assume, in addition,

Theorem 1.3 is a generalisation of results in [12]. For example it was proved in [12] that a transcendental meromorphic function has no unbounded (pre)periodic Fatou components if it satisfies (2). (ii) If / is a transcendental meromorphic function of order A = A(/) < 1/2 and 6(oo, / ) > 1 — cos 7rA, then for arbitrarily large r > 0, we have a R ^ r such that L(R,f) > r. In fact, we can take A(/) < a < 1/2 such that 6(oo,f) > 1 - cos7ra. From [6], the set F := {r > 1 : logL(r,/) has lower logari

LEMMA 2 . 1 . Let f(z) be transcendental and meromorphic in C with finite positive lower order \i — fi(f) and such that S — <5(oo, /) > 0. Define for r > 0 (4) D(r) := {e € [0, 2TT) : log+|/(rei8)| > ^T{r, /)}. Then there exists an unbounded sequence {r;} ofr such that for sufficiently small e > 0 we have a jo such that when j Js jo, (5) mes D(rj) ^ min< 2TT, — arcsin y - | — e. An open set is called hyperbolic if it has at least three boundary points in C. We define the hyperbolic metric on an

LEMMA 2 . 2 . Let f(z) be analytic in ft(ro;#i,02), U a hyperbolic domain and If there exists a point a € dU\{oo}, such that Cy(a) > 0, then there exists a constant d > 0 such that for sufficiently small e > 0, we have (6) |/(z)|=O(|z| d),2^oo, 2efi(r o;0i+£,0 2-e). PROOF: Write Q = £l(ro;9i,02)- Since f(Q,) C U, from the Schwarz-Pick Lemma we have (7) From the definition of Cu(d), we have (8) Xu(f(z))\f'(z)\ > Cu(a) j*'W z € SI On the other hand, since for e > 0 and z € QQ — Q(ro;6i + e,62 - e)

Theorem 1.1, an application of Lemma 2.1 implies that there exists a sequence {r;} of positive numbers such that mesD(rj) > a — t > 0, where D(rj) is defined as in (4). Obviously mes(£>(r,) n 5) = mes(D(r7)) \ (£ n £>(r;-)) ^ mes £>(r7-) -mesE^ K >0. Thus for each j we have mes I t = l =mes(5nD(rj)) -mes ((5 \(J/ 4) n £>(r,-)) 2 ~ 2 ' so there exists an open interval / = J;o c 5 such that for infinitely many j , (11) mes(D(rj) n /) > ^ - > 0. It is easy to see that we can assume that for each j

Theorem 1.1 follows. 3. PROOF OF THEOREM 1.3 In order to prove Theorem 1.3, we need the following result.

LEMMA 3 . 1 . Let D be a domain with at least three boundary points and f be meromorphic in C except possibly at most countably many essential sigularity points such that f(D) C D. Then one of the following mutually exclusive possibilities can occur. (1) There is a subsequence of {fn{z)} which converges to z in D. (2) fn(z) -*b&T),~D is the closure ofD.

Lemma 3.1 can be proved from the arguments of Heins [8]. We also need the following result, which is due to Zheng [12] and of independent significance.

LEMMA 3 . 2 . Let U be an unbounded hyperbolic domain and f : U —> U analytic. If fn{z)\u -+oo, n->oo, then there exists a curve 7 tending to infinity and a constant £ > 1, such that f(j)Cj and lA^\f(z)\^c\z\,Vzer PROOF OF THEOREM 1.3: Suppose that D is unbounded. Take a point a e D and a positive number M such that (16) \a\ < M and f, o • • • o /^(ff'(a))| < M, for j = 1,2,..., N and i = 0,1,... ,p. From (2) and (3) for arbitrarily large K > 3 we can take a sufficiently large R> M such that (17

Theorem 1.3 follows. REFERENCES [1] A. Baernstein, 'Proof of Edrei's spread conjecture', Proc. London Math. Soc. 26 (1973), 418-434. [2] I.N. Baker, 'Sets of non-normality in iteration theory', J. London Math. Soc. 40 (1965), 499-502. [3] W. Bergweiler, 'Iteration of meromorphic functions', Bull. Amer. Math. Soc. 29 (1993), 151-188. [4] A. Edrei, 'Meromorphic functions with three radially distributed values', Trans. Amer. Math. Soc. 78 (1955), 276-293. [5]