SOME PROPERTIES AND CHARACTERIZATIONS OF A-NORMAL FUNCTIONS

Let M be the set of all functions meromorphic on D={z∈ℂ:|z|l1}. For a∈(0,1], a function f∈M is called a-normal function of bounded (vanishing) type or f∈Na(N0a), if supz∈D(1−|z|)af#(z)l∞ (lim|z|→1(1−|z|)af#(z)=0). In this paper we not only show the discontinuity of Na and N0a relative to containment as a varies, which shows ∪0lal1Na⊂UBC0, but also give several characterizations of Na and N0a which are real extensions for characterizations of N and N0.

-logldp(z)l is the Green function of D with logarithmic singularity at w e D. Also assume that a (0, 1] and M is the class of functions meromorphic on D. For f M, let f#(z) [f'(z)[/(1 + If(z) [2), which is the spherical derivative of f.Further we say f is an a-normal function of bounded type if IlfllN" sup(1-[zl)=f(z) < , (1.1) zD and f is an a-normal function of vanishing type if lim (I -Izl)f(z) 0. (1. 2) The families of all a-normal functions of bounded and vanishing type are denoted by N and N, respectively.It is easy to observe that N' C N and that for a E (0, 1), N and N are proper subsets of N and No, which are the classical sets of normal and little normal functions, namely, N N and No NJ, respectively.There has been much interesting research on N and No (see [1][2][3]), and hence we look for N and N to have some analogous properties.In this paper, we first consider the continuity 504 J. XIAO of the families N and N as a varies, and we find that both are discontinuous morever that N and J /s<<1 N 0<<1/2 are proper subsets of D# and UBCo respectively, where D# is the family of functions f M satisfying =/D If# (z)]s din(z) < x, (1.3)   and UBCo is the family of functions f M satisfying lim f [f#(z)]Sg(z, w)din(z) O. (1.4) Here, it is worth while mentioning that D# C UBCo and that UBCo is an important meromor- phic counterpart of VMOA--the space of analytic functions with vanishing mean oscillation on D (see [4,8]).We then characterize functions in N and N' and obtain three criterions which are extensions of criteria for N and No.
2. CONTINUITY OF N AND N'.
Denote by D the class of functions f M with =0.
(..1) For a (0, I), it is easy to see that D C N. Furthermore, Theorem 2.1, together with N C N, N C N0 and [3,4] suggest that we coider the continuity of N and N;.For th ppo, we need a corollary which can be viewed an application of Theorem 2.1.COROLLARY 2.2 Let a, b (0, i].Then (i).U< N U< N. (n).N< N N< g, PROOF.(i).On the one hand, the relation: U< N$ U< g is clear.On the other hand, if f U< N, then f mu be in some N , saying, N , where a E (0, b).However, for a' (a,b) we have f N' by the proof of Theorem 2.1.So f U<N, and hence U<N =U<N .
(ii).This part can be proved similarly.Now, we c state the dcontinty of N and of N THEOREM 2.3 Let a, b E (0, 1].Then (i).U,<,, g c No .
Since lim_ (-) for b > a, f2 <Nd from [7, Theorem 1].But it follows that 6 N again from [7, Theorem 1].Note that we have here used a fact: f# equivalent to lf'[ once f is bounded and analytic on D. The above facts tell us that < N # N is true.

It clear that f is
Second, let us consider (2).For this, we pick fa(z) = bounded on D. Moreover f N by using [7,Theorem 1].However, lim_ 0 as b > a, and then e < N.That is to say, < N # N.
Ts completes the proof.
Finally, we discuss a special ce of Theorem 2.3.Theorem 2.3 implies that 0<<] N C N0. Noting the inclusion: UBCo C No [8], we will naturally ask what connection between N is a proper subset of 0<< N and UBCo It is a little bit surpring to us that 0<< UBCo.This result shows that there is a big gap from 0<<] N" or 0<<] N to N or N.
3. CHACTERIZATIONS OF N AND N.
In this section, we chartere functions in N d N for a (0, 1] in terms of the weighted average, the pudhyperbolic dk and the Green function, respectively.We use [EI to denote the meure of the set E D relative to din(z), i.e., IEI f din(z).
sup wED JD PROOF.We prove this theorem in accordance with the order (i) (ii) (iii) (i) (iv) (iii).
This completes the proof.
For N' we have a similar result.THEOREM 3.2 Let f M, a (0, 1] and p (1, oo).Then the following statements are equivalent: (i). f N.
Also, for this r: and all w 6 D, F-om the condition: f E N' it follows that there exists a pl (0, 1) such that for [w[ > pl,  Step 5. (iv) (iii).This is a simple consequence of (3.12).
This completes the proof.
It is an open question to which of the results from th paper are lid for a (1, ).Similar questions may also be ked about corresponding cls of harmonic functions (Cf [3]).