AN EXAMPLE OF A BLOCH FUNCTION

. A Bloch function is exhibited which has radial limits of modulus one almost everywhere but fails to belong to H p

Let E be the subset of the complex plane consisting of the closed unit 2 disc together with the Gaussian integers 2Z  Let G be the complement of E in in .Let g D + G be the analytic universal covering map of G given by the uniformlzation theorem (D denotes the unit disc).
PROPOSITION.The function g is an unbounded Bloch function with the proper- ties (ei0 le (1) g has a radial limit g at almost every point e of the unit circle.
(ii) the function g(e io) is of modulus one almost everywhere on the unit circle, (iii) g is the reciprocal of a singular inner function, and so g does not belong to any H p class.
Bloch functions on the unit disc may be defined as those analytic functions f on D for which the radii of the schlicht discs in the range of f are bounded above.The Bloch functions are somewhat anal- agous to functions in the disc algebra--Bloch functions can be characterized (see [i]) as those analytic functions which are uniformly continuous when D is given the hyperbolic metric and the Euclidean metric.
Since Bloch functions may be characterized (see [i]) as those analytic functions f on D for which the quantity f'(z) (i zl2) is bounded for z D it follows that the modulus of a Bloch function grows rather slowly--at most as fast as log(I/(l zl)) Because functions in the disc algebra and bounded functions have good boundary behaviour, it is natural to ask about boundary values of Bloch functions-- in particular about radial boundary values.(It is shown in [4] that a Bloch function has a radial limit at a point of the unit circle if and only if it has a non-tangential limit there.) In [5], Pommerenke gave an example of a Bloch function with radial limits almost nowhere.The example given here is constructed in a similar way, but it contrasts with Pommerenke's in that it shows that Bloch func- tions which have radial limits almost everywhere need not be particularly well-behaved.
The example answers a question posed by Joseph Cima (private commun- ication).He asked whether a Bloch function which has radial limits EXAMPLE OF A BLOCH FUNCTION 149 almost everywhere and has the additional property that the boundary function belongs to L p need be in H p The function g provides a negative answer to this question since g(e i@) L while g H p for + any 0 < p < In fact g does not belong to the class N (see [2] p. 25) which contains H p for every p PROOF.It is evident that g is an unbounded Bloch function.Also, to verify properties (i), (ii) and (iii), it is clearly sufficient to verify (iii).
To establish (iii), consider the analytic function f i/g on D The function f is bounded (by i) and is the universal covering map Being a bounded analytic function, f has radial limits almost everywhere on the unit circle.It is easy to see from the properties of covering maps that these radial limits are either of modulus i or else belong to K To complete the proof that f is a singular inner function, it is only necessary to show that the radial limit f(e belongs to K on a subset of the unit circle of measure zero.
But, for each k K it is true that the set of e i0 for which f(ei@) k has measure zero (see [2] p. 17).Since K is countable, it follows that the set of e io for which f(e belongs to K also has measure zero.The proof is now complete. The example may also be viewed as elucidating the almost total lack of relationships between the class B of Bloch functions on D and the subclasses H p and N + of the Nevanlinna class N (see [2] ).The only containment which holds between B and the other classes is the relation

H c__B
It is known that H p B for any 0 < p < and that N The example g given above belongs to 8 N but not to N

+
The fact N is shown by the example of Pommerenke's [5] mentioned above.
Finally, the example given here can be modified to show that there is no > 0 such that an analytic function f D + satisfying e i8 f(eiS) Lira f(r 1 almost everywhere on the unit circle must have a disc of radius in its range.(Merely replace 2 by 2 in the construction of g).This answers a question raised by J.S. Hwang.By contrast, he showed (see [3]) that a singular inner function (for example) must have a (Schlicht) disc of radius at least 2B/e in its range, where B denotes Bloch's constant.