ASYMPTOTIC VALUES AND THE GROWTH OF ANALYTIC FUNCTIONS IN SPIRAL DOMAINS

ASYMPTOTIC VALUES AND THE GROWTH OF ANALYTIC FUNCTIONS IN SPIRAL DOMAINS J . E . BRENNAN AND A . L . VOLBERG In this note we present a simple proof of a theorem of Hornblower which characterizes those functions analytic in the open unit disk having asymptotic values at a dense set in the boundary. Our method is based on a kind of ó-mollification and may be of use in other problems as well .


. Introduction
It is a well-known fact that if f is analytic in the open unit disk D and if lf 1 is subject to a sufhciently strong growth restriction, then f has radial limits almost everywhere en 8D.In 1963, however, G .R. MacLane [14] extended that principle te a much larger class of functions, where radial limits are completely inadequate for a description of boundary behavior .A function f analytic in D is said to have an asymptotic value A at the point (o E áD if there exists an are I' lying in D with one endpoint at (o such that f (z) -A as z , (o along I'. Here, A = oc is allowed and it is entirely possible for a given f to have more than one asymptotic value at a single point (o E áD (cf .[14], [16]) .The MacLane class A consists of those nonconstant analytic functions having asymptotic values at a dense set in OD.
The principal goal of MacLane's work is to obtain conditions on a given f sufficient to guarantee that f E A, and his results are based en the following simple fact : If f is analytic in D and y is an are on áD then either (i) f has asymptotic values at a dense set in y or; (ii) there exists a sequence of Koebe arcs y,z tending to a nontrivial subarc y' of y such that 1 f (z) 1 = c > 0 en y,, n = l, 2, . . .In particular, if as usual, it can be shown (cf .[14]) that f E A whenever (1 -r) log+ M(r)dr < oo.0 Nearly a decade later Hornblower [10] (cf.also [9]) improved MacLane's result, replacing (1 .2) with the weaker, and apparently sharp, criterion log+ log+ M(r)dr < oo.0 In addition to establishing the sufficiency of (1.3) he also proved that, corresponding to each e > 0, there are functions not in A for which (1 .4)log+ log+ M(r) < e _ 1 -r log ) g ~llr) On the other hand, more than fifty years ago Valiron [22] (cf.also [23, p . 191]) had actually shown that Hornblower's examples must have the property that and he further showed that equality in (1.5) is possible, from which it follows that, for any e > 0, there are functions not in A with Unfortunately, the work of Valiron seems to have been overlooked by subsequent authors and we are grateful to A. E. Eremenko for bringing it to our attention .
The examples constructed by both Valiron and Hornblower are rather long and quite technically involved .Our primary objective is to present a simple example of the Valiron-Hornblower type based on a kind of á-mollification .Although this work ovas carried out before ove became aovare of [22], it is, nevertheless, closer in spirit to the work of Valiron than to that of Hornblower.
Our second objective is to give a new proof of the sufficiency of (1 .3)based on an idea of E. M. Dyn'kin [7] .Hornblower's original proof in [10] relies (and so indirectly does ours) on a well-known theorem of Beurling, Levinson, Sjdberg and Wolf concerning the existente of a greatest subharmonic minorant to a given function (cf.[1], [2], [12], [13], [21], [25]) .A special case of the latter was obtained much earlier by Carleman [4], but his method is quite general in nature .Later MacLane [15] found another approach leading to the sufficiency of (1 .3)and in the process gave a new proof of the Beurling-Levinson-Sjdberg-Wolf theorem, which, incidentally, ovas also discovered by Gurarii [8] .Definitive results in this direction can be found in articles [5] and [6] of Domar (cf .aleo [11, p . 374-383]) .
There are, of course, other ways in which to describe or capture the boundary behavior of analytic functions subject to a growth restriction, the most noteworthy being in terms of distribution theory.It follows from the Schwartz program that if f is analytic in D then the functions f,, = f (reto) converge as r T 1 to a distribution supported on áD if and only if there exist constante C, k such that The corresponding result for functions with very rapid growth is due to Beurling [1] and is this : As r 1 1 the functions f,-converge to a generalized distribution if and only if r .(1 .8)log+ log+ M(r)dr < oo.0 We do not, at present, know of any direct connection between the existente of asymptotic and distributional boundary values .

Functions with no asymptotic values
In this section we shall outline a procedure for constructing analytic functions with a given growth, having no asymptotic boundary values .Later in Section 3 we shall present the more technical details associated with that construction .
To this end let M(r) be a given nondecreasing function defined for 0 < r < 1 such that M(r) > e and let us assume that f log log M(r)dr = +oo .0 Setting p(r) = log log M (1 -2r) we obtain a nonincreasing function on 0 < r <_ 1/2 with fo /2 p(r)dr = oo .With only mild regularity assumptions en M(r) (or equivalently on tt(r)) we arrive at our main theorem: Theorem 1 .If M(r) is given as above satisfying (2.1) and if, moreover, (1) rp(r) + 0 as r + 0 (2) p(r) > cp(2r), 0 < r < 1/2 and c > 1 i (3) fo rN,(r) dr < oo then there exists a function f analytic in D, not however belonging to ,A, such that It should be noted here that p(r) = r log lar is a typical function satisfying the conditions of Theorem 1 and it represents the situation studied by Hornblower [10] .Also, taking u = log 1f 1 we recover a theorem of Rippon [20] and Hayman [9], since u is subharmonic and, as we shall see, has no asymptotic boundary values on aD .
As a first step in the proof we define Evidently, X is a monotonically decreasing function and X(0+) = limX(y) = oo.We can, therefore, define ~b(x) = X -1 (x) for x > 1 and 0 can be extended to the entire interval (0, +oo) in such a way that 0(0+) = oo.The domain U bounded by the two curves ~b(x) and O(x + 27r) will be of special interest. (2.4) The map J(w) = eiw takes 0 onto SZ = D \ F, where is defined on SZ and grows like l a in the corresponding spiral domains St = J o G-1 (Bf ) .In particular, if a = 1 + c is Glose to 1 then (iii) La (z) --> oo as z -OD, z E S+ (iv) La(z) --> 0 as z -OD, z E S--Hence, La can have no asymptotic values at 8D.However, it has jump discontinuities all along I' and so does not directly suite our purpose .
Our main task will be to construct an analytic function sufficiently Glose to La that it inherits the same behavior in S+ and S_ .
Step 1 .We begin by mollifying La in the following way : For each z E D let D, be the disk with center at z and radius 0(1-J zJ).Here, 0 is a small, but fixed, positive constant to be specified later.Next consider the average (2.6) Although F is still not analytic in D, if Q is chosen properly we can arrange that (4) F(z) = La(z) for z E ro = J o G-1 (R) ( 5 ) IF(z)I <-CM(I z1) (6) l&F(z)j < const < oo.Because F(z) --> oo as z --> óD along Fo and lF(z)j is bounded on I', there can again be no asymptotic values on 8D.
Step 2 consists in selecting an analytic component of F defined by setting Since, by virtue of properly ( 6), IF -Al < const < oo all the requisite growth restrictions are preserved and A(z) is the function we are looking for .
To establish property (4) it is sufficient to prove that if z E Po then the disk D z of radius ,l(1 -Iz1) does not meet I' if 0 is small (and fixed) and z is near aD .To this end let us suppose that z E I'o and let ( = G(-¡ log z) be the corresponding point in II, which, of course, lies on the real axis 1(8 .Denote by Q the family of curves lying in II and separating ( from rl = 7r/2, that is, from the top ; Q* is the conjugate family defined relative to the bottom .The length-width ratio of Q is where l(Q) and l(Q*) denote the infimúm of the lengths of the rectifiable members of Q and Q*, respectively .Evidently, p(Q) is bounded and bounded away from zero as ( ---> +oo along I[8 .Since this is a conformally invariant property of p, (cf.McMillan [17], [18]), it follows from (3.1) that 0 can be chosen as indicated ; hence (4) .
Thus, it remains only to prove (6) or, equivalently, that IbF(z)j <_ K < oo throughout D .To accomplish this we first express La as the sum of an analytic function H and the Cauchy integral of a measure v supported on I': The important thing to note here is that dv(z) = j(z)dz where, for each z E I', the density j(z) = ±(La(z + ) -La (z )) from which we easily conclude that IL. (~) I = I exp exp aG(-i log ~) I < FYom this it follows easily that if ( E F then is the jump of Lo, across F, the sign being chosen to agree with the orientation of F. Introducing the kernel  provided, of course, that ( is close to aD .Therefore, adjusting the constant in (3.8) to a account for the fact that z E D., but perhaps z ¢ I', we conclude that then f E A.
everywhere in D, and so is trivially bounded, since M(r) -1/r°as r ---> 0 by way of assumption (i) .

Functions belonging to class A
Our goal here is to give a short proof of the following theorem of Homblower [10] .Theorem 2 .Let f (z) be analytic and nonconstant in D and let M log+-log+ M(r)dr < co 0 All known proos, including Homblower's, are based on a result of Levinson [12,p. 135], concerning the growth of harmonic measure in a cusp (cf.also Beurling [2,p. 381]) .We are able to avoid any xnention of harmonic measure by making use of the following result of Dyn'kin, the details of which can be found in [7] .
Lemma.Let M(r) be a monotonically increasing funetion defned for 0 < r < 1 such that (4 .1) is satisfied and let y', -y be two subares of áD with y' C -y .Then there exists a function 0 E Cl such that (1) 0 -1 on -y' (2) 0 -0 outside a neighborhood of "y To prove Theorem 2 let us suppose that f is analytic in D and that (4.1) is satisfied .Assuming that f 1 A there is an arc y C OD, no point of which is the endpoint of an asymptotic path for f.It follows, without loss of generality, from our remarks in Section 1 (cf .(ii)) that there exists a sequence of Koebe arcs y, --> y such that f is bounded on U-ym .Now choose "y' C_ -y and corresponding function 0 as in Dyn'kin's lemma .Extend each Figure 3 are ryn, to a simple closed curve enclosing a region D in such a way that D, ---> D as n -> oo .For each n = 1, 2, . . .let gn : Dn, --> D be a conformal map with g n (0) = 0. Applying Green's theorem to f gn0 we conclude that, for each z in some fixed neighborhood of y', in a neighborhood of "y'.This, together with the boundedness of f on U-yn , implies immediately that f is in fact bounded near y'.To see this just choose \1, , \2 E y' and note that (z -\1)(z -\2) f is bounded .As a result f must have radial limits almost everywhere en y', contrary to assumption.Therefore, f E .A .