ON BLOCH FUNCTIONS AND GAP SERIES

ON BLOCH FUNCTIONS AND GAP SERIES

The Nevanlinna class, denoted by N, consists of those functions f analytic in the unit disc U for which sup o< , « T(r, f) < oc, where T(r, f) denotes the Nevanlinna characteristic of f.Kennedy proved that if f E N then f i 1 (1) (1-r) exp(2T(r, f'» dr < oo and (2) liml Clog 1r o r -T(r,f')) =00 .
Both results are sharp .We note that (2) follows from (1) .Let B denote the space of Bloch functions .Two important subspaces of B are those denoted by BO and Bl .The space BO consists of those f E B such that (1 -Iz1) j'(z)j -> 0, as Iz1 -1, and Bl consists of those f E B such that if {z,} C U and 1f(z,,)1 -oo then (1 -1z,1)1 f'(z n)1 -0.It is well known that VMOA C Bo and BMOA C B. Kennedy's estimates are actually sharp for VMOA functions .In this paper we study the question of whether or not (1) and/or (2) remain true for a function f in B, Bl or BO .We prove that (2) need not be true for a Bloch function showing that the trivial estímate T(r, f) _< log i 1 r +0 (1) is the best that we can say in general .However (2) is true for any f E BO even though it may not satis£y (1) .We do not know whether or not (2) is true for any f E Bl but we can prove that it satisfies (2) with lim sup instead of lim .
Also, we generalize these results obtaining sharp comparison results between the integral means Mp (r, f) with T(r, f) for certain classes of functions f analytic in U.
It is well known that f' need not belong to N even if f is bounded .This was first proved by Rostman [6] who showed the existente of a Blaschke product whose derivative is not of bounded characteristic.Kennedy determined in [14] as closely as possible the restriction imposed on the growth of T(r, f') by (1.2) .He proved the following two theorems.
Then there exists f E N such that for all r sufficiently close to 1 .

. Introduction and main results
(1 -r) exp p(r) is decreasing.
1h(r)p(p)oo as r 1 -T(r,f') > M(r) Let us notice that, since T(r, f) is an increasing function of r, Theorem A implies that if f E N then 1 (1.5) log 1 r -T(r, f') -oo, as r --> 1 .
Also, since the function tt of Theorem B is increasing, (iii) shows that (i) is equivalent to (iv) log 1 1 r -p(r) T oo, as r T 1.
The author has recently obtained in [9] the analogues of Kennedy's results for analytic functions with finite Dirichlet integral in U.
The function f constructed by Kennedy to prove Theorem B is given by a power series ECkz' k with Hadamard gaps such that F-ICk 12 < oo.Such a function belongs to HP, 0 < p < oo, and, even more, to VMOA.This follows from Paley's multiplier theorem [3, p. 104] and the duality of Hl and BMOA [7, p . 270].Hence (1.3) and (1.5) are sharp (in the sense of Theorem B) for VMOA functions .
The question as to whether or not there exists a function f analytic and bounded in U with f satisfying the conclusion of Theorem B remains open.Kennedy pointed out in [14] that in dealing with this problem one could exclude functions f (z) = 1: akznk having Hadamard gaps.This is because if such a function is bounded in U then E Iakl < oo [20, vol. I p. 149 and 247] and so 1 (1 .6)expT(r, f')dr < oo 0 a stronger inequality than (1 .5) .Clunie proved in [2] that there exists a function f analytic and bounded in U not satisfying (1.6) .
A function f analytic in U is said to be a Bloch function if The space of all Bloch functions will be denoted by B .Two important subspaces of B are those denoted by BO and B I .The space BO consists of those f E B such that (1-IZI 2) I f'(z)I , 0, as Iz1 -> 1.Alternatively, Bo can be characterized as the closure of the polynomials in the Bloch norm [1, Th.
The space Bl was introduced in [10, p. 30] and [1, p. 36] where it was conjectured that if f(z) = Eñ=o a,zn E BI then a, ~0.This was disproved by Fernández [4], [5].If f E B then The first result in this paper asserts that this is essentially the best that we can say, showing that (1.5) and, hence, (1.3) need not be true for a Bloch function.However, we will prove that (1.5) holds for any f E Bo even though it may not satisfy (1.3) .
Theorem 1.For each integer q > 5 let Then 00 Then fq is a Bloch function and there exists a constant Cq such that Let us notice .that the function T of Theorem 2 can be taken to be and, hence, we obtain.

ON BLOCH FUNCTIONS AND GAP SERIES 407
Corollary 1.There exists f E Bo such that for all r suiciently close to 1 and, hence, satisfying If D is a B1-domain and f is the universal covering map of D then [12] f satisfies (1.8) and, hence, (1.5) .It is known that (1.8) may not be true for a function in B1 [4], [5] .We do not know whether or not (1 .5)remains true for any function f E Bl .We can prove the following result .
The early stages of this work benefited from conversations with A. Baernstein.He even told me that the conclusion of Theorem 1 should be true at least for sufficiently large values of q.It is a pleasure to express my gratitude .
Even though the motivation of this work was studying the possibility of extending Kennedy's results to Bloch functions, some of our results are more general than stated and, in fact, could be stated without making any reference to Bloch functions .
For f analytic in U and 0 < r < 1, define and Let us notice that, clearly 2. Proof of the main results Ixl=r For s > 0 and 0 < p < oo, let X9>p denote the space of those functions f analytic in U for which and let Xó'p denote the space of those functions f analytic in U for which Since M, (r, f) is an increasing function of p, we have (2.1)X',P C X'g,P and Xó, p C X''P, 0 < p < p < oo.
If p > 1 and f(z) = E°°o an z' E X',P (respectively X¿,p) then an applica- tion of Cauchy's formula easily gives an = O(ns) (respectively a, = o(ns)) .On the other hand, an argument similar to that used in [16,Example 1,p. 694] proves that if f(z) r_k o akznk is analytic in U and has Hadamard gaps then f EBg f' E X 1'' and f EBp ~¿f, EXó' , .
Hence theorems 1 and 2 will be corollaries of the more general results that we will prove for the spaces X' ,P and Xos,p .
If p < p' and f E Xs,P (respectively X`) then a result of Hardy and Littlewood (see [3, Th. 5.9]) shows that f E X",P (respectively Xo~'P) where The exponent s' is best possible .Using this result and arguing as in .[3,Th. 6.4] we can deduce that if 0 < p < 1 and f (z) = rñ=o a,z -E Xs,P then The function f(z) = (1 -z) -(s+ 1 /P) for which a,,, -r (s + P l ns -1 + 1/P shows that this estimate is sharp .Now, if p' < p and f E Xs,P then it is easy to see that the trivial result f E XS,P is the best that we can say in general .In fact, there exists f E X',' such that for every p E (0, oo] there exists a constant BP, s > 0 such that Indeed, let q > 2 be an integer and 00 f(z) = Egkszqk, Then, since f has Hadamard gaps, (2.2) shows that f E Xs , w .Now, it is a simple exercise to show that there exists a constant ,0s = ,6.,,q> 0 such that shows that the constant AP given there is of the form for some 5 q > 1 .This and (2.7) shows that 7s = -oo and hence (2.9) gives no information at all .However, we will prove in Theorem 4 that there exists f E X' ,1 satisfying (2.9) with a constant C3 in the place of -ys and, also, satisfying (2.4) with a constant B3 > 0 independent of p in the place of BP,3.Theorem 4. (i) Let s > 0, 0 < p < oo and f E X3,P, then (2.10) f EX s, P' , 0<p'<p, and (2.11) T(r, f) < slog 1 1 r +0(1) .
For s = 1, the conclusion of (ii) holds with f = fé for any integer q > 5.
Theorem 5 gives the analogous results for the spaces XOS, .
Then it is clear that (2.11) (respectively (2.16)) holds if f E X" (respectively if f E Xó'P) .
Proof of Theorem 4(ií): Let s > 0. Let q > 2 be an integer to be determined later and Then, for Iz1 = rn (2 .17) Then, since f is given by a power series with Hadamard gaps, (2.2) shows that f E X8,00 .Let rn =1q-n , n=1,2,3, . . .Let 17 E (0,1) to be determined later.Using the elementary fact i e-1 as j -> oo, we deduce that there exists N such that In order to obtain an upper bound for III we will use the following lemma which will be needed several times in this paper.Lemma 1.Let q > 2 be an integer and let s > 0. Let {ak} be a non- increasing sequence of positive numbers.Then, for any integer m > s, we Nave akgks (1 -q-n)," < (me-1)7nan 1 qnz-s -1 qns ' n = 1 2,3, . . .
Proof of lemma 1 : We will use the following elementary inequality Back to the proof of Theorem 4, take an integer m > s.Then, using Lemma 1 with ak = 1 for all k, we obtain (2 .22)III < (me -1 )m

~_1
-q q ns . 1 Then (2.17), (2.19), (2 .20)and (2 .22)show that if (z) j ~: (M.,m,q -?1e -1 )g ns, where, Now, take q so large that M,,m,q > 0 and then take 91 = qq > 0 so small that TS = Ms,..,qrle-1 > 0.  Let us notíce that in the above argument given s > 0 we can take m to be any integer greater than s and that different values of m would lead to different values of q for which M,,,,,, q > 0. It turns out that for s = 1 the choice of m which minimizes q is m = 3.If we set Aq -M1,3,q we have A q > 0 for q > 5.In fact, we have I\ q > A5 -0.061, q > 5.
Consequently, we obtain that for each integer q > 5 there exists a constant Cq such that T(r,fq)>log 1 r +Cq, 0<r<1.1-This finishes the proof of theorems 4 and 1 .E Proof of Theorem 5(ii): First let us notice that we may assume without loss of generality that oP satisfies also the following two conditions .t>x Then it is clear that <P2 >_ <D and it is easy to see that (1 .12),(1.13), (2.23) and (2.24) hold with OD2 in the place of <D.Hence we will assume that -D satisfies (2.23) and (2.24) in addition to the conditions of Theorem 5.
In order to show that there exists f E Xó,m satisfying (2.17), let us notice that if ~D is the function given in Theorem 5 then <p 1/2 satisfies also the conditions (1 .12)and (1.13) .Hence, if we apply the above argument with X1/2 in the place of 4>, we deduce that there exists f E Xó" such that MP(r,f) ?q`(r) 1/2 ( 1 -r) -9 , 0 < p < oo, for all r sufficiently to 1 .Since T(r) -3 0, as r -> 1, this implies (2.17).

Some further results and final remarks a)
The results that we have proved are comparison results between Mp (r, f) with Mp, (r, f) and T(r, f) for f in some of the spaces Xs,p or XO'p .It is well known (see e.g.[3,Th. 5.10]) that there exist functions f analytic in U with M,,.(r, f ) growing to infinity arbitrarily slowly which are not of bounded characteristic.This leads one to ask the following question: Let p(r) be a positive increasing function on 0 <_ r < 1 with y,(0) = 1 and u(r) -oo, as r -1, and let f be a function analytic in U satisfying Mp(r, f) = 0(p(r)), as r ~1.What can be said about the growth of Mp, (r, f) and T(r, f)?In particular, it seems natural to ask whether or not the analogue of Theorem 4(ii) is true in this setting, Le. does there exist a function f analytic in U with and a constant C such that We do not know the answer to this question .However, we do believe that the methods of this paper are not enough to construct such an f. b) First of all let us remark that some of the results that we are going to state below (Theorem 6, Corollary 2, and Theorem 7) could be stated in the general framework of the spaces XS,P and XÓ'P .However, for the sake of simplicity, we will state them in the setting of Bloch functions .
It seems natural to conjecture that the conclusion of Theorem 1 remains true for q = 2, 3, and 4.However, our argument does not prove this since, with the notation used in the proof of Theorem 1, we have A4 < 0.
A more general question would be characterizing those Bloch functions given by a power series with Hadamard gaps for which (1.5) or at least (1 .16) is true.The following theorem gives a partial answer to this question .Theorem 6.Let f be a Bloch function given by a power series Using Theorem 2, we obtain as an easy consequence of Theorem 6 the following result .
Corollary 2. Let f be a Bloch funetion given by a power series znj with w j+1 --> oo, as j -oo.nj f(z) = j=1 Then f E BO if and only if 1 z"'', with n, + --~oo, as j -oo.nj  This easily implies that r-+1 1 -r finishing the proof of Theorem 6.
lim inf (log -T(rj» < 00 c) So far we have proved in Theorem 2 that if a Bloch function f satisfies (1 .8)then it satisfies (1.5) .Furthermore, the functions f considered in theorems 3 and 6 satisfy not only (1 .16)but also (1.15).These facts might load one to ask whether or not the converse of Theorem 2(i) is true.Theorem 7 shows that the answer to this question is negative in a very strong sense.(1 -r) < 1jw(re' t )j < 3(1 -r), 0 < r < 1, ¡ti < 7rH.
Pommerenke proved in [16,Th. 2] that if a Bloch function f has radial limits almost everywhere on Iz1 = 1 then it satisfies (1.8) .Notice that the function f constructed to prove Theorem 7 is in fact analytic on the set {e" : 7rH < ¡ti < 7r} and consequently it has radial limits on a set of positive measure.The next result asserts that if a Bloch function f satisfies this last condition then it satisfies (1.5) .Proof.Since a Bloch function is normal [16, p. 689], [1, p. 12] the concepts of radial limits and angular limits are equivalent for f [15] (see also [17,Th. 9.3]) .By Privalov's theorem [19, p. 320] these angular limits are finite almost everywhere.Hence, if we set then ¡El > 0.
d) There are other questions that we could ask in this context .For instance, it seems natural to ask whether or not (1 .5) is true if f belongs to the closure of H°°in B. The answer to this question is afirmative.Actually, it is easy to see that (1.8) is preserved under convergence in the Bloch norm and hence Theorem 2 and [16, Th. 2] show the following: If f E CLB(B n N) (the closure of B n N in B) then f satisfies (1 .5) .
Acknowledgments.I wish to thank the referee for his helpful comments, specially for his remarks about the generality of our results.Originally we just stated our results in the setting of Bloch functions and not in the more general framework of theoem 4 and 5.
r, f') / = oo.---T he proof of Theorem 6 depends on the following two elementary lemmas whose proofs will be omitted .

Theorem 8 .
Let f be a Bloch function having radial limits on a set of positive mensure.Then rlim1 (log 1 1 r -T(r,f~) / =oo.
2.1] .The space Bl consists ofthose f E B such that if {zn } C U and I [4]nández gave in[4]examples of functions f E Bl not satisfying(1 .8).If D is a B1 -domain, Le. if every function g analytic in U with g(U) C D is in B,, and f is the universal covering map of D then Hayman,