Oiii iesu!:~ extend to the range 0 < p (: 2 known results of Zygmund and Hallenbeck and Samotij valid for p = I. AMS No. 30D50,30D55. Communicated: A.
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Sharp estimates on the radial growth of the derivative of bounded analytic functions
Daniel Girela a; María Del Mar Rodríguez a Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain Online Publication Date: 01 January 1996 To cite this Article: Girela, Daniel and Rodríguez, María Del Mar (1996) 'Sharp estimates on the radial growth of the derivative of bounded analytic functions', Complex Variables and Elliptic Equations, 28:3, 271 - 283 To link to this article: DOI: 10.1080/17476939608814858 URL: http://dx.doi.org/10.1080/17476939608814858
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Sharp Estimates on the Radial Growth of The Derivative of Bounded Analytic Functions DANIEL GIRELA* Analisis Matematico, Fac~iltadde Ciencias, iln!:vefsidadde 'eddlaga, 29077 Mdlaga, Spain
MARIA DEL MAR R O D R ~ U E Z i. 8.Geraid Brer;anl Alhaflri~!de k Torre?29130 F.4ilqp, Spain In this paper we prove that if 0 < p
< 2 and f
IS a
iunction atlaiytic
iii
the unit d:ss
thrn
for airnost every 6 iii :lie set ef :hex B such tha: / PAS r? noa-tangential limlt at eiC. in pariicuiar, this hoids for l i i i i i ~ cx:y ~t a in R i f f i~ in thc Nevanlinna class A' and we prove that this resuit is s h i p in a very strong sense. Namely, we prove that if O < p < 2 then given any posltive Func~iomF ( I j dcfiriec! 2:: [0,1) and tending to 0 as r --,1 there exists a function f analytic in A and continuous o n such that
/'(I - p)p-llf '(peis)lf'dp
+0
10
{
( ,
~ ( r )log r ) ' p
' 2 } ,
as r
-
1, for almost every d E R.
Oiii iesu!:~ extend to the range 0 < p (: 2 known results of Zygmund and Hallenbeck and Samotij valid for p = I. AMS No. 30D50,30D55 Communicated: A. Baernstein I1 (Received February 18, 1994)
1. INTRODUCTION AND MAIN RESULTS
Let A denote the unit disc {z E C : lzl < 1). For O < r define
< 1 and
~g(re'~)~PdO,0 < p
g analytic in A
< oo,
For 0 < p 5 oo the Hardy space HP consists of those functions g analytic in A for which
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Complex Variables, 19%, Vol. 28, pp. 271-283 Reprints available directly from the publish& Photocopying permitted by license only
D. GIRELA AND M. DEL MAR RODR~GUEZ
I f f is an analytic function in A and 8 E [ - r , ~ ] ,we define
m.
lne11 'v'(fleiaj rienuies iht: radiai va~iaiiu~l u i f oa iiie radius of A whi~iiiizriiiinates at the point e i e . Rudin proved in [7] that for f E Hw the set (8 E [-T,T] : ,,/r -lH\ < c e / majj be o: iiieasiiic z c i ~ i i dhe icf: open the ijuestion of w h ~ t h c ii: t J c may be empty. This has been recently disproved by Bourgain in [2]. On the other hand, Zygmund proved in [8] that if f is an analytic function in A
for almost every H in the set of those H such that f has a non-tangential limit at eio". In particuiai, j l j hoids for aimow cvcry 6 in i-?;,rj i f f is in ii". Halienbeck and S,!l?ntii"J havr it1 j S j that tllic rerrult ic chnrp in n very strang sense namely -'..'-p~i>vrd they proved the foiiowing resuit. T r r n n n c n r I nLj"nLI"L
AL [r J 4,
1
-
Thnnmmc I IIbVIbIII.,
[u,ij and tending to 0 as r on A such that
deJened Q,? i rhere e-xisrs ri fiincrivn J' anaijjiic in A and cnn.iinuou.v
2 Ul'U -nA
J
A1
7,
/L.A,nn n n r j n n r ; t f i ~ of r r p . f ; r , n U I I C l l UIIj y"0l.l.r J I 1 I"
(r)
""9.
for almost every 8 E [ - a , ~ ] .
A well known result of Littlewood and Paley [6] states that if 2 5 p < m and f E HP then
JLu- I
W - ~ I ~ W I 0 is arbitrary this finishes the proof.
3. PROOF OF THEOREM 2
Following the notation of [5, p. 441, let L denote the class of functions f analytic in A given by a power series with Hadamard gaps of the type
It is well known that L c HP for 0 < p < GO and, even more L c VMOA (see [I, p. 251 or 15, Lemma I]). In order to prove Theorem 2 we start proving the following weaker result.
THEOREM 3 Let 0 < p < 2 and let c(r) he a positive function defined on [O, 1) and tending to 0 as r -+ 1. Then there exists a function f E L which satisfies ( 8 )for almost every 19 E [-.rr,.rr]. The arguments that we are going to use to prove Theorem 3 are related to those used by Zygmund in [8, pp. 198-2001. We start with the following result about power series with Hadamard gaps.
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274
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275
RADIAL GROWTH
.%mfc!f Lemma 1 Using Lemma 6.5 in [9, Vol. I, p. 2031, we deduce that there exst 2 mtcra! number NO = iV0'o(Eiq)and a positive constant 14 = A ( E , y j siich that if g is a n analytic functior! in A as in ( i Z ) then
llsing Theorem 8.20 in [9, Vol. I, p. 21.51 with r = 4, we deduce that there exists a positive constant C = C ( q ) such that for 0 < p < 1
Then, using Hdder's inequality with the exponents l j and ~ 1/(1ing in mind (15),we obtain for 0 < p < 1
J,
t),
f
1~(~e")1'dQ = JE ~ g ( p ' * ) ~ ~ r ( g ( pj~"-')'dQ e10
r (J, ~ g ( p e ' do) ~ ) ~ (J ~E~
which, with (14), implies
p ' t ~ d o )
(16) and hav-
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276
D. GIRELA AND M. DEL MAR RODRIGUEZ
Finally. using (151, it is easy to see that (17) implies (13) with B = ( A ' c ~ ~ ( ' ~ ' ) ) ' ~ ( ' ~ ! . This finishes the proof of Lemma 1.
Proof of Lemma 3
We may assume without loss of generality that ~ ( r is) de-
Let (a;)r"_(, be defined by a! = 0 if 0 ( j - ,
< nl
and for every k
>1
Using (22), it is easy to check that
Notice that (24) shows that f t L. Let E be an arbitrary subset of [ - ~ , nwith ] positive Lebesgue measure. Setting q = 2, take the integer No and the positive constant B whose existence is ensured in i x m m a I . i,et M be a positive integer such that iv;, :2"' and iet
j=M
Then, w e have
Using (27); the definition of r; and the elementary inequality (1 - 2-!j-'))2'p 2 e-jp ( j > 1) we deduce, for n > M,
j=M+l
i n particular, we have
Now, using (231, (21) and (19) we deduce that, for
nk-1
>M,
which, with (28) and (20), implies
- p)P-llg1(peie)(pdpd0 = 00. limsup J Sr(l O l-Pl2 r-1 Now, having in mind (25) and (26), it is clear that (29) and (18) imply
- p)p-'l f '(peie)lf'dpd8 = 00. limsup J Jr(l O r-1
Since the set E is an arbitrary subset of [-r,n]with positive measure, we can deduce easily that
.
lim sup -1
J,V- p ) ~ - l I fl ( ~ e i e ) I P =d ~ .
' i ' 1
f \
for almost every ~ y ~ " ., 00
r i
8 E [-n,n].
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277
RADIAL GROWTH
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278
D. GIRELA AND M. DEL MAR RODR~GUEZ
Once that Theorem 3 has beer! proved our next step will be showing that (7) is sharp in ti". THEOREM4 Let 0 < p < 2 and let ~ ( rhe) a positive function defined on [0,1) and fs 25 y + 1. The,? !here -Slz Li' jfziyClic,z Jf c ;:.fZtCA Su"tFTGnc/Q\ qnr almost every 0 E [-T, T]. ?n+!A;ivn ".A&>&.
" V .
JLC*
\V/
JC'I
l;ir~qf$i Ezeorem 4 Using Theorem 3, we can i a ~ ae filnciiiin h E L such that
for almost every B E [-r, 7ij. Since, as remarked earlier, L !IBMOA. thrrr cxists a function b E L"(SA) such that
(see [i, pp. 22-Bjj. Then we couid prove 'Thcorem 4 with the arguments used in the proofs of Theorem 3 of [Is] and Theorem 1 z.f [41. Howewr, we shall give a different nrnnf nf Thenrcm 4 which fniinws thr argirments !ised hy Rl-rdin in j7, p 1 3 9 . Since h E H< h is in the Nevanlinna class and hence it can be written in the h = hllhz with hl,hz E H" (see e.g. [3, p. 161). Let H b e the set of those 0 E [ - T , T ] such that (30) holds and the radial limits I, .(Die\ - I i m h ./r,&"? ,, j = !,2, ,',\b
,-"l"",\"
r-1
"
'
exist and are different from 0. Using Riesz uniqueness theorem (3, p. 171, we easily see that the Lebesgue measure of the set H is 27r. Now, for j = 1,2, let
Let 0 E H. Since h,(e1') exists and h j ( e t e )# 0, ( j = 1,2), there exist po E (O,1) and two positive constants M I ,M2 such that
M I 5 jh,(pet")l
< Mz,
if
po
< p < 1,
j
=
i,2.
Then it is easy to see that, setting M = max(M;', M ~ M ; ' ) , we have
which, with (30), implies that
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RADIAL GROWTH
for at least one value oi
j t
{1,2). Coilsecjurfitly, we deduce that
Now for each s E (u. i j ciefinc liit f u i i i i i ~ n sg , 5j
and the sets G, by
-
- . ;, -, !i. I.> 2,
n. > 1,
D. GIRELA AND M. DEL MAR R O D ~ G U E Z
Now the definition of En, (38) and (42) show that if n > 2, 0 E En and pg is the one from the definition of En then
i
1
= n € ( p e ) log A -PC
Consequently, since in (43) we have pn-1 that if
\,
I
'-p/2\
l'p
) -
< pe and pn-1
+ 1,
as n -, oo,we deduce
then. limsup
J;(l
- ~ I P - l l f' ( P ~ ' "dIp~ = oo
for every B E E .
r-1
Notice that (41) implies that the Lebesgue measure of E is 27~.This completes the proof. References
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282
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283
14) D. wreia, Growth of the derivarivr. u l buurdcd a~iaij.iicfiiiicti~iis,Coi?ip!ex I%;;i;b!cs Thcov ,4pp!. 20 (1992), 221-227. IS] D. J. Hallenbeck and K. Samotij, On radial variation of bounded analytic functions, Complex Variables Theory Appl. 15 (19%). 43-52. [6] J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series (11), J. London i ' v i ~ i : i . S X 42 (1931); 52 YY. [7] W. Rudin, The radial variation of analytic functions, Duke Marh. J. 22 (1955), 235-242. [8] A. Zygmund, O n certain integrals, Trans. Amer Math. Soc. 55 (1944), 17&204. 191 A. Zygmund, Pigonometric series, Cambridge University Press, 1959.
