On Bloch functions and gap series

Publicacions Ma emá iques, Vol 35 (1991), 403-427 . Abs ac ON BLOCH FUNCTIONS AND GAP SERIES DANIELGIRELA Kennedy ob ained sha p es ima es o he g ow h o he Ne anlinna cha - ac e is ic o he de i a i e o a unc ion analy ic and wi h bounded cha ac e is ic in he uni disc . Ac ually, Kennedy's esul s a e sha p e en o VMOA unc ions . I is well known ha any BMOA unc ion is a Bloch unc ion and any VIVIOA unc ion belongs o he li le Bloch space . In his pape we s udy he possibili y o ex ending Kennedy's esul s o ce - ain classes o Bloch unc ions . Also, we p o e somemo e gene al esul s ob aining sha p compa ison esul s be ween he in eg al means M p ( , ) wi h T( , ) o ce ain classes o unc ions analy ic in he uni disc . The Ne anlinna class, deno ed by N, consis s o hose unc ions analy ic in he uni disc U o which sup o< , « T ( , ) < oc, whe e T ( , ) deno es he Ne anlinna cha ac e is ic o . Kennedy p o ed ha i E N hen i  1 (1)  (1- ) exp(2T( , '» d < oo and (2)  lim l Clog 1 - o  - T( , ')) =00 . Bo h esul s a e sha p . We no e ha (2) ollows om (1) . Le B deno e he space o Bloch unc ions . Two impo an subspaces o B a e hose deno ed by BO and B l . The space BO consis s o hose E B such ha (1 - Iz1) j'(z)j -> 0, as Iz1 - 1, and Bl consis s o hose E B such ha i {z  ,} C U and 1 (z,,)1 -oo hen (1 - 1z  ,1)1 '(z n )1 - 0 . I is well known ha VMOAC Bo and BMOAC B . Kennedy's es ima es a e ac ually sha p o VMOA unc ions . In his pape we s udy he ques ion o whe he o no (1) and/o (2) emain ue o a unc ion in B, B l o BO . We p o e ha (2) need no be ue o a Bloch unc ion showing ha he i ial es íma e T( , ) _< log i 1 +0(1) is he bes ha we can say in gene al . Howe e (2) is ue o any E BO e en hough i may no sa is£y (1) . We do no know whe he o no (2) is ue o any E Bl bu we can p o e ha i sa is ies (2) wi h lim sup ins ead o lim . Also, we gene alize hese esul s ob aining sha p compa ison esul s be ween he in eg al means M p ( , ) wi h T ( , ) o ce ain classes o unc ions ana- ly ic in U . 40 4  D . GIRELA Le be a unc ion analy ic in he uni disc U= {z E C : Iz1 < 1} . Then, he Ne anlinna cha ac e is ic T( , ) is de ined by (1 .1)  T ( , ) = 27 l°g+ 1 ( e'% d ,  0< < 1 . The Ne anlinna .class, deno ed by N, consis s o hose analy ic in U o which (1 .2)  sup  T ( , ) < oo . o<T<I I is well known ha ' need no belong o N e en i is bounded . This was i s p o ed by Ros man [6] who showed he exis en e o a Blaschke p oduc whose de i a i e is no o bounded cha ac e is ic . Kennedy de e mined in [14] as closely as possible he es ic ion imposed on he g ow h o T( , ') by (1 .2) . He p o ed he ollowing wo heo ems . (1 .3)  J I(1 - ) exp(2T( , ')) d < oo . 0 Theo em A . ([14, Th . I]) . Le E N . Then Theo em B . ([14, Th . II]) . Le p be a posi i e inc easing unc ion in (0,1) such ha l I (üi)  (1 - ) exp(2p( )) d < oc . 0 Then he e exis s E N such ha o all su icien ly close o 1 . 1 . In oduc ion and main esul s (1 - ) exp p( ) is dec easing . 1- h( ) - p(p) - oo as  1 - T( , ') > M( ) Le us no ice ha , since T ( , ) is an inc easing unc ion o , Theo em A implies ha i E N hen 1 (1 .5)  log 1  -T( , ') - oo, as --> 1 . ON BLOCH FUNCTIONS ANDGAP SERIES  405 Also, since he unc ion o Theo em B is inc easing, (iii) shows ha (i) is equi alen o (i )  log 1 1 - p( ) T oo, as T 1 . The au ho has ecen ly ob ained in [9] he analogues o Kennedy's esul s o analy ic unc ions wi h ini e Di ichle in eg al in U . The unc ion cons uc ed by Kennedy o p o e Theo em B is gi en by a powe se ies E Ckz' k wi h Hadama d gaps such ha F- ICk 12 < oo . Such a unc ion belongs o HP, 0 <p< oo, and, e en mo e, o VMOA . This ollows om Paley's mul iplie heo em [3, p . 104] and he duali y o H land BMOA [7, p . 270] . Hence (1 .3) and (1 .5) a e sha p (in he sense o Theo em B) o VMOA unc ions . The ques ion as o whe he o no he e exis s a unc ion analy ic and bounded in U wi h sa is ying he conclusion o Theo em B emains open . Kennedy poin ed ou in [14] ha in dealing wi h his p oblem one could exclude unc ions (z) =1 : akznk ha ing Hadama d gaps . This is because i such a unc ion is bounded in U hen E Iakl < oo [20, ol . I p . 149 and 247] and so 1 (1 .6)  expT( , ')d < oo 0 a s onge inequali y han (1 .5) . Clunie p o ed in [2] ha he e exis s a unc ion analy ic and bounded in U no sa is ying (1 .6) . A unc ion analy ic in U is said o be a Bloch unc ion i II IIB = SUP (1 - IZI 2 )I l (z)I + I ( 0 )j < oo . I=I<1 The space o all Bloch unc ions will be deno ed by B . Two impo an subspaces o B a e hose deno ed by BO and B I . The space BO consis s o hose E B such ha (1- IZI 2 ) I '(z)I , 0, as Iz1 -> 1 . Al e na i ely, Bo can be cha ac e ized as he closu e o he polynomials in he Bloch no m [1, Th . 2 .1] . The space Bl consis s o hose E B such ha i {z n } C U and I (z)( - o0 hen (1 - Izn1 2 )I '(zn)I  > 0 . Clea ly, BO C BI . I (z) = E°° o a,,zn E B hen sup Ia  ,I < oo [1] while i E BO hen a n --> 0 . Ac ually, he weake condi ion (1 .8)  12 ( , ') = o((1 - ) -2 ),  as ---> 1, is enough o conclude ha an  > 0 [16, p . 693] . He e, o g analy ic in U (1 .9)  12 ( , g) = 27 ,~ Ig( e") 12 d ,  0< < 1 . 40 6  D . GIRELA The space Bl was in oduced in [10, p . 30] and [1, p . 36] whe e i was conjec u ed ha i (z) = E ñ=o a  ,zn E BI hen a  , ~ 0 . This was disp o ed by Fe nández [4], [5] . Fe nández ga e in [4] examples o unc ions E Bl no sa is ying (1 .8) . I D is a B 1 -domain, Le . i e e y unc ion g analy ic in U wi h g(U) C D is in B,, and is he uni e sal co e ing map o D hen Hayman, Pa e son and Pomme enke p o ed in [12] ha sa is ies (1 .8) bu Fe nández p o ed in [5] ha he e exis s a unc ion analy ic in U whose ange lies in a B 1 -domain o which (1 .8) is no ue . The in eg al means and adial g ow h o B 1 - unc ions we e s udied by he au ho in [8] . Impo an examples o Bloch unc ions a e gi en by powe se ies wi h Hada- ma d gaps, Le . powe se ies (z) = Eñ=o akznk analy ic in U wi h nk+1 > " 1 nk o some cons an A > 1 . Fo such an we ha e [1, p . 19] E B i and only i sup jakI < oo and [4], [16] E Bo e-* E B I e* ak -+ 0 . I is well known ha VMOA C Bo and BMOA C B . The main objec o his pape is s udying whe he o no (1 .3) and (1 .5), which a e sha p o VMOA unc ions, emain ue o unc ions in he spaces B, BI, o Bo . I EB hen (1 .10)  T( , ') < log 1 1 +O(1) . The i s esul in his pape asse s ha his is essen ially he bes ha we can say, showing ha (1 .5) and, hence, (1 .3) need no be ue o a Bloch unc ion . Howe e , we will p o e ha (1 .5) holds o any E Bo e en hough i may no sa is y (1 .3) . Theo em 1 . Fo each in ege q > 5 le Then 00 q(z) = E z qk ,  ~ z1 < 1 . k-o Then q is a Bloch unc ion and he e exis s a cons an C q such ha (1 .11)  T( , ')>log 1 +C q , 0< <1 . - 1- Theo em 2 . (i) Le be a Bloch unc ion sa is ying 12 ( , ~) = o((1 - )as log  1  - T ( , ') ---> oo,  as 1- (ii) This esul is bes possible, e en in BO, in he ollowing sense . Le <D be a posi i e unc ion in [T, oo) sa is ying (1 .12)  -  lim -¿(x) = 0 . x-oo (1 .13)  <D (2x) - ~D (x),  as x ---> oo . Le 0< <1 . 1- Then he e exis s E BO such ha (1 .14)  T( , ') > log 1 1 + log T ( ) o all su ícien ly close o 1 . Le us no ice . ha he unc ion T o Theo em 2 can be aken o be and, hence, we ob ain . ON BLOCH FUNCTIONS ANDGAP SERIES  407 Co olla y 1 . The e exis s E Bo such ha o all suicien ly close o 1 and, hence, sa is ying I D is a B1-domain and is he uni e sal co e ing map o D hen [12] sa is ies (1 .8) and, hence, (1 .5) . I is known ha (1 .8) may no be ue o a unc ion in B1 [4], [5] . We do no know whe he o no (1 .5) emains ue o any unc ion E Bl . We can p o e he ollowing esul . Theo em 3 . Le E B l . Then 1/2 T ( ) = (log 1 e )  0 < < 1, - T( , ') > log  1  - 1 loglog  e 1- 2 1- 1 (1 - ) exp(2T( , ')) d = oo . 0 (1 .15)  limin (1 - ) 2 I2( , ') = 0 -1 and (1 .16)  limsup Clog  1  - T( , ') / = oo . -+1  1 -T The ea ly s ages o his wo k bene i ed om con e sa ions wi h A . Bae ns ein . He e en old me ha he conclusion o Theo em 1 should be ue a leas o su icien ly la ge alues o q . I is a pleasu e o exp ess my g a i ude . 408  D . GIRELA E en hough he mo i a ion o his wo k was s udying he possibili y o ex ending Kennedy's esul s o Bloch unc ions, some o ou esul s a e mo e gene al han s a ed and, in ac , could be s a ed wi hou making any e e ence o Bloch unc ions . Fo analy ic in U and 0 < < 1, de ine and Le us no ice ha , clea ly 2 . P oo o he main esul s Ip ( , ) = 27 ,~  ( e")¡' d ,  0 <p< oo, MP( , ) = Ip( , ) 1/p  0 < p < oo, M .( , ) = maxI (z)j . Ixl= Fo s > 0 and 0 < p < oo, le X 9 >p deno e he space o hose unc ions analy ic in U o which M p ( , ) = O((1 - ) - S), as ---> 1, and le Xó'p deno e he space o hose unc ions analy ic in U o which M P ( , ) = o((1 - ) -S ), as --> 1 . Since M, ( , ) is an inc easing unc ion o p, we ha e (2 .1)  X' ,P C X'g , P and Xó , p C X''P,  0 < p < p < oo . I p > 1 and (z) = E°° o a nz' E X' , P ( espec i ely X¿,p) hen an applica- ion o Cauchy's o mula easily gi es a n = O(ns) ( espec i ely a, = o(ns)) . On he o he hand, an a gumen simila o ha used in [16, Example 1, p . 694] p o es ha i (z) _k o akznk is analy ic in U and has Hadama d gaps hen (2 .2)  E XS,' <~ :> ak = O(n'), as k (2 .3)  E X ó ` <--¿ ak = o(nk), as k --> oo . EBg ' E X 1 '' and EBp ~¿ , EXó' , . Hence heo ems 1 and 2 will be co olla ies o he mo e gene al esul s ha we will p o e o he spaces X' , P and Xos,p . ON BLOCH FUNCTIONS AND GAP SERIES  409 I p < p' and E Xs , P ( espec i ely X`) hen a esul o Ha dy and Li lewood (see [3, Th . 5 .9]) shows ha E X",P ( espec i ely Xo~'P) whe e 1 1 s'=s+-- ; . p p The exponen s' is bes possible . Using his esul and a guing as in . [3, Th . 6 .4] we can deduce ha i 0 < p < 1 and (z) = ñ=o a  ,z - E Xs , P hen The unc ion (z) = (1 - z) - (s+ 1/P) o which a,,, - (s + Pl ns -1 + 1 /P shows ha his es ima e is sha p . Now, i p' < pand E Xs , P hen i is easy o see ha he i ial esul E XS,P is he bes ha we can say in gene al . In ac , he e exis s E X',' such ha o e e y p E (0, oo] he e exis s a cons an B P , s >0 such ha (2 .4)  M P (T, ) > BP .s (1 - T) -s ,  1 < <1,  0 <p< oo . _  2- Indeed, le q > 2 be an in ege and 00 (z) = E gkszqk,  Iz1 < 1 . k=0 Then, since has Hadama d gaps, (2 .2) shows ha E Xs , w . Now, i is a simple exe cise o show ha he e exis s a cons an ,0s = ,6 .,,q > 0 such ha (2 .5)  M2 ( , ) >_ i . (1 - )-3, 1 < < 1 . 2 _ This implies (2 .4) o 2 <_ p < oo wi h B P , s = J6 s . Using Theo em 8 .20 o [20, ol . I, p . 215] we deduce ha o each p E (0, 2) he e exis s a cons an A P = A P , q > 0 such ha (2 .6)  M, ( , )>APM2( , ),  0< <1,  0<p<2, which, wi h (2 .5) implies (2 .4) o 0 < p < 2 wi h (2 .7)  B P , s = QsAP . Since (2 .8)  logM P ( , ) 1 2~  log ( e")¡ d , as p 10, 410  D . GIRELA (2 .4) shows ha , o 2 < < 1, (2 .9)  T ( , ) > 27 J-  log I ( e 2 % d > s log 1 1 + ys whe e An examina ion o he p oo o Theo em 8 .20 in [20, ol . I, p . 215] shows ha he cons an A P gi en he e is o he o m o some 5 q > 1 . This and (2 .7) shows ha 7s = -oo and hence (2 .9) gi es no in o ma ion a all . Howe e , we will p o e in Theo em 4 ha he e exis s E X' ,1 sa is ying (2 .9) wi h a cons an C 3 in he place o -ys and, also, sa is ying (2 .4) wi h a cons an B 3 > 0 independen o p in he place o B P , 3 . Theo em 4 . (i) Le s > 0, 0 < p < oo and E X3,P, hen (2 .10)  EX s, P ' ,  0<p'<p, and (2 .11)  T( , ) < slog 1 1 +0(1) . (ii) This esul is bes possible in he ollowing sense . The e exis s E X -, ' and wo cons an s C s E R and B 9 > 0 such ha (2 .13)  M, ( , ) > B 3 (1 - ) -s ,  0< < 1,  0 <P :5 oo, and -ys = lim in log B P , 9 . P -0 A = 6(P-2)/p p q (2 .14)  T ( , ) > 2  log 1 ( e") ¡ d > s log 1 1 + Q,  0 < < 1 . Fo s = 1, he conclusion o (ii) holds wi h = é o any in ege q > 5 . Theo em 5 gi es he analogous esul s o he spaces XOS, . Theo em 5 . (i) Le s > 0, 0 <p< oo and E Xoq'P . Then (2 .15)  E Xó'P ,  0 < p' < p, and ON BLOCH FUNCTIONS AND GAP SERIES  411 (2 .16)  s log  1 1  -T ( , ) ----+ oo,  as - (ii) This esul is bes possible in he ollowing sense . Le D be a posi i e unc ion in [1, oo) sa is ying (1 .12) and (1 .13) and le Then he e exis s E Xó` such ha (2 .17) and ~Y( )=~~ i ~, 0< <1 . 1- ( ) ) M . ( , ) --, oo, as ---> 1, o e e y p E (0, o0], (2 .18)  T ( , ) > -  log 1 ( e z )1 d > s log 1 1 + log T ( ) 27 ,  - o all suicien ly Glose o 1 . P oo o Theoo ms 4(i) and 5(i) : We ha e al eady p o ed (2 .10) and (2 .15) . Also, (2 .11) and (2 .16) a e ob ious o p = oo . Now, le be a unc ion analy ic in U and 0 <p< oo . Using he a i hme ic- geome ic inequali y, we ob ain T( , ) = -, log+ 1 ( e")1 d < p - J_, log(I ( e")¡P+ 1) d Hence < p log C2~ ~~(I ( e i )I P + 1~ d =p log(I P ( , ) + 1) . 1  1  1 slog 1- -T( , )> plog(1- )sP(IP( , )+1) . Then i is clea ha (2 .11) ( espec i ely (2 .16)) holds i E X" ( espec i ely i E Xó'P) . P oo o Theo em 4(ií) : Le s > 0 . Le q > 2 be an in ege o be de e mined la e and (z) = k=0 k ks z e -i  l z l < 1 . 41 8 Le Then (2 .38) implies , Z I < L ,(z) 12 dxdy = , L l (z) I 2 dxdy + F < + m2)2 ~  I ,(z)I2 (1  JIZI< (1+ I (z)1 2 ) 2 which implies D . GIRELA F = {z : I z1 < and i (z) l < M}, G = {z : I z1 < and l (z) j > M} . d IZI< z dxd lim in d  d I /1  )1 2  y = 0 -1 d 1- L , (z) 12 dx dy 2 dxdy +  y  2 dxdy ~ IZI< (1 - IZI) < (1 + M 2 ) 2 7 S( , ) + 27 77 2 1- Hence, using (2 .37), we ob ain lim in (1 - )  I ' (z) 12 dxdy < 27 12 . il .l< - Since 17 > 0 is a bi a y, we ha e lmin (1- ) L 1(z)12dxdy=0 ,ZI< and his is equi alen o (1 .15) . Now, an a gumen simila o ha used in he p oo o Theo em 5(i) shows ha (1 .15) implies (1 .16) . This inishes he p oo o Theo em 3 . 3 . Some u he esul s and inal ema ks a) The esul s ha we ha e p o ed a e compa ison esul s be ween M p ( , ) wi h M p , ( , ) and T( , ) o in some o he spaces Xs , p o XO'p . I is well known (see e .g . [3, Th . 5 .10]) ha he e exis unc ions analy ic in U wi h M,, . ( , ) g owing o in ini y a bi a ily slowly which a e no o bounded cha ac e is ic . This leads one o ask he ollowing ques ion : Le p( ) be a posi i e inc easing unc ion on 0 <_ < 1 wi h y,(0) = 1 and u( ) - oo, as - 1, and le be a unc ion analy ic in U sa is ying M p ( , ) = 0(p( )), as ~ 1 . Wha can be said abou he g ow h o M p , ( , ) and T ( , )? In pa icula , i seems na u al o ask whe he o no he analogue ON BLOCH FUNCTIONS AND GAP SERIES  419 o Theo em 4(ii) is ue in his se ing, Le . does he e exis a unc ion analy ic in U wi h and a cons an C such ha We do no know he answe o his ques ion . Howe e , we do belie e ha he me hods o his pape a e no enough o cons uc such an . b) Fi s o all le us ema k ha some o he esul s ha we a e going o s a e below (Theo em 6, Co olla y 2, and Theo em 7) could be s a ed in he gene al amewo k o he spaces XS,P and XÓ'P . Howe e , o he sake o simplici y, we will s a e hem in he se ing o Bloch unc ions . I seems na u al o conjec u e ha he conclusion o Theo em 1 emains ue o q = 2, 3, and 4 . Howe e , ou a gumen does no p o e his since, wi h he no a ion used in he p oo o Theo em 1, we ha e A4 < 0 . A mo e gene al ques ion would be cha ac e izing hose Bloch unc ions gi en by a powe se ies wi h Hadama d gaps o which (1 .5) o a leas (1 .16) is ue . The ollowing heo em gi es a pa ial answe o his ques ion . Theo em 6 . Le be a Bloch unc ion gi en by a powe se ies (3 .1) Then (z) = M . ( , ) = 0(p( )), as -- 1, T( , )>logp( )+C, 0< <1? lim Su ) Clog 1 1  - T ( , ') / = oo . -,1 - Fu he mo e, i limsuplajj > 0 hen j_oo lim in ~log 1 1 - T ( , ')~ < oo . Using Theo em 2, we ob ain as an easy consequence o Theo em 6 he ol- lowing esul . Co olla y 2 . Le be a Bloch une ion gi en by a powe se ies zn j wi h w j+1 --> oo, as j - oo . nj (z) = j=1 Then E BO i and only i 1 z"'', wi h n, +  --~ oo, as j - oo . nj lim 1 (log 11 -  - T( , ') / = oo . ---~ The p oo o Theo em 6 depends on he ollowing wo elemen a y lemmas whose p oo s will be omi ed . 42 0 Lemma 3 . [14, p . 339] Le {Sk} be a sequence o posi i e numbe s such ha Sk+11Sk -> oo as k - oo . Then, as k ---> oo, Sk+ll k - oo and k/Sk -+ oo, as k ---> oo . Then, as k -> oo, P oo o Theo em 6 : Le be a Bloch unc ion gi en by (3 .1) . Since E B, he e exis s K > 0 such ha (3 .2)  laj j < K,  j = 1, 2, 3, . . . Le {mj} be an inc easing sequence o posi i e numbe s such ha (3 .3) k-1  00 Si = O( k) and  E  S .,- 1 = O(Skl) . j=1  j=k+1 = O( k) and  E S~ 1 = O( kl) . j=1  j=k+1 D . GIRELA Lemma 4 . Le {Sk} and { k} be wo sequences o posi i e numbe s such ha m' , oo and nj+l - oo, as j --> oo . nj m i Fo example, we can ake mj = (njnj+l)1/2 . Le Iz1 = 1 - mk1 . Then 0o  k (3 .4)  Iz '(z)j = E njzn- 1 < K E nj + K E nj(1 _ M k l)n, . j=1 j=1 j=k+1 Using (3 .3) and Lemma 4, we ob ain (3 .5) = O(mk), as k , oo . Now, (2 .21) wi h m = 3, (3 .3) and Lemma 4 imply (3 .6)  nj ( 1 - mk 1)n'  (3e-1)3mk  n,-2 j=k+1  j=k+1 _ (3e-1)3mko(mk2) = O(mk), as k -, oo . Then (3 .4), (3 .5) and (3 .6) show ha and hence This implies ha l '(z)l > Iz '(z)I Lemma 3 implies ha ON BLOCH FUNCTIONS ANDGAP SERIES  421 sup  ¡z '(z)1 = o(mk), as k - oo, IzI-I-Mk 1 logmk -T(1- mk  oo, as k - oo . -~I 1_ Asume now ha is gi en by (3 .1), sa is ies (3 .2) and limsuplaki > 0 . Then k-+oo he e exis s M > 0 such ha he se T = {k : laki > M} is in ini e . Take k E T and le k = 1 - nk I . Then, o Iz 1 = k k-I 00 >Mnk(1-7Lk I ) nk -KEn~-K E nj(1-nk1)--, =I_II_III . j-I j=k+I Using (2 .18), we see ha he e exis s a cons an C > 0 such ha I > Cnk . Finally, (2 .21) and Lemma 3 show ha Consequen ly, we ob ain ha and, hence Izln k I '(z)j ? Cnk- o(nk), as k - a oo(k E T), T( k, ') > log(nk - o(nk)) + O(1), as k -> oo(k E T) . This easily implies ha -+1  1 - inishing he p oo o Theo em 6 . lim in (log  - T( j» < 00 c) So a we ha e p o ed in Theo em 2 ha i a Bloch unc ion sa is ies (1 .8) hen i sa is ies (1 .5) . Fu he mo e, he unc ions conside ed in heo ems 3 and 6 sa is y no only (1 .16) bu also (1 .15) . These ac s migh load one o ask whe he o no he con e se o Theo em 2(i) is ue . Theo em 7 shows ha he answe o his ques ion is nega i e in a e y s ong sense . 422  D . GIRELA Theo em 7 . Le 0 < H < 1 . Then he e exis s a Bloch unc ion such ha (3 .7)  l min (1- ) 2 I2( , ') > 0 and (3 .8)  lim  T ( , ~ )  = H . li 1 log 11 1 P oo . We will use he ollowing esul due o Spech [18, Th . III] (see also [13, Lem . 1 and 2]) on he con o mal mapping o ce ain nea ly ci cula egions . The e exis s a simply connec ed domain D in he plane wi h (3 .9)  U U {ei : 7 H < I i < 7 } C D and such ha i w deno es he con o mal mapping om D on o U wi h w(0) = 0 and w'(0) > 0, hen (3 .10)  lw'(z) - 11 < 2,  z E D, and (3 .11)  2 (1 - ) < 1 - jw( e' )j < 3(1 - ),  0 < < 1,  ¡ i < 7 H . Le q > 12 be an in ege and de ine Le . (z) = e(w(z)) . Since he Bloch space is p ese ed unde subo dina ion (see e .g . [17, p . 35]), i ollows ha is a Bloch unc ion . Then he e exis s a cons an C such ha , o e e y , (3 .12)  log + J '( e i % < log 1 + C,  0< < 1 . 1- Now, (3 .9) implies ha 00 (Z) = W(Z)q", n=1 log + i ' ( e i ) 1 z E U, lim  1  = 0,  7 H < ¡ i < 7 , --~1  log 1= and hence, by he domina ed con e gen e heo em, lim 1  log + j i ( i e i ) I d = 0 -~1 27 .RH<, ,<7,  log 1 1 which, wi h (3 .12), easily implies which implies ha Thenwe ha e ON BLOCH FUNCTIONS AND GAP SERIES (3 .13)  lim sup T( , ') <_ H . -+1 log 1-T Le n = 1 - q - n . Then, o z =  ,e i wi h ¡ i < 7 H, we ha e 00 (3 .14)  w (z) l (z) I = gnw(z)en k=i n-1 00 > gnjw(z)I9  - 1 : q k -  1 :  gklw(z)I9k = I - II - III . k=1 k=n+l Using (3 .11) . we ob ain jw( e' )1 > 3 - 2,  ¡ i < 7 H, I = gnp(z)I9  > qn (1 - 3q-n)9" . Hence, since (1 - 3j-1)j > e -4 (j >- 12), we ha e 1 (3 .15)  I > e-4qn = e_4  . 1 - n Now ake q so big ha q 1 1 < 4e -4 and (3e -1 ) 3 82 11 '< 4e_ n_1 (3 .16)  11 =  qk <  qn =  1  1  < 1e-4  1 k=1 q-1 g-11- n 4 1 - n . and, by Schwa z's lemma and Lemma 1, (3 .17)  III =  gklW(Z)I9k <-  E q k(1 _ q-n)," k=n+1 k=n+1 < (3e -1 ) 3  1  qn = (3e -1 ) 3  1  1  < 1e-4  1 q 2 -1  q 2 -11 - n 4 1 - n Hence (3 .14), (3 .15), (3 .16) and (3 .17) show ha (3 .18)  I w () l (z) I > 2e-41  1 n'  z = ne i  I l < 7 H . 423 424  D . GIRELA No ice ha Iw(z)I < 1 and ha (3 .10) implies ha Iw'(z)I > 1/2 . Then (3 .18) easily implies (3 .19)  I '(z)I ? 4e-41 l n ,  z = neQe  ¡ i < 7 H . I is clea ha (3 .19) shows ha he e exis s a cons an C > 0 such ha C Consequen ly, 12 ( ., ' ) ? (1 - n)2 Since 12 ( , ') is an inc easing unc ion o , i ollows ha , o n < < ,, + 1, 12 12  _ C C 1 ( , ' ) ?  ( n, ')  (1 - C n) 2 q 2 ( 1 - n+1)2 .> q2 (1 - )2' This p o es (3 .7) . Finally, (3 .19) shows ha o n big enough, T( n , ~) > 1  1 log + I '( neZc) I d > H log 21 iel<, x 4(1 - n) lim in T ( , ) > H -1 lo g I -T which, wi h (3 .13), implies (3 .8) . We should ema k ha he condi ion H > 0 is needed in Theo em 7 . In ac , i is a simple exe cise o p o e ha i E B and T ( , ') = o (log 1 1 ), as -> 1, hen 12 ( , ) = o((1 - ) -2 ), as  . 1 . Pomme enke p o ed in [16, Th . 2] ha i a Bloch unc ion has adial limi s almos e e ywhe e on Iz1 = 1 hen i sa is ies (1 .8) . No ice ha he unc ion cons uc ed o p o e Theo em 7 is in ac analy ic on he se {e" : 7 H < ¡ i < 7 } and consequen ly i has adial limi s on a se o posi i e measu e . The nex esul asse s ha i a Bloch unc ion sa is ies his las condi ion hen i sa is ies (1 .5) . Theo em8 . Le be a Bloch unc ion ha ing adial limi s on a se o posi i e mensu e . Then lim 1 (log 1 1 - T( , ~) / =oo . P oo . Since a Bloch unc ion is no mal [16, p . 689], [1, p . 12] he concep s o adial limi s and angula limi s a e equi alen o [15] (see also [17, Th . ON BLOCH FUNCTIONS ANDGAP SERIES  42 5 9 .3]) . By P i alo 's heo em [19, p . 320] hese angula limi s a e ini e almos e e ywhe e . Hence, i we se hen ¡El > 0 . Now, E = { E [-7 , 7 ] : has a ini e angula limi a e i } , (3 .20)  log 1 1 - T ( , ') = (LE¡ II log 1 1 + ( 27 2~JEI log 1 1 - 27 ,  log+ I ' ( ei ) 1 d ) = I + II . [_i ,7 1_E I ¡El = 27 , hen II = 0 . O he wise -  log + I '( e' ) 1 d 27  E I II = 27 - ¡El  log  1  -  1   log+(I '( e' )I d 27  (  1 -  27 - IEI  ~_,~ , j_E > 27 - ¡El Clog  1  - 1  log(I '( e' )I + 1) d l -  27  1 -  27 - IEI  [,,,1 _ E 27 _, , . 1 ] _ E log (1 - )(L '( e i )I + 1) d Hence, since E B, he e exis s a cons an C such ha (3 .21)  II>C, 0< <1 . A guing as in he p oo o [16, Th . 2] we ob ain ha (1- ) 2  I '( ei ) I2 d - 0, as - 1, E and hen an a gumen like ha used in he p oo o Theo em 5(i) p o es ha I --> oo, as -> 1 . This, wi h (3 .21), shows ha log  1 1  -T ( , ') -+ oo, as --~ 1 . - d) The e a e o he ques ions ha we could ask in his con ex . Fo ins ance, i seems na u al o ask whe he o no (1 .5) is ue i belongs o he closu e o H°° in B . The answe o his ques ion is a i ma i e . Ac ually, i is easy o see ha (1 .8) is p ese ed unde con e gence in he Bloch no m and hence Theo em 2 and [16, Th . 2] show he ollowing : I E CLB(B n N) ( he closu e o B n N in B) hen sa is ies (1 .5) . Acknowledgmen s . I wish o hank he e e ee o his help ul commen s, specially o his ema ks abou he gene ali y o ou esul s . O iginally we jus s a ed ou esul s in he se ing o Bloch unc ions and no in he mo e gene al amewo k o heoem 4 and 5 . 42 6  D . GIRELA Re e ences 1 .  J .M . ANDERSON, J.G . CLUNIE AND CH . POMMERENKE, On Bloch unc- ions and no mal unc ions, J . Reine Angew . Ma h . 270 (1974), 12-37 . 2 .  J.G . CLUNIE, On he de i a i e o a bounded unc ion, P oc . London Ma h . Soc . (3) 14A (1965), 58-68 . 3 .  P . L . DUREN, "Theo y o HP spaces," Academic P ess, New Yo k, 1970 . 4 .  J .L . FERNÁNDEZ, On he coe icien s o Bloch unc ions, J . London Ma h . Soc . (2) 29 (1984), 94-102 . 5 .  J .L . FERNÁNDEZ, On he g ow h and coe icien s o analy ic unc ions, Ann . Ma h . 120 (1984), 505-516 . 6 .  O . FROSTMAN, Su les p odui s de Blaschke, Kungl . Fysiog . Süllsk . i Lund Fó h . 12, 15 (1942), 169-182 . 7 .  J .B . GARNETT, "Bounded Analy ic Func ions," Academic P ess, New Yo k, 1981 . 8 .  D . GIRELA, In eg al means and adial g ow h o Bloch unc ions, Ma h . Z . 195 (1987), 37-50 . 9 .  D . GIRELA, On analy ic unc ions wi h ini e Di ichle in eg al, Complex Va iables Theo y Appl . 12 (1989), 9-15 . 10 . W . K . HAYMAN, "Resea chp oblems in unc ion heo y," London Uni e - si y P ess, London, 1967 . 11 . W . K . HAYMAN, "Me omo phic unc ions," Ox o d Uni e si y P ess, Lon- don, 1975 . 12 . W .K . HAYMAN, S .J . PATTERSON AND CH . POMMERENKE, On he co- e icien s o ce ain au omo phic unc ions, Ma h . P oc . Camb idge Phil . Soc . 82 (1977), 357-367 . 13 .  P . B . KENNEDY, A p ope y o bounded egula unc ions, P oc . Roy . I ish Acad . 60, sec . A (1959), 7-14 . 14 .  P . B . KENNEDY, On he de i a i e o a unc ion o bounded cha ac e is ic, Qua . J . Ma h . Ox o d (2) 15 (1964), 337-341 . 15 . O . LEHTO AND K .J . VIRTANEN, Bounda y beha iou and no mal me o- mo phic unc ions, Ac a Ma h . 97 (1957), 47-65 . 16 . CH . POMMERENKE, On Bloch unc ions, J . London Ma h . Soc . (2), 2 (1970), 689-695 . 17 . CH . POMMERENKE, "Uni alen unc ions," Vandenhoeck und Rup ech , Gó ingen, 1975 . ON BLOCH FUNCTIONS AND GAP SERIES  42 7 18 . E .J . SPECHT, Es ima es on he mapping unc ion and i s de i a i es in con o mal mapping o nea ly ci cula egions, T ans . Ame . Ma h . Soc . 7 1 (1951),183-196 . 19 .  M . TSU,Ii, "Po encial Theo y in modem unc ion heo y," Chelsea Pu . Co ., New Yo k, 1975 . 20 . A . ZYGMUND, "T igonome ic Se ies," Camb idge Uni e si y P ess, Cam- b idge, 1959 . Análisis Ma emá ico Facul ad de Ciencias Uni e sidad de Málaga 29071 Málaga SPAIN P ime a e sió ebuda el 15 de Feb e de 1990, da e a ue si,ó ebuda el 8 de Mai,g de 1990