DERIVATIVES OF INNER FUNCTIONS IN BERGMAN SPACES INDUCED BY DOUBLING WEIGHTS

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 4, 7, ERIVATIVES OF INNER FUNCTIONS IN BERGMAN SPACES INUCE BY OUBLING WEIGHTS Fernando Pérez-González, Jouni Rättyä and Atte Reijonen Universidad de La Laguna, eartamento de Análisis Matemático P.O. Box, 456, 38 La Laguna, Tenerife, Sain; University of Eastern Finland, eartment of Physics and Mathematics P.O. Box, 8 Joensuu, Finland; jouni.rattya@uef.fi University of Eastern Finland, eartment of Physics and Mathematics P.O. Box, 8 Joensuu, Finland; atte.reijonen@uef.fi Abstract. We find a condition for the zeros of a Blaschke roduct B which guarantees that B belongs to the Bergman sace A induced by a doubling weight, and show that this condition is also necessary if the zero-sequence of B is a finite union of searated sequences. We also give a general necessary condition for the zeros when B A, and offer a characterization of when the derivative of a urely atomic singular inner function belongs to A.. Introduction and main results Let H denote the sace of analytic functions in the unit disc = {z C: z < } of the comlex lane C. A function : [,, integrable over, is called a weight. It is radial if z = z for all z. For < < and a weight, the weighted Bergman sace A = consists of f H such that f fz zdaz <, A where daz = dxdy is the normalized Lebesgue area measure on. As usual, A π α stands for the classical weighted Bergman sace induced by the standard radial weight z = z α, where < α <. For f H and < r <, set π / M r,f = fre it dt, < <, π and M r,f = max z =r fz. For <, the Hardy sace H consists of f H such that f H = su <r< M r,f <. A function Θ H is an inner function if it has unimodular radial limits almost everywhere on the boundarytof the unit disc. The question of when the derivative of an inner function belongs to the Hardy or the Bergman saces has been a subject of research since 97 s. Membershi of the derivative in the Hardy sace H and its Banach enveloe B, with < <, was studied in [, 3, 4, 7,, 33]. erivatives of inner functions in the weighted Bergman sace A α has been studied in [, 9, ], see htts://doi.org/.586/aasfm Mathematics Subject Classification: Primary 3J, 3J5; Secondary 3H. Key words: Bergman sace, Blaschke roduct, doubling weight, inner function. This research was suorted in art by Ministerio de Economía y Cometitivivad, Sain, roject MTM P and roject MTM RET; by Academy of Finland roject no. 689; by North Karelia Regional fund of Finnish Cultural Foundation; by Väisälä Fund of Finnish Academy of Science and Letters, and by Faculty of Science and Forestry of University of Eastern Finland roject no

2 736 Fernando Pérez-González, Jouni Rättyä and Atte Reijonen [, 3, 4, 5, 6, 7,, 3, 34] for recent develoments. See also the monograhs [9] and [5]. Many known results on the classical weighted Bergman sace A α were recently generalized in [6] to the setting of A induced by a normal weight. Recall that a radial weight is called normal if there exist real numbers a and b and r, such that r r ր, r a r ց, b for r > r [35]. Normal weights are essentially constant in hyerbolically bounded sets [6, Lemma ], hence they cannot oscillate too much, and in articular they do not have zeros. The urose of this note is to continue the line of investigation of [6] with the difference that we consider weights that are less regular. The class of radial weights such that z = z sds admits the doubling roerty z C + z gives a sufficiently general setting for our uroses. Since the definition of deends on integrals, it does not require any local smoothness for. The oint of dearture of this study is the recent oerator theoretic result which tells, in articular, when the Schwarz Pick lemma may be alied to the derivative of an inner function in the norm of the Bergman sace A without causing any essential loss of information. More recisely, if < < and, then by the main result in [3] the asymtotic equality Θz. Θ A z is valid for all inner functions Θ if and only if r su <r< r r zdaz s ds <. s Writing if the suremum above is finite, an immediate consequence of this result is that each subroduct of a Blaschke roductb such thatb A with also has its derivative in A. We also deduce that, for, the derivative of a finite roduct n j= Θ j of inner functions belongs to A if and only if Θ j A for all j =,...,n. Therefore in this case we may consider different tyes of inner functions searately. Before roceeding further, more definitions on weights are in order. We say that if there exist C = C, α = α > and β = β α such that. r C α t r C t β r t, r t <. t It is known that the existence of β such that the right-hand inequality is satisfied is equivalent to by [9, Lemma ], and therefore = >. It is easy to see that the left-hand inequality is equivalent to the existence of K = K > and C = C > such that the doubling roerty r C r K is satisfied for all r <. For details and more, see [3]. For a given sequence {z n } in for which n z n converges, the Blaschke roduct associated with the sequence {z n } is defined as Bz = n z n z n z n z z n z.

3 erivatives of inner functions in Bergman saces induced by doubling weights 737 A sequence {z n } in is called searated or uniformly discrete if there exists δ > such that inf z k z n k n z k z n = δ. Writing J if r s su ds <, <r< r r s the main result of this study on Blaschke roducts reads as follows. Theorem. Let < < and. Let B be the Blaschke roduct associated with a finite union of searated sequences {z n }. If either < and, or < < and J, then.3 B A z n z n. To give some insight to the hyotheses let us take a look at the case < < which J. Roughly seaking the containment in says that the r integral s/ s ds must grow as a negative ower of r, and J if s/ r s ds tends to as a ositive ower of r. In the case of the standard weight z = z α, the requirement J reduces to the chain of inequalities < α <. For this secial case the result is well known, and is generalized in [6, Theorem ] for an aroriate subclass of normal weights. Theorem in turn generalizes the last-mentioned result. In Section we first establish shar uer bounds for B A, when B is a general Blaschke roduct, by using standard techniques. To show that A -norm of B dominates the sum in.3 is more involved and the true difficulty in roving Theorem stems from the fact that does not admit any local smoothness. We circumvent the roblem by using maximal functions, their boundedness and Carleson measures for A. Therefore our reasoning is substantially different from that of [6, Theorem ]. The roof of Theorem is resented in Section where we also oint out that the hyothesis can be relaxed to if < and B is a Carleson Newman Blaschke roduct. The significant difference between the classes and is that contains the so-called raidly increasing weights that induce Bergman saces A lying in a sense much closer to the Hardy saces H than any of the standard weighted Bergman saces A α [8]. The canonical examle of a smooth weight in α \ is v α z = z log z e for each < α <. It is natural to search for necessary conditions for the zeros {z n } of a Blaschke roduct B when its derivative belongs to A. It is known by [3] that n z n β < for all β > +α/ α if B A α with < α < /. This result was recently generalized in [3] to other values of < : If B A α, where 3/+α <, then n z n β < for all β > + α / α. The case a of the next result gives an analogue of these results for A. Theorem. Let be a radial weight, and let B be the Blaschke roduct associated with a sequence {z n }.

4 738 Fernando Pérez-González, Jouni Rättyä and Atte Reijonen a Let <. If there exist ε > and a constant C = C,ε, > such that r.4 r C ε t, r t <, t then B A z n ε z n γ for all γ >. ε b Let < <. If there exist ε >, < γ < and a constant +ε C = C,ε,,γ such that γ ε r r.5 C t r C ε t, r t <, t t then B z A +ε n zn γ. If z = z α, then z n ε z n γ z n α+ ε +γ. Since γ > /ε, we have α + /ε + γ > + α /ε, where ε α by the hyothesis.4. Thus [3, Theorem ] follows from Theorem. Further, b shows that if B A α with α < and > max{, + α}, then n z n β < for all β > /α. This is a natural counterart of [3, Theorem 6] for >. The roof of Theorem is given at the end of Section. The argument we emloy uses ideas from the roofs of [3, Theorem 6] and [3, Theorem ]. The resence of a general weight instead of the standard weight causes technical obstructions in the argument, but also allows us to make certain arts of the roof more simle and transarent. Therefore Theorem can be considered as a streamlined generalization of [3, Theorem 6] and [3, Theorem ]. Singular inner functions are of the form S σ z = ex T z +w z w dσw, z, where σ is a ositive measure on T, singular with resect to the Lebesgue measure. If the measure σ is urely atomic, then this definition reduces to the form Sz = z +ξ n z +ξ n ex γ n = ex γ n, z, z ξ n n z ξ n where ξ n T are distinct oints and γ n > satisfy n γ n <. This tye of functions are known as urely atomic singular inner functions associated with {ξ n } and {γ n }. If there exist ε > and an index j such that ξ j ξ n > ε for all n j, then S is said to be associated with a measure having a searate mass oint. In the case where the roduct has only one term, S is called an atomic singular inner function. In Section 3 we consider urely atomic singular inner functions. A useful auxiliary result for our uroses is a combination of the first corollary of [, Theorem 5] and [7, Theorems and 4.4.8]: If < < and S is a singular inner function, then.6 π Sreit dt r n, log e r, <, = r /, >, for r <. An immediate consequence of this estimate and. is that there does not exist singular inner functions S such that S A if and either = and rlog and r dr = or > r r dr =. Regarding.6,

5 erivatives of inner functions in Bergman saces induced by doubling weights 739 we will show that urely atomic singular inner functions associated with measures whose masses γ n satisfy n γ n <, where = min{,}, obey instead of only. Further, it will turn out that for these are the only singular inner functions satisfying instead of in.6. As a consequence of these deductions and., we obtain the following theorem which is the last of the main results of this study. Theorem 3. Let < < and = min{,}. Let be a radial weight, and let S be a urely atomic singular inner function satisfying γ n <. Moreover, assume that either or S is associated with a measure having a searate mass oint. Sz zdaz a If <, then S A and z <. b If =, then the following statements are equivalent: i S A ; Sz ii zdaz < ; z iii r log dr <. r c If >, then the following statements are equivalent: i S A ; Sz ii zdaz < ; z iii r r dr <. Theorem 3 is based on Theorems 8 and, to be roven in Section 3, which show that M r,s π and Sreit dt/ r are comarable under aroriate hyotheses. Since these results concern the L -means of S, they immediately give information on the question of when S belongs to the Hardy sace H. At this oint it is also worth observing that for all inner functions Θ the quantities Θ H and su <r< π Θreit dt/ r are comarable, see, for examle, [3, Theorem ]. We have not found the statement of Theorem 3 even in the secial case of the classical weighted Bergman saces A α in the existing literature. Although, by using π the estimates for Sreit dt and M / r,s established in [, 5], where S is as in Theorem 3, one may easily rove some articular cases of our theorem. Moreover, in view of the main result in [6] our result does not come as a surrise in the case of atomic singular inner functions.. Blaschke roducts We begin with uer bounds for B A when B is any Blaschke roduct. For short, we write log if e r e su log r log sds <. <r< r s Proosition 4. Let B be the Blaschke roduct associated with a sequence {z n }, and let be a radial weight. a If < <, then B A z n.

6 74 Fernando Pérez-González, Jouni Rättyä and Atte Reijonen b If = and log, then B A c If < and, then B A z n e log z n z n. d If < < and J, then Proof. Since we have B A z n z n. z n z n. B z Bz = z n z n zz n z, B z n z = z n zz n z k= z n z n z z n z z k z k z z k z k z z n z z n z B nz ϕ z n z, where B n z = z k z k z k n and ϕ z k z k z az = a z for all a,z. If <, then az hx = x is sub-additive, and hence B z zdaz z n z daz, z n z where, by direct calculations, z daz z n z, < <, log e z n r =, r z n r dr, The assertions in the cases a c now follow by dividing the integrals into two arts, from zero to z n and the rest, then by estimating in a natural manner and finally using the hyotheses. If, then the Schwarz Pick lemma and a similar deduction as in the case = yield B z zdaz B z z daz z z n and the assertion d follows similarly as in the revious cases. r r z n r dr, Let Nfz = su ζ Γz fζ denote the maximal function related to the lens tye regions { Γz = ζ : argz argζ < } ζ z, z \{},

7 erivatives of inner functions in Bergman saces induced by doubling weights 74 with vertexes inside the disc. The Hardy Littlewood maximal theorem [, Theorem 3.,. 55] shows that N: A L is bounded and there exists a constant C >, indeendent of, such that. f A Nf L C f A, f H, see [8, Lemma 4.4] for details. With these rearations we are ready to rove our main result on Blaschke roducts. Proof of Theorem. By Proosition 4 it suffices to show that is dominated by a constant times B A z n z n. Note that for Proosition 4 we have to assume either < and or < < and J. In the remaining art of the roof only the hyothesis is needed. It is worth noting that this art uses some ideas from the roof of [3, Theorem ]. is searated with a searation constant δ j. Let r < min{δ j : j =,...,M} such that for given j, the discs zn j = {z : zn j z < r zn } j are airwise disjoint. Then Let {z n } = M j= {zj n }, where each {zj n } Bz z zj n z j nz z zj n z j n < r, z zj n, and hence su Bz r <, j =,...,M. z zn j Since by the hyothesis, is essentially constant in each disc zn j by.. This and the obvious inequality Brξ r B sξ ds, valid for almost every ξ T, now yield z n z n = M j= j= z j n z j n zn j M Bz z n j daz j= zn j zn j + M Bz z daz z + M M Bz z daz z + B s z z ds daz. z z + z Consider first the case <. By [3, Lemma 4], the inner integral is dominated by a constant times z NB s z s ds. z

8 74 Fernando Pérez-González, Jouni Rättyä and Atte Reijonen This estimate together with Fubini s theorem and. gives z n z B z z s ds z n s + = B z dµ, z, where dµ, z = z z s ds daz. s + daz The right-hand side is bounded by a constant times B if A A is continuously embedded into L µ,, that is, if µ, is a -Carleson measure for A. By [9, Theorem ] this is the case if and only if µ, Sa Sa for all Carleson squares Sa = {z : argz arga < a, z a } with a \{}. Since both µ, and are radial, this condition is equivalent to. r t t s ds s + dt r, < r <. Fubini s theorem shows that the left-hand side equals to r r s s ds+ s + s ds, where, by an integration by arts and the hyothesis, r s s ds = r r + r + s s ds r r + r, r and r s s ds = r s s β ds r s β r β r r ds r s β by the first inequality in.. It follows that. is satisfied, and hence the case < is roved. Let now < < and. Two integrations by arts show that the condition is self-imroving in the sense that if, then ε for all ε > sufficiently small, see the roof of [3, Lemma 3] for details. Hence we may choose ε = ε,, such that ε, and define hz = z ε. Then Hölder s inequality and Fubini s theorem yield B s z z ds daz + z = z π z z B s z hs dt z ds daz z z ht z + s B se iθ dt r dθhs ht r +rdrds r

9 erivatives of inner functions in Bergman saces induced by doubling weights 743 z B z hz r dt ht B z z ε z r drdaz r + r dr r + ε daz. Since the -Carleson measures for A are indeendent of by [9, Theorem ],. with relaced by ε imlies that µ ε, is a -Carleson measure for A. The assertion in the case < < follows, and the roof is comlete. The Blaschke roductb with the zero-sequence{z n } is called a Carleson Newman Blaschke roduct if the measure µ = z n δ zn is a -Carleson measure for H. This is equivalent to {z n } being a finite union of uniformly searated sequences which is the same as B being a finite roduct of interolating Blaschke roducts. An equivalent quantitative condition is.3 su a ϕ a z n <, see [, 3, 4]. Recall that {z n } is uniformly searated, if there exists a constant δ > such that inf z k z n n N z k z n = δ. k n The following result shows that in Theorem we may omit the hyothesis if < and B is a Carleson Newman Blaschke roduct. Proosition 5. Let < and, and let B be the Carleson Newman Blaschke roduct associated with {z n }. Then z n z n B. A Proof. It is well known that the Carleson Newman Blaschke roduct B satisfies.4 Bz ϕ zn z, z. We sketch a roof of this fact for the convenience of the reader. Since r logr for < r, log Bz ϕ z n z, and hence Bz ex ϕ z n z. This together with.3 and the fact that e x /x is decreasing yields.4. By combining. and.4 we deduce B A ϕzn z z z daz z n and the assertion is roved. s z n s ds z n z n,

10 744 Fernando Pérez-González, Jouni Rättyä and Atte Reijonen If Θ is an inner function, then there exists a Blaschke roduct B Θ associated with a uniformly searated sequence {z n } such that Θz B Θ z for all z [7, 8]. B Θ is called an aroximating Blaschke roduct of Θ. By using Proosition 5, we obtain the following result. Corollary 6. Let < < and such that r r, as r, and let Θ be an inner function. Then Θ A if and only if Θ is a finite Blaschke roduct. Proof. Since Θ H if Θ is a finite Blaschke roduct, it suffices to rove the only if art of the assertion. Let Θ be an inner function and assume first that its aroximating Blaschke roduct B Θ has infinitely many zeros {z n }. Then., Proosition 5 and the hyothesis r r, as r, yield Θ A Θz zdaz z B Θ A z n =. z n BΘ z zdaz z Hence Θ A only if B Θ is a finite Blaschke roduct. But if B Θ has finitely many zeros, then Θ z Θz z B Θz z B Θz, z, and hence Θ is continuous u to the boundary [, Theorem 3.]. Therefore Θ is a finite Blaschke roduct, and the assertion is roved. We next establish a generalization of [6, Corollary ] and [9, Theorem 7b]. For q > and a weight, we write q z = z z q for all z. Corollary 7 shows that, under aroriate hyotheses, the quantities Θ A and Θ A +q are q finite at the same time for each inner function Θ. Corollary 7. Let < <, < q < and, and let Θ be an inner function. If a < < and J, or b +q and, or c < +q +q and J, then Θ A Θ +q Aq +q. Proof. We begin with showing that if and < q <, then.5 q z z z q, z. Since for each radial we have q z z z q for all z, it suffices to show that q r r r q for all r <. To see this, let C = C, α = α > and β = β α be the constants aearing in.. Let r < and choosea=a such that C /α < a <. Set r = r andr n+ = r n +a r n

11 erivatives of inner functions in Bergman saces induced by doubling weights 745 for all n N {}. Then r n, as n, and hence. yields rn+ q r = s s q ds r n+ q r n r n+ n= r n n= = r q a qn+ r n r n+ n= r q n= a qn+ r n C rn+ r n = C a α r q a qn+ r n n= α C C a α r r q a qn+ rn r = C C a α r r q a nq+β+q r r q, and.5 follows. Let B Θ be the aroximating Blaschke roduct of Θ with zeros {z n }. Let first < <. By using.5 it is easy to see that the conditions and J are equivalent to q +q and q J +q, resectively. Therefore., Theorem and.5 yield Θ A B Θ A z n z n n= n= β q z n z n +q B Θ +q Aq +q Θ +q Aq +q and thus the assertion is roved for < <. The other two cases follow in a similar manner. We end this section with the roof of Theorem. Proof of Theorem. Let < < and ε > be as in the statement of the theorem. Assume, without loss of generality, that {z n } is ordered by increasing moduli and z n for all n. An integration by arts shows that if and only if r r s r s ds, r +. But since.4 is satisfied by the hyothesis, r s r ds s + r ε r ds r s + +ε r, r <, and thus. Further, by the roof of [3, Theorem 6], Bz = B z n z z n z n z, z, where B n z = n j= z j z, n N, z, z j z,

12 746 Fernando Pérez-González, Jouni Rättyä and Atte Reijonen and hence. imlies.6 B A B n z z n rdaz. z n z We next estimate B n aroriately downwards close to the boundary. To do this, let ρ ε,γ z = z ε+min{, } z γ, z, and set γ = inf{γ : ρ n z n < }, where ρ n = ρ ε,γ z n. Since {z n } is a Blaschke sequence, γ. Let γ > γ = γ ε so that ρ n z n converges. As in the roof of [3, Theorem ], note that and let r z j z j r z j ρ r z j + z j ρ + z j ρ+, < r <, < ρ <, r j = z j + z j ρ j + z j = z j z j ρj, j N. ρ j+ + z j ρ j+ Then z j < r j < for all j N. Moreover, for z R n = max j n r j, we have n n z z j B n z z j= j z n R n z j z j ρ j z j ρ j z j= j R n j= j=.7 ρ j z j = ex ρ j log z j ex. z j Let j= <. Then.6, Minkowski s inequality and.7 yield B A B n z z n zdaz z n z z n zdaz \,R n z n z rdr z n. z n r Recall that z j < r j R j, and hence B A z n R n z n z n = R n j= z n z n inf j n r j z n ε. ε Rn z n Since r j z j z j ρ j / and {z n } is ordered by increasing moduli, we obtain B A z n z n inf j n z j z j ρ ε j z n z n z ε n inf z j ρ j. j n

13 erivatives of inner functions in Bergman saces induced by doubling weights 747 Now that j= ρ j z j converges and z j for all j, there exists δ > such that inf z j ρ j = inf z j z j ρ j z j δ. j N j N Therefore B A z n z n inf z j ρ j log j n z j ρ j z n z n inf ρ j z j j n z n +γ ε z n ε. ε ε If B A, then +γ ε > γ, and by letting γ γ, we deduce γ. ε The assertion in the case < follows. Let >. Then we can dro inside the sum in.6 without using Minkowski s inequality. Hence, by deducting as above, we obtain B z A n R n z n z ε n inf ρ j z j. j n Since the left-hand inequality of.5 is equivalent with the asymtotic inequality we have r +ε r γ t +ε t γ, r t <, B z A n +γ ε z n +ε. If B A, then +γ ε > γ, and by letting γ γ, we deduce γ. +ε The assertion in the case < < follows, and the roof is comlete. 3. Purely atomic singular inner functions Recall that urely atomic singular inner functions are of the form Sz = z +ξ n z +ξ n ex γ n = ex γ n, z, z ξ n n z ξ n n where ξ n T are distinct oints and n γ n <. If the roduct has only one term, with γ = γ and ξ = ξ, then we write S = S γ,ξ. Theorem 8. Let < < and = min{,}. Let S be the urely atomic singular inner function associated with {ξ n } and γ = {γ n } l. Then 3. for < r <. π Sreit dt r h r =, log r, <, = r /, >,

14 748 Fernando Pérez-González, Jouni Rättyä and Atte Reijonen Proof. By.6, it suffices to show that π Sreit dt r h r. We begin with an estimate for S = S γ,ξ, where ξ = e iθ. Since 3. we obtain π 3.3 e is = coss = s S γ,ξ re it dt = k= k sk k! s π 4! + π4 6! π6 s, π s π, 8! 3 = = π π π π 3 = r ex γ dt re it θ ex γ r ds re is r ex γ ds r +r e is r ex γ ds r + rs 3 r γ r γ r γ 6γ r r γ r γ r r +rπ /3 x 3 x 3 e x dx x r γ e x dx x r γ, < r <. To rove the general case, we may assume that {γ n } is non-increasing. Write π S n = S γn,ξ n for short. If Sreit dt r for some >, then the same clearly holds for all. Therefore it suffices to rove the assertion for <. Since Sz = S n z S n z, z, we obtain π Sre it dt π = π S n re it dt S n re it dt = γ n>γ r By 3.3, we have 3.4 π = I r+i r. I r r π S n re it dt+ γ n>γ r γ n γ γn r S n re it dt γ n γ r + γn r γ π S n re it dt x 3 e x dx. x r γ n

15 erivatives of inner functions in Bergman saces induced by doubling weights 749 If γ n γ r, then I 3 r = γ γn r x 3 e x dx x r γ n γ γn r dx γ x 3 n r h r, n N, because e s s for s,. If γ r < γ n < γ r, then γ r γn ex γ ny r dy r I 3 r = γ n r γ n r γ n r r r + y 3 dy y 3 y ex γ ny r dy +γ y 3 y γ n r h r, n N. Hence I 3 r γ n r h r. By an analogous manner, we can also show that I 4 r = γn r e x dx γ x 3 γ n x r γ n r h r, γ n > γ r. Now, by using the estimates for I 3 and I 4 in 3.4, we obtain I r γ l r h r. Further, since π π S n re it r dt = ex γ n dt ξ n re it we have γ n r π I r r h r γ n γ r dt re it γ n r h r, n N, γ n γ l r h r. By combining the estimates for I and I we deduce the assertion. Note that Theorem 8 in the case= has been roved earlier in [], but the roof there is based on different methods. The following corollary shows that Theorem 8 is shar for. Corollary 9. If S is a singular inner function, then the following statements are equivalent: a S is a urely atomic singular inner function associated with γ l; b M / / r,s log r, as r ; π c Sreit dt r log r, as r ; d there exists < < such that π Sreit dt r, as r ; π e Sreit dt r, as r, for each < <. Proof. The statements a and b are equivalent by [5, Theorem.]. Moreover, a imlies c e by Theorem 8, and c imlies b by the Schwarz-Pick lemma.

16 75 Fernando Pérez-González, Jouni Rättyä and Atte Reijonen To comlete the roof, it suffices to show that d imlies a. If a does not hold, then the roof of [5, Theorem.] yields r π Sre it dt, r, for each < <, and this clearly contradicts d. Thus the assertion is roved. or If an inner function Θ satisfies either log r M / / r,θ +, r, π Θreit dt +, > r, r, then it is a Blaschke roduct. In the first case, the assertion is a direct consequence of [5, Theorem.]. In the latter case, the assertion follows by.6 and a secial case of the Beurling factorization according to which every non-constant inner function is either a Blaschke roduct, a singular inner function or a roduct of the revious ones []. Theorem 3 for can be roved by using Theorem 8 and., as will be shown later. To deal with the remaining case, we will use the following result which shows that M r,s h r for each urely atomic singular inner function S associated with a measure having a searate mass oint. Theorem. Let < < and = min{,}. Let S be the urely atomic singular inner function associated with{ξ n } andγ = {γ n } l, and having a searate mass oint in its inducing measure. Then there exists r = r,s, such that π 3.5 Sreit dt M r r,s h r, r < r <, where h is as in Theorem 8. Proof. Letξ j be a searate mass oint and writes = S n, wheres n = S γn,ξ n. It suffices to show that Mr,S Mr,S j for r close enough to one because then this together with Theorem 8, the Schwarz Pick lemma and the main result of [6] yields π h r Sreit dt M r r,s M r,s j h r, r < r <. To rove Mr,S Mr,S j, note first that S z = γ n ξ n z ξ n ex z +ξ k γ k z ξ k k= γ n ξ n = z ξ n ex z γ k z ξ k k= γ j z ξ j γ n z ex γ z ξ n k. z ξ k n j k=

17 erivatives of inner functions in Bergman saces induced by doubling weights 75 For each k N, write ξ k = e iθ k, where θ k < π. By the hyothesis, there exists ε = εs,π such that θ j θ k > ε for all k j. Let first t θ j < ε and r >. Then 3. yields ex r γ k = ex r γ re it e iθ k k j k r +r e it θ k k j and ex 6 k j γ j γ k rt θ k fre it = re it e iθ j n j ex γ n re it e iθn 48 ε k j Set M = MS = 4/ε γ l. Then, for r > max γj, we obtain 4M γ k ex 48 ε γ l γ j r +rt θ j 4 ε γ l. { }, γj 4M fre it γ j r +rt θ j γ j 6 re it e. iθ j { } γj If α = min, ε, then, by combining the estimates above, we deduce 4M M r,s θj +α θ j α S re it dt θj +α θ j α and t θ j < S γ j,ξ j re it dt M r,s γ j,ξ j for r close enough to one deending on and S. Thus the assertion is roved. With these results in hand we can easily establish Theorem 3. Proof of Theorem 3. Let us begin with the case. By multilying 3. by rr and integrating with resect to r we obtain Sz zdaz h z rrrdr, \, where S and h are as in Theorem 8. The assertion in Theorem 3 for now follows by.. If S is associated with a measure having a searate mass oint, then the assertion can be roved by an analogous manner using 3.5. The statement in Theorem 3 for is actually valid for each radial weight and each urely atomic singular inner function S with γ l. This is an immediate consequence of Theorem 8, the Schwarz Pick lemma and [5, Theorem.]. Further, it is worth observing that S H if and only if <. Acknowledgements. This aer is a art of the third author s Ph thesis. The authors thank the thesis reviewers Igor Chyzhykov and Toshiyuki Sugawa, and the oonent José Ángel Peláez for careful reading of the manuscrit and valuable comments.

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