OON THE HARDY NUMBER OF COMB DOMAINS
CHRISTINA KARAFYLLIA
Abstract.
Let H p ( D ) be the Hardy space of all holomorphic functionson the unit disk D with exponent p >
0. If D (cid:54) = C is a simply connecteddomain and f is the Riemann mapping from D onto D , then the Hardynumber of D , introduced by Hansen, is the supremum of all p for which f ∈ H p ( D ). Comb domains are a well-studied class of simply connecteddomains that, in general, have the form of the entire plane minus an in-finite number of vertical rays. In this paper we study the Hardy numberof a class of comb domains with the aid of the quasi-hyperbolic distanceand we establish a necessary and sufficient condition for the Hardy num-ber of these domains to be equal to infinity. Applying this condition,we derive several results that show how the mutual distances and thedistribution of the rays affect the finiteness of the Hardy number. By aresult of Burkholder our condition is also necessary and sufficient for allmoments of the exit time of Brownian motion from comb domains to beinfinite. Introduction
A classical problem in geometric function theory is to find geometric con-ditions for a holomorphic function on the unit disk to belong in Hardy spaces(see e.g. [1], [11], [12], [14], [17], [19] and [21]). In this paper we study thisproblem in the case of conformal mappings from the unit disk onto a combdomain. The Hardy space with exponent p > H p ( D ) and is defined to be the set of all holomorphic functions, f , on theunit disk D that satisfy the conditionsup
Primary 30H10; Secondary 30C35.
Key words and phrases.
Hardy number, Hardy space, comb domain. a r X i v : . [ m a t h . C V ] J a n CHRISTINA KARAFYLLIA mapping from D onto D . The Hardy number of D , or equivalently of f , isdefined by h ( D ) = sup { p > f ∈ H p ( D ) } . We note that this definition is independent of the choice of the Riemannmapping onto D . It is known that every conformal mapping on D belongsto H p ( D ) for all p ∈ (0 , /
2) [8, p. 50]. This implies that h ( D ) lies in[1 / , + ∞ ].There is no general method for computing the Hardy number but there aresome ways to estimate it for certain types of domains. In [11] Hansen gave alower bound for the Hardy number of an arbitrary region and improved thisbound for simply connected domains. Moreover, he determined the exactvalue of the Hardy number of starlike [11] and spiral-like regions [12]. In [20]Poggi-Corradini studied the Hardy number of Kœnigs mappings. He alsoproved [19] for a certain class of functions, which give a geometric model forthe self-mappings of D , that the Hardy number is equal to infinity if andonly if the image region does not contain a twisted sector. Furthermore, in[9] and [17] Ess´en, and Kim and Sugawa, respectively, gave a descriptionof the Hardy number of a plane domain in terms of harmonic measure. In[15] the current author gave a formula for the Hardy number of a simplyconnected domain in terms of hyperbolic distance. Finally, Burkholder [7]studied the Hardy number of a domain in relation with the exit time ofBrownian motion (see also [5]). More precisely, if D is a simply connecteddomain, then we define the number (cid:101) h ( D ) to be the supremum of all p > p -th moment of the exit time of Brownian motion is finite.Then Burkholder proved in [7] that (cid:101) h ( D ) = h( D ) / . (1.1)Comb domains furnish an interesting class of simply connected domainsand thus they have been studied from various points of view. For example,they have been studied in relation with the angular derivative (see [13],[16] and references therein), the harmonic measure [3] and the semigroupsof holomorphic functions [4]. Moreover, in [6] Boudabra and Markowskystudied the moments of the exit time of planar Brownian motion from combdomains.Let { x n } n ∈ Z be a strictly increasing sequence of real numbers such that x = 0 and inf n ∈ Z ( x n − x n − ) > . Also, let { c n } n ∈ Z be a sequence of positive numbers such that for someconstants c , c > c ≤ c n ≤ c for every n ∈ Z . We consider comb domains of the form (see Fig. 1) D = C \ (cid:91) n ∈ Z { x n + iy n : | y n | ≥ c n } . N THE HARDY NUMBER OF COMB DOMAINS 3
Figure 1.
Comb domainSince we want to find conditions for the Hardy number of such combdomains to be equal to infinity, we can simplify the problem in the followingway. First, we observe that if D c = C \ (cid:91) n ∈ Z { x n + iy : | y | ≥ c } and D c = C \ (cid:91) n ∈ Z { x n + iy : | y | ≥ c } , then D c ⊆ D ⊆ D c and hence h( D c ) ≤ h( D ) ≤ h( D c ) (see [11]). More-over, since the Hardy number is invariant under affine mappings (see [11]),we have h( D c ) = h( D c ). So, it follows that h( D ) = + ∞ if and only ifh( D c ) = h( D c ) = + ∞ . Therefore, it suffices to study comb domains ofthe form D c = C \ (cid:91) n ∈ Z { x n + iy : | y | ≥ c } , where c >
0. However, we can do more simplifications. We observe that if D + c = D ∩ { z : Re z > − x } and D − c = D ∩ { z : Re z < − x − } , then h( D c ) = + ∞ if and only if h( D + c ) = + ∞ and h( D − c ) = + ∞ . Thisfollows from Proposition 8 in [6] and (1.1). Furthermore, h( D − c ) = h( − D − c ).Therefore, it suffices to study the Hardy number of comb domains of the form D + c . Finally, since the Hardy number is invariant under affine mappings,without loss of generality, we suppose that c = 1 and the infimum of thedifferences x n − x n − is greater than 1. So, henceforth we consider combdomains C of the form C = { z : Re z > − x } \ (cid:91) n ∈ N ∪{ } { x n + iy : | y | ≥ } , CHRISTINA KARAFYLLIA where x = 0 and { x n } n ∈ N is a strictly increasing sequence of positive num-bers such that lim n → + ∞ x n = + ∞ and inf n ∈ N ( x n − x n − ) > . First, we establish a necessary and sufficient condition for h( C ) to beequal to infinity by studying the Euclidean distances between the rays. Forevery n ∈ N , we denote these distances by α n = x n − x n − . Theorem 1.1.
Let C be a comb domain of the form described above. Then h( C ) = + ∞ if and only if lim n → + ∞ n (cid:80) i =1 log α i log x n = + ∞ or, equivalently, lim n → + ∞ n (cid:80) i =1 log α i log n (cid:80) i =1 α i = + ∞ . An immediate consequence is that if the sequence α n is bounded thenh( C ) = + ∞ . So, we actually study the case of α n being unbounded. Byapplying Theorem 1.1 we can examine how the mutual distances and thedistribution of the rays affect the finiteness of the Hardy number. First,we consider the case of α n growing at a subexponential rate and prove thath( C ) is always equal to infinity. Theorem 1.2. If lim n → + ∞ log α n n = 0 , then h( C ) = + ∞ . This result is stronger than the corollary of the main theorem of Boudabraand Markowsky in [6], where they approach the problem by studying themoments of the exit time of the Brownian motion. In fact, their maintheorem implies that if α n grows at most polynomially in n then the Hardynumber is infinite. However, it does not cover all subexponential sequences.For the proof of Theorem 1.2 see Section 4.Next, we explain why the assumption in Theorem 1.2 cannot be relaxed.Theorem 1.2 covers all the cases when α n grows at a subexponential rate,even those in which the sequence α n oscillates very rapidly. For example,one can take α n = 2 when n is odd and α n = n p when n is even and p > α n is of exponential type, i.e. α n = e cn for every n ∈ N ,and hence there are no sharp oscillations, then Theorem 1.1 implies that theHardy number is equal to infinity. However, if we allow wild oscillations andsuppose that α n ≤ e n , then the Hardy number might be finite. Actually, weconstruct such an example in Theorem 1.4. Therefore, in order to obtain ageneral result in case α n ≤ e n , we need to suppose that there are no wildoscillations. By imposing that lim n → + ∞ α n = + ∞ N THE HARDY NUMBER OF COMB DOMAINS 5 and thus preventing sharp oscillations of α n , we prove that h( C ) = + ∞ . Infact, a more general result is true. Theorem 1.3.
Let { b n } n ∈ N be an increasing sequence of positive numberssuch that inf n> ( b n − b n − ) > . Let α n ≤ e b n for every n ∈ N . If lim n → + ∞ log α n b n − b n − = + ∞ , then h( C ) = + ∞ . An interesting case, as we already remarked, is when b n = n . Corollary 1.1.
Let α n ≤ e n for every n ∈ N . If lim n → + ∞ α n = + ∞ , then h( C ) = + ∞ . Next, we prove that the assumption in Theorem 1.3 is sharp. In otherwords, if there are wild oscillations of α n , then the Hardy number might befinite. Theorem 1.4.
Let { b n } be as Theorem 1.3. There is a comb domain C such that α n ≤ e b n for every n ∈ N , lim inf n → + ∞ log α n b n − b n − < + ∞ and h( C ) < + ∞ . Theorem 1.3 covers a variety of cases such as α n being comparable to e n p for some p > α n being comparable to e e nk for some k < α n beingcomparable to e e n/ log n . However, it does not apply if α n is comparable to e e n . In this case, despite the fact that there are no wild oscillations of α n ,Theorem 1.1 implies that h( C ) is finite. Theorem 1.5. If α n is comparable to e e n for every n ∈ N , then h( C ) < + ∞ . Therefore, the Hardy number of C might be finite when the sequence α n oscillates very quickly or if it goes to infinity rapidly enough like α n beingcomparable to e e n . Remark 1.1.
Note that by (1.1) all the results above concerning the Hardynumber provide us with information about the finiteness of the moments ofthe exit time of Brownian motion from comb domains. In fact, (cid:101) h ( D ) is equalto infinity if and only if h( D ) is equal to infinity. Remark 1.2.
By Corollary 1.1 the conformal mapping from D onto thecomb domain C with x n = e n belongs to every H p ( D ) space. However, itdoes not belong to BM OA (see [22]). So, it is an example which ensuresthat
BM OA (cid:40) (cid:92) p> H p ( D ) . CHRISTINA KARAFYLLIA
In Section 2, we introduce some preliminaries such as notions and resultsin hyperbolic geometry and their connection with the Hardy number. InSection 3, we prove Theorem 1.1 and applying this, in Section 4, we proveall the other theorems stated above.2.
Preliminary results
Hyperbolic distance.
The hyperbolic distance between two points z, w in the unit disk D (see [2, p. 11-28]) is defined by d D ( z, w ) = log 1 + (cid:12)(cid:12)(cid:12) z − w − z ¯ w (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) z − w − z ¯ w (cid:12)(cid:12)(cid:12) . It can also be defined on any simply connected domain D (cid:54) = C in thefollowing way: If f is a Riemann mapping of D onto D and z, w ∈ D ,then d D ( z, w ) = d D (cid:0) f − ( z ) , f − ( w ) (cid:1) . Also, for a set E ⊂ D , we define d D ( z, E ) = inf { d D ( z, w ) : w ∈ E } .2.2. Quasi-hyperbolic distance.
Let D (cid:54) = C be a simply connected do-main. The hyperbolic distance between z , z ∈ D can be estimated by thequasi-hyperbolic distance which is defined by δ D ( z , z ) = inf γ : z → z (cid:90) γ | dz | d ( z, ∂D ) , where the infimum ranges over all the paths γ connecting z to z in D and d ( z, ∂D ) denotes the Euclidean distance of z from ∂D . It is known [2, p.33-36] that 12 δ D ≤ d D ≤ δ D . (2.1)2.3. Hardy number and hyperbolic distance.
In [15] the current au-thor proves that the Hardy number of a simply connected domain can befound with the aid of hyperbolic distance in the following way.
Theorem 2.1.
Let D be a simply connected domain containing the origin.If F r = r∂ D ∩ D for r > , then h ( D ) = lim inf r → + ∞ d D (0 , F r )log r . The Stolz–Cesaro theorem.
Next, we state a generalized form ofthe Stolz–Cesaro theorem which we apply in Section 4. For the proof see[18] and [10, p. 263–266].
Theorem 2.2.
Let { b n } n ∈ N be a sequence of positive numbers such that + ∞ (cid:80) n =1 b n = + ∞ . For any real sequence { a n } n ∈ N , it is true that lim sup n → + ∞ a + a + · · · + a n b + b + · · · + b n ≤ lim sup n → + ∞ a n b n N THE HARDY NUMBER OF COMB DOMAINS 7 and lim inf n → + ∞ a + a + · · · + a n b + b + · · · + b n ≥ lim inf n → + ∞ a n b n . A necessary and sufficient condition
In this section we give a necessary and sufficient condition for the Hardynumber to be equal to infinity. First, we prove two auxilary lemmas whichgive a lower and an upper estimate for h( C ). Lemma 3.1.
Let C be a comb domain of the form described in Section 1.If K = 4 log((1 + √ / , then h( C ) ≤ lim inf n → + ∞ n (cid:80) i =1 log α i log x n + nK log x n + 4 . Proof.
Let r >
0. There exists a number n ∈ N such that x n − < r ≤ x n .It is true that d C (0 , r ) = d C (0 , x n − ) + d C ( x n − , r ) . (3.1)Applying (2.1) and letting m i − denote the midpoint of the interval [ x i − , x i ]for every i ∈ N , we infer that d C (0 , x n − ) ≤ δ C (0 , x n − ) = 2 (cid:90) x n − dxd ( x, ∂C ) = 2 n − (cid:88) i =1 (cid:90) x i x i − dxd ( x, ∂C )= 4 n − (cid:88) i =1 (cid:90) m i − x i − dxd ( x, ∂C ) = 4 n − (cid:88) i =1 (cid:90) m i − x i − dx (cid:113) x − x i − ) = 4 n − (cid:88) i =1 arcsinh ( m i − − x i − ) = 4 n − (cid:88) i =1 arcsinh (cid:18) x i − x i − (cid:19) = 4 n − (cid:88) i =1 log (cid:32) α i (cid:114)(cid:16) α i (cid:17) + 1 (cid:33) . (3.2)Recall that the domain C has the property that inf n ∈ N α n >
1, which impliesthat, for every i ∈ N , α i > . Therefore, (cid:114)(cid:16) α i (cid:17) + 1 = (cid:115)(cid:16) α i (cid:17) + 4 (cid:18) (cid:19) ≤ α i √ (cid:32) α i (cid:114)(cid:16) α i (cid:17) + 1 (cid:33) ≤ log (cid:16) α i (cid:16) √ (cid:17)(cid:17) = log α i + K , (3.3) CHRISTINA KARAFYLLIA where K = 4 log((1 + √ / d C (0 , x n − ) ≤ n − (cid:88) i =1 log α i + ( n − K. (3.4)Now, in order to find an upper estimate for d C ( x n − , r ), we consider thefollowing cases.Case 1: If r ∈ (cid:16) x n − , x n − + x n (cid:105) , then d C ( x n − , r ) ≤ δ C ( x n − , r ) = 2 (cid:90) rx n − dx (cid:112) x − x n − ) = 2 arcsinh ( r − x n − ) ≤ r. Case 2: If r ∈ (cid:16) x n − + x n , x n (cid:105) , then d C ( x n − , r ) ≤ d C ( x n − , x n ) ≤ δ C ( x n − , x n ) = 2 (cid:90) x n x n − dxd ( x, ∂C )= 4 (cid:90) xn − xn x n − dx (cid:112) x − x n − ) = 4 arcsinh (cid:18) x n − x n − (cid:19) ≤ r. Therefore, it follows that in both cases, d C ( x n − , r ) ≤ r. (3.5)Recall that r > x n − . So, by (3.1), (3.4) and (3.5) we derive that d C (0 , r )log r ≤ n − (cid:80) i =1 log α i log x n − + ( n − K log x n − + 4 arcsinh r log r . This in conjunction with Theorem 2.1 givesh( C ) ≤ lim inf r → + ∞ d C (0 , r )log r ≤ lim inf n → + ∞ n (cid:80) i =1 log α i log x n + nK log x n + 4 and the proof is complete. (cid:3) Lemma 3.2.
Let C be a comb domain of the form described in Section 1.Then h( C ) ≥ lim inf n → + ∞ n (cid:80) i =1 log α i log x n − . Proof. If F r = r∂ D ∩ C , by Theorem 2.1 we haveh( C ) = lim inf r → + ∞ d C (0 , F r )log r = lim n → + ∞ d C (0 , F x in r n )log r n , (3.6) N THE HARDY NUMBER OF COMB DOMAINS 9 where { r n } is an increasing sequence of positive numbers and F x in r n is thecomponent of F r n lying in the vertical strip { z : x i n < Re z < x i n +1 } . Since C is symmetric with respect to the real axis, without loss of generality, wesuppose that F x in r n lies on the upper half-plane (see Fig. 2). If h( C ) = + ∞ ,the result is trivial. Hence, we suppose that h( C ) < + ∞ and take thefollowing cases. Figure 2.
The component F x in r n .Case 1: For infinitely many n , F x in r n is the component of F r n containing r n . By passing to a subsequence we assume that this is the case for all n . If m j − denotes the midpoint of the interval [ x j − , x j ], then we have d C (cid:0) , F x in r n (cid:1) = d C (0 , r n ) ≥ d C (0 , x i n ) ≥ δ C (0 , x i n ) = 12 (cid:90) x in dxd ( x, ∂C )= 12 i n (cid:88) j =1 (cid:90) x j x j − dxd ( x, ∂C ) = i n (cid:88) j =1 (cid:90) m j − x j − dx (cid:113) x − x j − ) = i n (cid:88) j =1 arcsinh (cid:18) x j − x j − (cid:19) = i n (cid:88) j =1 log (cid:32) α j (cid:114)(cid:16) α j (cid:17) + 1 (cid:33) ≥ i n (cid:88) j =1 log α j . (3.7) Since x i n < r n < x i n +1 , by (3.7) and (3.6) it follows thath( C ) ≥ lim inf n → + ∞ i n (cid:80) j =1 log α j log x i n +1 = lim inf n → + ∞ i n +1 (cid:80) j =1 log α j log x i n +1 − log α i n +1 log x i n +1 ≥ lim inf n → + ∞ i n +1 (cid:80) j =1 log α j log x i n +1 − ≥ lim inf n → + ∞ n (cid:80) j =1 log α j log x n − . (3.8)Case 2: For infinitely many n , F x in r n is not the component of F r n containing r n . By passing to a subsequence we suppose that this is the case for all n .First, suppose that, if K = 2 / (3h( C )), then x i n +1 ≤ K r n log r n (3.9)for infinitely many n . Recall that m i n denotes the midpoint of the interval[ x i n , x i n +1 ] and z i n = m i n + i (cid:113) r n − m i n (see Fig. 2). We have d C (0 , F x in r n ) ≥ d C ( m i n + i, z i n ) ≥ δ C ( m i n + i, z i n ) = 12 (cid:90) (cid:113) r n − m in dxd ( x, ∂C )= (cid:113) r n − m i n − x i n +1 − x i n ≥ (cid:113) r n − x i n +1 − x i n +1 ≥ K log r n r n (cid:115) r n − K (cid:18) r n log r n (cid:19) − , where we applied (3.9). This in combination with (3.6) implies thath( C ) ≥ K lim inf n → + ∞ (cid:114) r n − K (cid:16) r n log r n (cid:17) − r n = 1 K = 32 h( C ) , which is a contradiction. Therefore, x i n +1 > K r n log r n (3.10)for all but finitely many n . So, working as in Case 1, we have d C (0 , F x in r n ) ≥ d C (0 , x i n ) ≥ i n (cid:88) j =1 log α j . N THE HARDY NUMBER OF COMB DOMAINS 11
By this and (3.6), it follows thath( C ) ≥ lim inf n → + ∞ i n (cid:80) j =1 log α j log r n = lim inf n → + ∞ i n (cid:80) j =1 log α j log x i n +1 log x i n +1 log r n ≥ lim inf n → + ∞ i n (cid:80) j =1 log α j log x i n +1 log K + log r n − log(log r n )log r n = lim inf n → + ∞ i n (cid:80) j =1 log α j log x i n +1 ≥ lim inf n → + ∞ n (cid:80) j =1 log α j log x n − , where we applied (3.10) and (3.8). Consequently, in any case we obtain thedesired result. (cid:3) Next, we prove our main theorem.
Proof of Theorem 1.1.
Suppose that h( C ) = + ∞ . Iflim inf n → + ∞ n log x n < + ∞ , then by Lemma 3.1 we deduce thatlim n → + ∞ n (cid:80) i =1 log α i log x n = + ∞ . Now, suppose that lim inf n → + ∞ n log x n = + ∞ . Recall that inf n ∈ N α n = l >
1. So, we have n (cid:80) i =1 log α i log x n > n log x n log l and thus lim n → + ∞ n (cid:80) i =1 log α i log x n = + ∞ in both cases. The other direction is direct by Lemma 3.2. (cid:3) Consequent results
In this section we prove several results derived by Theorem 1.1. They areall stated in Section 1. First, we show that if the sequence α n grows at asubexponential rate, then the Hardy number is equal to infinity. Proof of Theorem 1.2.
Let A n = max ≤ j ≤ n α j . First, we prove that our assump-tion implies that lim n → + ∞ log A n n = 0 . Suppose, on the contrary, that it is false. Then there are a constant δ > { A k n } n ∈ N of { A n } n ∈ N such that, for every n ∈ N ,log A k n k n ≥ δ. For every n ∈ N there is an m n ∈ N such that A k n = α m n and 1 ≤ m n ≤ k n .Case 1: If there is a constant K > m n ≤ K for every n ∈ N ,then 0 ≤ log A k n k n = log α m n k n ≤ max ≤ i ≤ K log α i k n . So, taking limits as n → + ∞ , we derive thatlim n → + ∞ log A k n k n = 0 , which is a contradiction.Case 2: If m n → + ∞ , then there is a subsequence { m l n } n ∈ N such that m l n → + ∞ and m l n is strictly increasing with respect to n . Thus, δ ≤ log A k ln k l n = log α m ln k l n = log α m ln m l n m l n k l n ≤ log α m ln m l n . Taking limits as n → + ∞ , we infer that δ ≤
0, which is a contradiction.Therefore, lim n → + ∞ log A n n = 0 . (4.1)Recall that inf α n = l >
1. Since α n ≤ A n for every n ∈ N and { A n } n ∈ N isan increasing sequence, we have n (cid:80) i =1 log α i log n (cid:80) i =1 α i ≥ n log l log n (cid:80) i =1 A i ≥ n log l log n + log A n = log l log nn + log A n n . Taking limits as n → + ∞ , by (4.1) we deduce thatlim n → + ∞ n (cid:80) i =1 log α i log n (cid:80) i =1 α i = + ∞ . N THE HARDY NUMBER OF COMB DOMAINS 13
Thus, Theorem 1.1 implies that h( C ) = + ∞ . (cid:3) The following corollary of Theorem 1.2 is the corollary of Theorem 4 in[6, p. 3]. Let { x n } n ∈ Z be an increasing sequence of distinct real numberswithout accumulation point in R and { c n } n ∈ Z be an associated sequence ofpositive numbers, and let D = C \ (cid:91) n ∈ Z { x n + iy n : | y n | ≥ c n } . Corollary 4.1.
Let α n = x n − x n − . Suppose that inf n ∈ Z α n > and { c n } n ∈ Z is bounded. If + ∞ (cid:88) j =1 (max | n |≤ j α n ) θ j < + ∞ for every θ ∈ (0 , , then (cid:101) h ( D ) = + ∞ .Proof. By assumption, there is a constant c > c n ≤ c for every n ∈ Z . Thus, D ⊆ C \ (cid:91) n ∈ Z { x n + iy : | y | ≥ c } := D c and h( D ) ≥ h( D c ). By this and (1.1), it suffices to prove that h( D c ) = + ∞ or, equivalently, h( D − c ) = h( D + c ) = + ∞ (see Section 1). Without loss ofgenerality, we suppose that c = 1 and inf α n >
1. We have + ∞ (cid:88) j =1 α j θ j ≤ + ∞ (cid:88) j =1 (max | n |≤ j α n ) θ j < + ∞ for every θ ∈ (0 , θ ∈ (0 , n → + ∞ α n θ n = 0and hence for every θ ∈ (0 ,
1) there is an n ( θ ) ∈ N such that for n ≥ n , α n θ n/ < α n n <
12 log 1 θ .
Set ε = (1 /
2) log(1 /θ ). So, for every ε > n ( ε ) ∈ N such thatfor n ≥ n , log α n n < ε. By Theorem 1.2, we deduce that h( D + c ) = + ∞ . Working with α n for n < D − c ) = + ∞ and thus, it followsthat h( D c ) = + ∞ . (cid:3) The following theorem implies that if the sequence log α n grows at asubexponential rate and there are no wild oscillations of α n , then the Hardynumber is equal to infinity. Proof of Theorem 1.3.
Since { b n } n ∈ N is an increasing sequence, we havelog n (cid:88) i =1 α i ≤ log n (cid:88) i =1 e b i ≤ log (cid:16) ne b n (cid:17) = log n + b n . (4.2)By assumption, inf n> ( b n − b n − ) = k >
0. This implies that b n = n (cid:88) i =2 ( b i − b i − ) + b ≥ ( n − k + b . (4.3)So, for every n ∈ N , we have0 ≤ log nb n ≤ log n ( n − k + b and thus lim n → + ∞ log nb n = 0 . By this and (4.2), we obtain the following estimateslim inf n → + ∞ n (cid:80) i =1 log α i log n (cid:80) i =1 α i ≥ lim inf n → + ∞ n (cid:80) i =1 log α i b n b n log n + b n = lim inf n → + ∞ n (cid:80) i =1 log α i b + n (cid:80) i =2 ( b i − b i − ) ≥ lim inf n → + ∞ log α n b n − b n − = + ∞ . In the last inequality we applied Theorem 2.2. Therefore, Theorem 1.1implies that h( C ) = + ∞ . (cid:3) Next, we prove that the condition of Theorem 1.3 is sharp.
Proof of Theorem 1.4.
Fix a c > { b k m } m ∈ N be a subsequence of { b m } m ∈ N such that b k m ≥ m − (cid:88) i =1 b k i (4.4)for every m ≥
2. Moreover, we observe that (4.3) implies that k m b k m ≤ k + k − b kb k m . (4.5)We consider a comb domain with α n = (cid:26) c, n / ∈ { k m : m ∈ N } e b km , n = k m for some m ∈ N . N THE HARDY NUMBER OF COMB DOMAINS 15
Applying (4.4) and (4.5), we have the following estimates I k m := k m (cid:80) i =1 log α i log k m (cid:80) i =1 α i = ( k m − m ) log c + m (cid:80) i =1 b k i log (cid:18) ( k m − m ) c + m (cid:80) i =1 e b ki (cid:19) ≤ k m log c + 2 b k m b k m ≤ log c (cid:18) k + k − b kb n (cid:19) + 2 . This implies thatlim inf n → + ∞ n (cid:80) i =1 log α i log n (cid:80) i =1 α i ≤ lim inf m → + ∞ I k m ≤ k log c and hence by Theorem 1.1 we derive that h( C ) < + ∞ . Finally, it followsthat lim inf n → + ∞ log α n b n − b n − ≤ lim inf m → + ∞ log α k m +1 b k m +1 − b k m ≤ log ck < + ∞ and the proof is complete. (cid:3) Finally, we prove that if α n is comparable to e e n , then the Hardy numberis finite. Proof of Theorem 1.5.
By assumption there are constants c , c > n ∈ N , c e e n ≤ α n ≤ c e e n . So, it follows thatlim inf n → + ∞ n (cid:80) i =1 log α i log n (cid:80) i =1 α i ≤ lim inf n → + ∞ n (cid:80) i =1 log α i log α n ≤ lim inf n → + ∞ n log c + n (cid:80) i =1 e i log c + e n = ee − n → + ∞ e n log c + e n = ee − < + ∞ . By Theorem 1.1 we deduce that h(C) < + ∞ . (cid:3) References [1] A. Baernstein and D. Girela and J. ´A. Pel´aez,
Univalent functions, Hardy spaces andspaces of Dirichlet type , Illinois J. Math. (2004), 837–859.[2] A.F. Beardon and D. Minda, The hyperbolic metric and geometric function theory ,Quasiconformal mappings and their applications (2007), 9–56.[3] D. Betsakos,
Harmonic measure on simply connected domains of fixed inradius , Ark.Mat. (1998), 275–306.[4] , On the asymptotic behavior of the trajectories of semigroups of holomorphicfunctions , J. Geometric Analysis (2016), 557–569. [5] D. Betsakos and M. Boudabra and G. Markwoski, On the probability of fast exits andlong stays of planar Brownian motion in simply connected domains , J. Math. Anal.Appl. (to appear).[6] M. Boudabra and G. Markowsky,
On the finiteness of moments of the exit timeof planar Brownian motion from comb domains , Ann. Acad. Sci. Fenn. Math. (toappear), arXiv:2101.06895.[7] D. L. Burkholder,
Exit times of Brownian motion, harmonic majorization, and Hardyspaces , Advances in Mathematics (1977), 182–205.[8] P.L. Duren, Theory of H p Spaces , Academic Press, New York-London, 1970.[9] M. Ess´en,
On analytic functions which are in H p for some positive p , Ark. Mat. (1981), 43–51.[10] O. Furdui, Limits, Series, and Fractional Part Integrals , Problems in mathematicalanalysis, Problem Books in Mathematics, Springer, New York, 2013.[11] L.J. Hansen,
Hardy classes and ranges of functions , Michigan Math. J. (1970),235–248.[12] , The Hardy class of a spiral-like function , Michigan Math. J. (1971), 279–282.[13] J.A. Jenkins, On comb domains , Proc. Amer. Math. Soc. (1996), 187–191.[14] C. Karafyllia,
Hyperbolic distance and membership of conformal maps in the Hardyspace , Proc. Amer. Math. Soc. (2019), 3855–3858.[15] ,
On the Hardy number of a domain in terms of harmonic measure and hy-perbolic distance , Ark. Mat. (2020), 307–331.[16] N. Karamanlis, On the Angular Derivative of Comb Domains , Comput. MethodsFunct. Theory (2019), 613-–623.[17] Y.C. Kim and T. Sugawa, Hardy spaces and unbounded quasidisks , Ann. Acad. Sci.Fenn. Math. (2011), 291–300.[18] G. Nagy, The Stolz-Cesaro theorem
Hardy spaces and twisted sectors for geometric models , Trans.Amer. Math. Soc. (1996), 2503–2518.[20] ,
The Hardy class of Kœnigs maps , Michigan Math. J. (1997), 495–507.[21] F. P´erez-Gonz´alez and J. R¨atty¨a, Univalent functions in Hardy, Bergman, Bloch andrelated spaces , J. d’ Anal. Math. (2008), 125–148.[22] D. Stegenga and K. Stephenson,
A geometric characterization of analytic functionswith bounded mean oscillation , J. London Math. Soc (2) (1981), 243–254. Institute for Mathematical Sciences, Stony Brook University, Stony Brook,NY 11794, U.S.A.
Email address ::