On Hilbert lemniscate theorem for a system of continua
aa r X i v : . [ m a t h . C V ] M a y On Hilbert lemniscate theorem for a system of continua
V. V. ANDRIEVSKII
Abstract
Let K be a compact set in the complex plane consisting of a finite numberof continua. We study the rate of approximation of K from the outside bylemniscates in terms of level lines of the Green function for the complement of K . Keywords:
Hilbert’s theorem, Green’s function, equilibrium measure, quasi-conformal curve, lemniscate.
MSC:
1. Introduction and main results
Let K ⊂ C be a compact set in the complex plane C consisting of disjointclosed connected sets (continua) K j , j = 1 , , . . . , ν , i.e., K = ν [ j =1 K j ; K j \ K k = ∅ for j = k ; diam( K j ) > S ) is the diameter of S ⊂ C . We always assume that Ω := C \ K is connected. Here, C := C ∪ {∞} is the extended complex plane.According to the Hilbert lemniscate theorem (see [12, p. 159]), for any openneighborhood U of K , there exists a polynomial p such that | p ( z ) ||| p || K > , z ∈ C \ U, (1.1)where || f || S denotes the uniform norm of f : S → C on S ⊂ C . Certainly, thedegree of p depends on U .Let P n , n ∈ N := { , , · · ·} be the set of all polynomials of degree at most n .Denote by g Ω ( z ) the Green function for Ω with pole at ∞ . It will be convenientfor us to extend the Green function to K by setting it equal to zero there. Let s n ( K ) , n ∈ N be the infimum of s > p ∈ P n such that(1.1) holds with U = U s := { z : g Ω ( z ) < s } . A result by Siciak [14, Theorem 1] for the Fekete polynomials yields that s n ( K ) = O (cid:18) log nn (cid:19) as n → ∞ (1.2) 1cf. [3, Theorem 1], [9, Theorem 2], [11, Theorem 2.2]). Theorem 1
Under the above assumptions, s n ( K ) = O (cid:18) (log log n ) n (cid:19) as n → ∞ . (1.3)We also would like to demonstrate that if more information is known about thegeometry of K , (1.3) can be further improved in the following way. A Jordancurve L ⊂ C is called quasiconformal (see [1], [10, p. 100] or [8]) if for every z , z ∈ L , diam( L ( z , z )) ≤ Λ L | z − z | , where L ( z , z ) is the smaller subarc of L between z and z ; and Λ L ≥ L . A quasidisk is a Jordan domain bounded bythe quasiconformal curve. Theorem 2
If each K j is a closed quasidisk, then s n ( K ) = O (cid:18) n (cid:19) as n → ∞ . (1.4)See [3, Theorem 2] for a special case of this result.Our proof for Theorem 2 yields insights that can be leveraged to obtain otherresults. Specifically, that for sufficiently large n there exists polynomial p n ∈ P n such that G ( p n ) := { z : | p n ( z ) | ≤ } consists of exactly ν disjoint Jordandomains and K ⊂ G ( p n ) ⊂ U C/n holds with a constant C = C ( K ) > p n may be chosen such that all its zeros belongto K .It is worth pointing out that (1.4) is optimal in the following sense. Let K be aclosed quasidisk, i.e. ν = 1 , for which there exist ζ ∈ ∂K, δ > < β < ζ , radius δ and opening βπ is a subsetof Ω . Then, according to [3, Theorem 3], s n ( K ) ≥ εn , n ∈ N is true with some constant ε = ε ( K ) > ν = 1 it is natural to approximate K by lemniscatesgiven by Faber polynomials F n = F n ( K ) . It was shown in [3, pp. 300-301] that2or the quasidisk K constructed by Gaier [7], the inequality (1.1) does not holdfor p = F n , U = U α log n/n , some constant α = α ( K ) and an infinite number of n ∈ N (cf. (1.2)-(1.4)).In what follows, we use following notation. d ( S , S ) := inf z ∈ S ,z ∈ S | z − z | , S , S ⊂ C . For a Jordan curve L ⊂ C , denote by int ( L ) the bounded connected componentof C \ L .For a (Borel) set S ⊂ C , denote by | S | its linear measure (length) and by σ ( S ) its two-dimensional Lebesgue measure (area).In what follows, we denote by c, c , . . . positive constants that are either ab-solute or they depend only on K . For the nonnegative functions a and b wewrite a (cid:22) b if a ≤ c b , and a ≍ b if a (cid:22) b and b (cid:22) a simultaneously.
2. Construction of auxiliary polynomials
In this section we review (in more general setting) the construction of themonic polynomials suggested in [15, 16, 4, 6]. For the convenience of the reader,we repeat the relevant material from these papers without proofs, thus makingour exposition self-contained.We start with some general facts from potential theory which can be found, forexample, in [17, 12, 13]. The Green function g ( z ) = g Ω ( z ) has a multiple-valuedharmonic conjugate ˜ g ( z ) . LetΦ( z ) := exp( g ( z ) + i ˜ g ( z )) ,K s := { z : g ( z ) = s } , s > . Note that cap( K s ) = e s cap( K ) . (2.1)Here cap( S ) is the logarithmic capacity of a compact set S ⊂ C .Let s > < s < s , the set K s = ∪ νj =1 K js consists of ν mutually disjoined Jordan curves, where K js is the curve surrounding K j .Moreover, we fix a positive number s ∗ < s /
10 so small that for each j =1 , . . . , ν , d ( ζ , K j ) ≤ d ( ζ , K js ) , ζ ∈ int( K js ∗ ) . (2.2) 3et µ = µ K be the equilibrium measure of K and let ω j := µ ( K j ) . The function φ j := Φ /ω j ( ζ ) is a conformal and univalent mapping of Ω j := int( K js ) \ K j ontothe annulus A j := { w : 1 < | w | < e s /ω j } as well as K js = { ζ ∈ Ω j : | φ j ( ζ ) | = e s/ω j } , < s < s . Note that for µ s := µ K s , µ s ( K js ) = µ ( K j ) = ω j , < s < s . Furthermore, for an arc γ = { ζ ∈ K js : θ ≤ arg φ j ( ζ ) ≤ θ } , < θ − θ ≤ π, we have µ s ( γ ) = ( θ − θ ) ω j π . Assuming that m ∈ N is sufficiently large, i.e. m > / (min j ω j ) we let m j := ⌊ mω j ⌋ , j = 1 , , . . . , ν − ,m ν := m − ( m + . . . + m ν − ) , where ⌊ a ⌋ means the integer part of a real number a .Therefore, 0 ≤ m ν − mω ν = ν − X j =1 ( mω j − m j ) ≤ ν − . (2.3)Next, for 0 < s < s ∗ , we represent each K js as the union of closed subarcs I js,k , k = 1 , . . . , m j such that I js,k \ I js,k +1 =: ξ js,k , k = 1 , . . . , m j − , and I js,m j ∩ I js, =: ξ js,m j =: ξ js, are points of K js ordered in a positive direction,as well as µ s ( I js,k ) = ω j m j , k = 1 , . . . , m j . Consider also ψ j := φ − j ,˜ D js,k := (cid:26) t = re iη : ψ j ( e s/ω j + iη ) ∈ I js,k , ≤ e s/ω j − r ≤ e s/ω j − (cid:27) ,D js,k := ψ j ( ˜ D js,k ) , D js := m j [ k =1 D js,k , D s := ν [ j =1 D js . emma 1 Let m, q ∈ N and c := 640 π max j e s ∗ /ω j . Then for m ≥ m := ⌊ cq/s ∗ + 10 ν/ min j ω j ⌋ , s = cq/m < s ∗ , j = 1 , . . . , ν and k = 1 , . . . , m j , we have d ( ξ js,k , K j ) c q ≤ | ξ js,k − ξ js,k − | ≤ diam( I js,k ) ≤ | I js,k | ≤ d ( I js,k , K j )10 q . (2.4) Moreover, if q = 1 then σ ( D js,k ) ≥ d ( ξ js,k , K j ) c (2.5) as well as diam( D js,k ) ≤ d ( ξ js,k , K j ) . (2.6)For the proof of Lemma 1, see Section 3.Next, we construct the points ζ js,k,l , l = 1 , . . . , q as follows. For s = cq/m asin Lemma 1 and u = 1 , . . . , q , let m js,k,u := 1 µ s ( I js,k ) Z I js,k ( ξ − ξ js,k ) u dµ s ( ξ ) . Consider the system of equations q X l =1 ( r js,k,l ) u = qm js,k,u =: ˜ m js,k,u , u = 1 , . . . , q. We interpret r l := r js,k,l as the roots of the polynomial z q + a q − z q − + . . . + a whose coefficients satisfy Newton’s identities˜ m u + a q − ˜ m u − + . . . + a q − u +1 ˜ m = − ua q − u , u = 1 , . . . , q, (2.7)where ˜ m u := ˜ m js,k,u satisfy | ˜ m u | ≤ qd u , d = d js,k := diam ( I js,k ) .According to (2.7), | a q − u | ≤ q u d u , u = 1 , . . . , q, which implies | r js,k,l | ≤ qd js,k . (2.8)See [6, Section 2] for more details.Let ζ js,k,l := ξ js,k + r js,k,l . By virtue of (2.4) and (2.8), | ζ js,k,l − ξ js,k | d ( ξ js,k , K j ) ≤ | r js,k,l | d ( I js,k , K j ) ≤ , (2.9) 5s well as for ξ ∈ I js,k , | ξ − ξ js,k | d ( ξ js,k , K j ) ≤ d js,k d ( I js,k , K j ) ≤ . (2.10)By [5, p. 23, Lemma 2.3], which is an immediate consequence of Koebe’s one-quarter theorem, and (2.2), we have the following lemma. Recall that ψ j isdefined in A j := { τ : 1 < | τ | < e s /ω j } . Let E j := { w : 1 < | w | < e s ∗ /ω j } . Lemma 2
Let w ∈ E j , τ ∈ A j and ξ = ψ j ( w ) , ζ = ψ j ( τ ) . Then d ( ξ, K j ) | w | − ≤ | ψ ′ j ( w ) | ≤ d ( ξ, K j ) | w | − . (2.11) Moreover, if either | τ − w | ≤ ( | w | − / or | ζ − ξ | ≤ d ( ξ, K j ) / , then | τ − w || w | − ≤ | ζ − ξ | d ( ξ, K j ) ≤ | τ − w || w | − . (2.12)Next, we claim that for z ∈ K j s , s < s ∗ , | z − ξ js,k | ≥ d ( ξ js,k , K j ) . (2.13)Indeed, if we assume, contrary to (2.13), that | z − ξ js,k | < d ( ξ js,k , K j ) , then, according to the left-hand side of (2.12), e s/ω j − e s/ω j ≤ | φ j ( z ) − φ j ( ξ js,k ) |≤ e s/ω j − | z − ξ js,k | d ( ξ js,k , K j ) < e s/ω j − , which contradicts to the obvious inequality e x − e x ≥ e x − , x ≥ . Hence, (2.13) is proven.A major component of the proof of (1.3) and (1.4) is the polynomial P n ( z ) := ν Y j =1 m j Y k =1 q Y l =1 ( z − ζ js,k,l ) , n = qm, s = cqm ≤ s ∗ . z ∈ K s , mg Ω s ( z ) + m log cap( K s ) = m Z K s log | z − ξ | dµ s ( ξ )= ν X j =1 m j X k =1 (cid:18) m − m j ω j (cid:19) Z I js,k log | z − ξ | dµ s ( ξ )+ ν X j =1 m j X k =1 µ s ( I js,k ) Z I js,k log | z − ξ | dµ s ( ξ )=: Σ ( z ) + Σ ( z ) . Since Z K s | log | z − ξ || dµ s ( ξ ) ≤ | log diam( K s ) | + Z K s log diam( K s ) | z − ξ | dµ s ( ξ ) ≤ | log diam( K s ) | − g Ω s ( z ) − log cap( K s ) (cid:22) , according to (2.3), for z ∈ K s , we obtain | Σ ( z ) | (cid:22) Z K s | log | z − ξ || dµ s ( ξ ) (cid:22) . Next, for the same z ∈ K s ,log | P n ( z ) | − q Σ ( z )= ν X j =1 m j X k =1 q X l =1 log | z − ζ js,k,l | − µ s ( I js,k ) Z I js,k log | z − ξ | dµ s ( ξ ) ! = ν X j =1 m j ω j m j X k =1 q X l =1 Z I js,k log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z − ζ js,k,l z − ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ s ( ξ ) . Moreover, (2.4), (2.8), (2.13) and Taylor’s theorem [2, pp. 125-126] implylog z − ζ js,k,l z − ξ ! = log − ζ js,k,l − ξ js,k z − ξ js,k ! − log − ξ − ξ js,k z − ξ js,k ! = q X u =1 u ξ − ξ js,k z − ξ js,k ! u − ζ js,k,l − ξ js,k z − ξ js,k ! u ! + B js,k,l ( z ) , where | B js,k,l ( z ) | (cid:22) qd js,k d ( z, I js,k ) ! q +1 . q X l =1 Z I js,k (cid:0) ( ξ − ξ js,k ) u − ( ζ js,k,l − ξ js,k ) u (cid:1) dµ s ( ξ )= q X l =1 (cid:18) ω j m j m js,k,u − ( r js,k,l ) u ω j m j (cid:19) = ω j m j q X l =1 (cid:0) m js,k,u − ( r js,k,l ) u (cid:1) = ω j m j ˜ m js,k,u − q X l =1 ( r js,k,l ) u ! = 0 , for z ∈ K s , we have | log | P n ( z ) | − ng Ω s ( z ) − n log cap( K s ) | ≤ q | Σ ( z ) | + | log | P n ( z ) | − q Σ ( z ) |(cid:22) q + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν X j =1 m j ω j m j X k =1 q X l =1 Z I js,k ℜ q X u =1 u ξ − ξ js,k z − ξ js,k ! u − ζ js,k,l − ξ js,k z − ξ js,k ! u ! + B js,k,l ( z ) (cid:1) dµ s ( ξ ) (cid:12)(cid:12) ≤ q + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν X j =1 m j ω j m j X k =1 q X u =1 u ( z − ξ js,k ) u q X l =1 Z I js,k (cid:0) ( ξ − ξ js,k ) u − ( ζ js,k,l − ξ js,k ) u (cid:1) dµ s ( ζ )+ q X l =1 Z I js,k B js,k,l ( z ) dµ s ( ξ ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:22) q ν X j =1 m j X k =1 qd js,k d ( z, I js,k ) ! q +1 . To summarize, according to (2.1) and the identity g Ω s = g Ω ( z ) − s, z ∈ C \ U s , for z ∈ K s , s < s ∗ , we have | log | P n ( z ) | − ng Ω ( z ) − n log cap( K ) |(cid:22) q + q ν X j =1 m j X k =1 qd js,k d ( z, I js,k ) ! q +1 . (2.14) 8 . Distortion properties of φ j and ψ j Proof of Lemma 1.
Only the first and the last inequalities in (2.4) are nottrivial. According to our assumption12 ≤ mω j m j ≤ , j = 1 , . . . , ν. Let η j < η j < . . . < η jm j = η j + 2 π be determined by t js,k := φ j ( ξ js,k ) =exp( s/ω j + iη jk ) , i.e., η jk − η jk − = 2 π/m j .Let c := max j e s ∗ /ω j . For t ∈ ˜ I js,k := φ j ( I js,k ) , | t − t js,k | ≤ e s/ω j | e iη jk − e iη jk − |≤ c πm j ≤ cqmω j = 132 sω j < (cid:0) e s/ω j − (cid:1) . Since (2.12) implies for ξ ∈ I js,k , | ξ − ξ js,k | d ( ξ js,k , K j ) ≤ | t − t js,k | e s/ω j − < , (3.1)by (2.11), for t ∈ ˜ I js,k , we have | ψ ′ j ( t ) | ≤ d ( ξ js,k , K j ) e s/ω j − ≤ d ( I js,k , K j ) e s/ω j − . Hence, | I js,k | ≤ e s/ω j Z η jk η jk − | ψ ′ j ( e s/ω j + iθ ) | dθ ≤ c d ( I js,k , K j ) e s/ω j − η jk − η jk − ) ≤ πc cq ω j mm j d ( I js,k , K j ) ≤ πc cq d ( I js,k , K j ) ≤ d ( I js,k , K j )10 q , which proves the last inequality in (2.4).The first inequality in (2.4) follows from (2.12) and (3.1): | ξ js,k − ξ js,k − | d ( ξ js,k , K j ) ≥ | ψ j ( ξ js,k ) − ψ j ( ξ js,k − | )16( e s/ω j − ≥
116 2 π πm j ω j mc cq ≥ c q . ζ ∈ D js,k and ζ ∗ := ψ j (( e s/ω j − φ j ( ζ ) / | φ j ( ζ ) | ) by virtue of (2.12), wehave d ( ζ , I js,k ) ≤ | ζ − ζ ∗ | < d ( ζ ∗ , K j ) . Therefore, by (2.4), | ζ − ξ js,k | ≤ | ζ − ζ ∗ | + | ζ ∗ − ξ js,k |≤ (cid:0) | ζ ∗ − ξ js,k | + d ( ξ js,k , K j ) (cid:1) + | ζ ∗ − ξ js,k |≤ (cid:18)
540 + 14 (cid:19) d ( ξ js,k , K j ) , which yields (2.6).Moreover, (2.11) implies for τ = φ ( ζ ) and ζ ∈ D js,k , | ψ ′ j ( τ ) | ≥ d ( ζ , K j ) e s/ω j − ≥ d ( ξ js,k , K j ) e s/ω j − σ ( D js,k ) = Z Z ˜ D js,k | ψ ′ j ( τ ) | dσ ( τ ) ≥ d ( ξ js,k , K j ) ( e s/ω j − σ ( ˜ D js,k )= 164 d ( ξ js,k , K j ) e s/ω j − | ˜ I js,k | ≥
164 2 πm j ω j c s d ( ξ js,k , K j ) ≥ d ( ξ js,k , K j ) c . ✷ Since by Lemma 1 and (2.13) for z ∈ K s ,2 qd js,k d ( z, I js,k ) ≤ , according to (2.14) we have | log | P n ( z ) | − ng Ω ( z ) − n log cap( K ) |(cid:22) q + q (cid:18) (cid:19) q ν X j =1 m j X k =1 d js,k d ( z, I js,k ) ! . Furthermore, by Lemma 1, (2.5) and (2.13), for z ∈ K s , ν X j =1 m j X k =1 d js,k d ( z, I js,k ) ! (cid:22) ν X j =1 m j X k =1 σ ( D js,k ) d ( z, D js,k ) (cid:22) Z Z D s dσ ( ζ ) | ζ − z | . d ( z, K s ) (cid:23) s , usingpolar coordinates with center at z , we obtain Z Z D s dσ ( ζ ) | ζ − z | (cid:22) log diam( K s ) d ( z, K s ) , (cid:22) log s s , which yields | log | P n ( z ) | − ng Ω ( z ) − n log cap( K ) | (cid:22) q + 2 − q log m. (3.2) Proof of Theorem 1 . Without loss of generality, we assume that n is suffi-ciently large. First, let n = mq , where q = q m := ⌊ m ⌋ . Then by (3.2),for z ∈ K cq/m , | log | P n ( z ) | − ng Ω ( z ) − n log cap( K ) | (cid:22) (log log m ) ≍ (log log n ) . Thus, the maximum principle implies that for P ∗ n := P n cap( K ) n , s = cq/m = cq /n and z ∈ C \ U s ,exp (cid:0) ng Ω ( z ) − c (log log m ) (cid:1) ≤ | P ∗ n ( z ) | ≤ exp (cid:0) ng Ω ( z ) + c (log log m ) (cid:1) . Therefore, || P ∗ n || K ≤ || P ∗ n || K s ≤ exp (cid:0) c (log log n ) (cid:1) . At the same time, for z ∈ C \ U δ , δ = 3 c (log log n ) /n , | P ∗ n ( z ) | ≥ exp (cid:0) c (log log m ) (cid:1) > || P ∗ n || K , which proves (1.3) for n = mq m .For arbitrary (sufficiently large) n we find m ∈ N such that mq m ≤ n < ( m + 1) q m +1 . Since 1 ≤ nmg m ≤ ( m + 1) q m +1 mq m → n → ∞ , we obtain s n ( K ) ≤ s mq m ( K ) (cid:22) (log log m ) mq m (cid:22) (log log n ) n , which completes the proof of (1.3). ✷ From now on we assume that each K j , j = 1 , . . . , ν is a quasidisk. Since ∂ Ω j consists of two quasiconformal curves, φ j can be extended to a Q j -quasiconformalhomeomorphism of a neighborhood of Ω j to a neighborhood of A j with some Q j ≥ emma 3 Let ζ k ∈ Ω j , w k := φ j ( ζ k ) , k = 1 , , . Then:(i) the conditions | ζ − ζ | ≤ C | ζ − ζ | and | w − w | ≤ C | w − w | areequivalent; besides, the constants C and C are mutually dependent and dependon Q j and K ;(ii) if | ζ − ζ | ≤ C | ζ − ζ | , then C (cid:12)(cid:12)(cid:12)(cid:12) w − w w − w (cid:12)(cid:12)(cid:12)(cid:12) /Q ≤ (cid:12)(cid:12)(cid:12)(cid:12) ζ − ζ ζ − ζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12)(cid:12) w − w w − w (cid:12)(cid:12)(cid:12)(cid:12) Q , where Q := max j Q j and C = C ( C , Q, K ) > . For ζ ∈ Ω j let ˜ ζ := ψ j ( φ j ( ζ ) / | φ j ( ζ ) | ) . We fix ζ ∗ j ∈ K js . By Lemma 3, for ζ ∈ K js and z ∈ K j s , s < s ∗ , d ( ζ , K j ) | ζ − z | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ − ˜ ζζ − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:22) (cid:18) s | φ j ( ζ ) − φ j ( z ) | (cid:19) /Q , (3.3) d ( ζ , K j ) ≤ | ζ − ˜ ζ | ≍ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ − ˜ ζζ − ζ ∗ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:22) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ j ( ζ ) − φ j ( ˜ ζ ) φ j ( ζ ) − φ j ( ζ ∗ j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) /Q ≍ s /Q . (3.4)Furthermore, by virtue of Lemma 1, for z ∈ K s and q ∈ N , Z K js d ( ζ , K j ) q | ζ − z | q +1 | dζ | = m j X k =1 Z I js,k d ( ζ , K j ) q | ζ − z | q +1 | dζ |(cid:23) m j X k =1 d ( I js,k , K j ) q | I js,k | d ( z, I js,k ) q +1 (cid:23) m j X k =1 d js,k d ( z, I js,k ) ! q +1 . Therefore, according to (2.14) for z ∈ K r s , r = 1 , . . . , ν , | log | P n ( z ) | − ng Ω ( z ) − n log cap( K ) |(cid:22) ν X j =1 ,j = r Z K js d ( ζ , K j ) q | dζ | + Z K rs d ( ζ , K r ) q | ζ − z | q +1 | dζ | . Let q := ⌊ Q ⌋ and w = φ r ( z ) . Then by (2.11), (3.3) and (3.4), Z K js d ( ζ , K j ) q | dζ | (cid:22) s Z | τ | = e s/ωj d ( ψ j ( τ ) , K j ) q +1 | dτ |(cid:22) s ( q +1) /Q − (cid:22) .
12s well as Z K rs d ( ζ , K r ) q | ζ − z | q +1 | dζ | (cid:22) s Z | τ | = e s/ωr (cid:18) d ( ψ r ( τ ) , K r ) | ψ r ( τ ) − ψ r ( w ) | (cid:19) q +1 | dτ |(cid:22) s Z | τ | = e s/ωr s ( q +1) /Q | τ − w | ( q +1) /Q | dτ | (cid:22) . Thus, the maximum principle implies that for P ∗ n ( z ) := P n ( z ) cap( K ) n and s = cq/m = cq /n, n = qm, m > m ,1 c e ng Ω ( z ) ≤ | P ∗ n ( z ) | ≤ c e ng Ω ( z ) , z ∈ C \ U s . (3.5) Proof of Theorem 2 . Without loss of generality, we assume that n is suffi-ciently large. First, let n = mq be such that (3.5) holds. We have || P ∗ n || K ≤ || P ∗ n || K s ≤ c exp (cid:0) cq (cid:1) . If we let c := 2 log c + 11 cq , then for z ∈ C \ U c /n , we get | P ∗ n ( z ) | ≥ c exp( c ) > c exp (cid:0) cq (cid:1) , which shows that s n ( K ) ≤ c /n .If mq < n ≤ m ( q + 1) , then s n ( K ) ≤ c mq ≤ c m ( q + 1) ≤ c n . Hence, in both cases we have (1.4). ✷ Acknowledgements.
The author is grateful to M. Nesterenko for his helpfulcomments.