Inequalities for sums of Green potentials and Blaschke products

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1 Submitted exclusively to the London Mathematical Society doi: /0000/ Inequalities for sums of reen potentials and Blaschke products Igor. Pritsker Abstract We study inequalities for the ima of reen potentials on a compact subset of an arbitrary domain in the complex plane. The results are based on a new representation of the pseudohyperbolic farthest-point distance function via a reen potential. We also give applications to sharp inequalities for the supremum norms of Blaschke products. 1. reen potentials Let C be a domain possessing the reen function g (z, ζ) with pole at ζ. For the positive Borel measures ν k, k = 1,..., m, with compact supports in, define their reen potentials [1, p. 96] by U ν k (z) := g (z, ζ) dν k (ζ), z. Note that reen potentials are superharmonic and nonnegative in. Suppose that ν := m ν j is a unit measure. We study inequalities of the following type U ν k A + B U ν k, (1.1) where is a compact set of positive logarithmic capacity cap(). Observe that the opposite inequality is always true with A = 0 and B = 1: U ν k U ν k, which suggests that optimal constants in (1.1) should satisfy A 0 and B 1. This problem was considered for logarithmic potentials in [11], where an analog of (1.1) was proved with constants B = 1 and A expressed through the farthest-point distance function and potential theoretic quantities. We generalize those ideas to reen potentials and the hyperbolic setting below. Furthermore, we prove a sharp version of (1.1), and provide applications to the inequalities for Blaschke products in supremum norms on. Since has positive capacity by our assumption, we can consider its reen equilibrium measure µ and the minimum reen energy (Robin s constant) V, see [13, p. 132]. It is well known that µ is a positive unit Borel measure supported on, whose potential satisfies U µ (z) V, z, and U µ (z) = V q.e. on, (1.2) see Theorem 5.11 [13, p. 132]. The equality in (1.2) holds quasi everywhere (q.e.) on, which means that it holds up to an exceptional set of zero logarithmic capacity. Hence Fubini s 2000 Mathematics Subject Classification 31A15 (primary), 30C15 (secondary). Research was partially supported by the National Security Agency, and by the Alexander von Humboldt Foundation.

2 Page 2 of 15 IOR. PRITSKR theorem gives that U µ (z) dµ (z) = U µ (z) dµ(z) V for any positive unit Borel measure µ supported in. This immediately implies that and that for ν = m ν j we have U ν k 0 V + U µ V, (1.3) U ν k. While this basic version of (1.1) may be useful, it is clearly not sharp as equality is never attained in the above inequality. A sharp version of (1.1) requires more sophisticated tools. For z, ζ in the unit disk D, consider the pseudohyperbolic metric δ D (z, ζ) = z ζ / 1 ζz. It has a standard extension to an arbitrary simply connected domain by conformal invariance. However, we make an even more general definition of the pseudohyperbolic metric for any domain C possessing the reen function g (z, ζ) with pole at ζ : δ (z, ζ) := e g (z,ζ), z, ζ. Thus the function log δ (z, ζ) = g (z, ζ) is superharmonic in each variable z, ζ. If we define the farthest-point pseudohyperbolic distance function for a compact set by d (z) := sup δ (z, ζ), ζ then the function log d (z) = ζ g (z, ζ) is superharmonic in, see Theorem of [12, p. 38]. It is clear from Harnack s inequality applied to reen function that log d (z) is continuous in, provided is not a singleton. Furthermore, we have the following representation (Riesz decomposition) of this superharmonic function as a reen potential. Theorem 1.1. Let C be a domain with the reen function g (z, ζ), and let be a compact set, which is not a single point. Then log d (z) = g (z, ζ) = g (z, ζ) dσ(ζ), z, (1.4) ζ where σ is a positive measure supported on, whose total mass satisfies σ () 1. (1.5) If is simply connected, then strict inequality holds in (1.5), and the support of σ has points of accumulation on. Furthermore, if is the closure of a Jordan domain H, then supp σ H. If = {ζ} is a singleton, then we obviously have that log d (z) = g (z, ζ), i.e. σ = δ ζ is the unit point mass at ζ. However, for all nontrivial compact sets the total mass σ () of the representing measure is less than 1, at least when is simply connected. We conjecture that strict inequality holds in (1.5) for all domains possessing reen functions. This is different from the case of uclidean metric in R 2, see [8] and [7], where the corresponding Riesz measure has unit mass. It turns out that σ () depends on and in a rather complicated way. We give an explicit example of this measure below. It would be of interest to study relations between the properties of σ and the hyperbolic geometry of in. The first representation of type (1.4) for the uclidean farthest-point distance function

3 RN POTNTIALS AND BLASCHK PRODUCTS Page 3 of 15 appeared in [2], where only finite sets were considered. The general uclidean case of an arbitrary compact set was treated in [8], and more detailed studies of the Riesz measure were accomplished in [7] and [4]. Theorem 1.1 provides important tools for proving a sharp version of (1.1) of the following form. Theorem 1.2. Let be a compact set, cap() > 0. Suppose that ν k, k = 1,..., m, are positive Borel measures compactly supported in, such that the total mass of m ν k is equal to 1. We have that U ν k C + σ() U ν k, (1.6) where C := g (z, ζ) dµ (ζ) V σ() (1.7) z cannot be replaced by a smaller constant independent of m. We mention another equivalent form of the best constant: ( ) C = U µ (z) V dσ(z) 0, (1.8) where the above inequality is immediate from (1.2). An illustration of Theorems 1.1 and 1.2 with explicit measure and constants is given in the following. and Corollary 1.3. If = D := {z : z < 1} and = D r := {z : z r}, 0 < r < 1, then dσ D D r (z) = Furthermore, (1.6) holds with r(1 r 2 )(1 z 2 ) 2π z (r z + 1) 2 dxdy, z = x + iy D, (1.9) (r + z ) 2 C D D r σd D r (D) = 1 r < 1. (1.10) 1 + r = log 1 + r2 2r + 1 r log r. (1.11) 1 + r We remark that analogs of Theorems 1.1 and 1.2 can be proved in R n, n 3, see [1] and [6] for the corresponding theory of reen potentials. 2. Blaschke products We use a well known connection of Blaschke products with reen potentials of discrete measures in the unit disk D. iven a finite Blaschke product of degree n B(z) = e iθ n z z j 1 z j z, {z j} n D,

4 Page 4 of 15 IOR. PRITSKR define the normalized counting measure in the zeros of B by ν := 1 n δ zj, k = 1,..., n, n and observe that U(z) ν = 1 log B(z), k = 1,..., n. n This simple idea yields several interesting applications of general results stated in the previous section. Let be the supremum norm on a compact set D. Theorem 2.1. Let D be a compact set of positive capacity. If B k, k = 1,..., m, are finite Blaschke products of the corresponding degrees n k, then D m m σ (D) B k e ncd B k, (2.1) where n = m n k. The constant C D := z log cannot be replaced by a smaller value independent of m. 1 ζz z ζ dµd (ζ) V D σ D (D) (2.2) The above result is a generalization of Theorem 2.1 of [8] on products of polynomials in uniform norms. Such inequalities for products of polynomials have been studied since at least 1930s, see [8], [9] and [10] for history and references. However, (2.1) appears to be the first inequality of this type for Blaschke products. An example with explicit constants, for being a concentric disk, is given below. Corollary 2.2. If = D r := {z : z r}, 0 < r < 1, in Theorem 2.1, then m ( ) 2r 2r/(r+1) n m B k Dr 1 + r 2 1 r 1+r B k D r. (2.3) The constant C D defined in (2.2) is asymptotically sharp, i.e., it cannot be decreased so that (2.1) still holds true for all Blaschke products as specified in Theorem 2.1. However, it is of interest to find an exact constant in (2.3), which is attained for each n N. Such a constant was found in Mahler s inequality for products of polynomials, cf. [5]. In fact, there are arrays of Blaschke products for which this best constant C D is attained asymptotically in the sense of (2.4) below. It is a natural problem to study and characterize such extremal arrays of Blaschke products. The following two theorems address the problem by essentially stating that extremal arrays of Blaschke products are described by the equidistribution of their zeros according to the reen equilibrium measure for in D. Theorem 2.3. Let D be a compact set, cap() > 0. Suppose that B k,l, k = 1,..., m l, are finite Blaschke products of the corresponding degrees n k,l, and set n l := m l n k,l. Let τ nl be the normalized counting measure in the zeros of the product m l B k,l. If we have an array

5 RN POTNTIALS AND BLASCHK PRODUCTS Page 5 of 15 of Blaschke products B k,l that satisfies where C D lim n l is defined by (2.2), then ( ml B k,l m l B k,l σd (D) = e CD, (2.4) ) 1/nl lim n l m l 1/n l B k,l = e V D. (2.5) Furthermore, if (2.4) holds and one the following conditions is satisfied: (i) is a compact set with empty interior, and D \ is connected; (ii) = H, where H is a Jordan domain; then the measures τ nl converge to µ D in the weak* topology. We also have a partial converse for the latter theorem. Theorem 2.4. Let D be a regular compact set. Suppose that B k,n (z) = e iθ z z k,n k,n, k = 1,..., n, n N, 1 z k,n z are Blaschke factors. Let τ n be the normalized counting measure in the zeros of the product n B k,n. If the measures τ n converge to µ D in the weak* topology as n, then where C D lim is defined by (2.2). ( n B k,n n B k,n σd (D) ) 1/n = e CD, (2.6) All results of this section may be easily transplanted into the upper half-plane Π +, by using the standard conformal mapping φ(z) = (z i)/(z + i) of Π + onto D. Note that finite Blaschke products on Π + take the following form: B(z) = e iθ n z z j z z j, {z j } n Π +. Proof of Theorem Proofs It is clear from the definition that log d (z) = ζ g (z, ζ). Since g (z, ζ) is superharmonic in as a function of z, which holds for every ζ, we have that u(z) := ζ g (z, ζ) is a superharmonic function in by Theorem of [12, p. 38]. The reen function g(z, ζ) is positive for any z and any ζ. Furthermore, we have for its limiting boundary values that lim g(z, ζ) = 0 z ξ for any ζ and quasi every ξ, see [1], [12] and [13]. Hence u(z) is a nonnegative function in that satisfies lim u(z) = 0 z ξ

6 Page 6 of 15 IOR. PRITSKR for quasi every ξ. The generalized Maximum Principle for harmonic functions now implies that the greatest harmonic minorant of u(z) in is the function h(z) 0. We now conclude by Corollary of [1, p. 108] (or by Theorem 1.24 of [6, p. 109]) that u(z) = g (z, ζ) dσ(ζ) = U σ (z), z, where σ is the Riesz representation measure supported on. Let K be a regular compact set. Then we have from (1.2) (see also [13, p. 132]) that U µ K (z) = V K, z K. Using Fubini s theorem, we obtain that U σ (z) dµ K(z) = U µ K (z) dσ (z) On the other hand, U σ which implies by (1.2) that VK σ(k) It immediately follows that K U µ K (z) dσ (z) = VK σ(k). (z) = u(z) = g (z, ζ) g (z, ζ), ζ, ζ U σ (z) dµ K(z) g (z, ζ) dµ K(z) = U µ K (ζ) V K. σ (K) 1 for any regular compact subset K of, and (1.5) holds because can be exhausted by such subsets. We now assume that is simply connected, and prove the rest of statements in this theorem. Since is not a singleton, it contains at least two distinct points ζ 0 and ζ 1. Consider a conformal mapping φ of onto the unit disk D, such that φ(ζ 0 ) = 0 and φ(ζ 1 ) = a (0, 1). Recall that g (z, ζ) = g D (φ(z), φ(ζ)) = log 1 φ(ζ)φ(z), z, ζ. φ(z) φ(ζ) Hence d (z) = dd φ() (φ(z)), z, and all reen potentials satisfy corresponding conformal invariance property. Let D r := {w : w r}, 0 < r < 1, and consider K r := φ 1 (D r ). It is clear that K r is a compact set such that K r for r close to 1. Recall that the Martin kernel of D relative to the origin is defined by M(w, t) := g D (w, t)/g D (0, t), w, t D, see Chapter 8 of [1]. It is known that the limiting boundary values of the Martin kernel in the unit disk coincide (up to a constant) with those of the Poisson kernel: lim M(w, t) = 1 w 2 t e iθ e iθ, w D, θ [0, 2π), w 2 see xample in [1]. In particular, we obtain that lim M(a, t) = lim g D(a, t)/g D (0, t) = 1 a t 1 t a < 1, which means that there are ɛ, δ > 0 such that g D(w, t) g D (w, a) (1 ɛ)g D (w, 0), w + 1 < δ. t φ() Using conformal invariance of reen potentials, we obtain from xample 5.13 of [13, p. 133] that dµ K r (φ 1 (re iθ )) = dθ/(2π) and VK r = log r. We now let w = re iθ = φ(z), and estimate

7 RN POTNTIALS AND BLASCHK PRODUCTS Page 7 of 15 for r sufficiently close to 1 that 2π g (z, ζ) dµ K ζ r (z) = g D(w, t) dθ 0 t φ() 2π g D (w, a) dθ w+1 <δ 2π + (1 ɛ) g D (re iθ, 0) dθ θ π <δ 2π + g D (re iθ, 0) dθ θ π δ 2π ( = 1 ɛδ ) log 1 ( π r = 1 ɛδ ) VK π r. w+1 δ Arguing as in the proof of (1.5), we obtain that g (z, ζ) dµ K ζ r (z) = U σ (z) dµ K r (z) = U µ Kr (z) dσ (z) Kr (z) dσ (z) = VK r σ(k r ) K r U µ by Fubini s theorem and (1.2). It follows that V K r σ (K r ) ( 1 ɛδ ) VK π r g D (w, 0) dθ 2π for all r (0, 1) sufficiently close to 1. Consequently, we obtain that σ () (1 ɛδ/π) < 1 after letting r 1. Recall our notation u(z) = log d (z) = g (z, ζ) = U σ (z), z. ζ For any z, the above imum is attained on at a point ζ z : u(z) = ζ g (z, ζ) = g (z, ζ z ), z, by lower semicontinuity of g (z, ). Note that u(z) is harmonic in \ supp σ, and let A be any connected component of \ supp σ. iven a fixed z A, we have that u(z) = g (z, ζ z ) and u(t) g (t, ζ z ), t. Hence v(t) := u(t) g (t, ζ z ) is a subharmonic function that attains its maximum in A at z, which implies that u(t) = g (t, ζ z ) for all t A by the Maximum Principle. Consider = H, where H is a Jordan domain. If we assume that supp σ H =, then there exists ζ H such that u(t) = g (t, ζ), t H. But lim t ζ u(t) = lim t ζ g (t, ζ) =, which means that u(ζ) = by superharmonicity of u. This is clearly impossible, and we conclude that supp σ H. If we assume that supp σ is a compact subset of, then we can find r (0, 1) such that supp σ is contained in the interior of K r, and also K r. Hence the argument from the previous paragraph gives ζ such that u(z) = g (z, ζ) for all z K r. We thus obtain that u(z) dµ K r (z) = g (z, ζ) dµ K r (z) = U µ Kr (ζ) = V K r by (1.2) and the regularity of K r. On the other hand, we have u(z) dµ K r (z) = U σ (z) dµ K r (z) = U µ Kr (z) dσ (z) = VK r σ(k r ) < VK r. This contradiction implies that supp σ must have limit points on. We need the following analog of Bernstein-Walsh estimate for polynomials (cf. Theorem of [12, p. 156]).

8 Page 8 of 15 IOR. PRITSKR Lemma 3.1. For any unit Borel measure µ with compact support in, we have U µ (z) U µ U µ (z) V, z. Proof. Consider the function h(z) := U µ (z) U µ (z), which is superharmonic in \. Superharmonicity of U µ (z) and (1.2) imply that lim h(z) U µ (ζ) V z ζ We also have for the boundary values of h(z) on that lim h(z) = 0 z ζ U µ V, ζ. for q.e. ζ by Theorem 5.1(iv) [13, p. 124]. Recall from (1.3) that U µ V. Hence the Minimum Principle for superharmonic functions gives that h(z) U µ V for all z \. But the same inequality also holds on by (1.2). We shall also use points minimizing discrete reen energy, which are usually called Fekete points. More precisely, an n-tuple of points {ξ k,n } n is called the nth Fekete points of if g (z j, z k ) = g (ξ j,n, ξ k,n ), {z k } n where the imum is taken over all n-tuples {z k } n. The following facts are well known. Lemma 3.2. Let {ξ k,n } n be the nth Fekete points of a compact set. Then the discrete energies of Fekete points are monotonically increasing to V : 2 g (ξ j,n, ξ k,n ) V as n, (3.1) n(n 1) and lim 1 z n n g (z, ξ k,n ) = V. (3.2) Furthermore, if cap() > 0 then the normalized counting measures in Fekete points converge to the reen equilibrium measure in the weak* topology: τ n := 1 n δ ξk,n µ n as n. (3.3) Proof. The result of (3.1) is standard, and can be found in many books, see [12, p. 153] for logarithmic potentials, and see [6, p. 160] for Riesz potentials. A very general result of this kind that covers reen potentials was obtained in [3]. Since we do not have precise references for (3.2) and (3.3), we sketch their proofs here. Note that τ n is a unit measure, so that (1.3) gives 1 z n n g (z, ξ k,n ) = z U τn (z) V.

9 RN POTNTIALS AND BLASCHK PRODUCTS Page 9 of 15 On the other hand, we have for the (n + 1)-tuple (z, ξ 1,n,..., ξ n,n ) that +1 g (ξ j,n+1, ξ k,n+1 ) n g (z, ξ k,n ) + g (ξ j,n, ξ k,n ) by the extremal property of Fekete points. Hence monotonicity in (3.1) yields that n n(n + 1) g (z, ξ k,n ) g (ξ j,n+1, ξ k,n+1 ) g (ξ j,n, ξ k,n ) n(n + 1) = n(n + 1) n(n 1) 2 (n 1) +1 which immediately implies that V 1 n 2 g (z, ξ k,n ) z n n(n 1) g (ξ j,n, ξ k,n ) g (ξ j,n, ξ k,n ), g (ξ j,n, ξ k,n ) g (ξ j,n, ξ k,n ) V as n. Thus (3.2) is proved. Since τ n, n N, is a sequence of positive unit Borel measures supported on, we can use weak* compactness and assume that τ n τ as n along a subsequence N N. It is clear that τ is a positive unit Borel measure supported on, and that τ n τ n converges to τ τ in the weak* topology along the same subsequence. Let K M (z, ζ) := min (g (z, ζ), M). Then K M (z, ζ) is a continuous function in z and ζ, and K M (z, ζ) increases to g (z, ζ) as M. Using the Monotone Convergence Theorem, we obtain for the reen energy of τ that ( ) g (z, ζ) dτ(z) dτ(ζ) = lim lim K M (z, ζ) dτ n (z) dτ n (ζ) M N lim lim 2 M N n 2 K M (ξ j,n, ξ k,n ) + M n lim 2 lim M N n 2 g (ξ j,n, ξ k,n ) = V, where we used (3.1) on the last step. Since the reen energy of a probability measure supported on attains its minimum V only for the equilibrium measure µ, see Theorem 5.10 of [13, p. 131], we obtain that τ = µ. The latter argument holds for any subsequence N N, which means that (3.3) is also proved. Proof of Theorem 1.2. Since reen potentials are superharmonic in, each potential U ν k attains its imum on at a point c k : U ν k = U ν k (c k), k = 1,..., m. Let ν := m ν k, so that ν is a unit measure with potential U(z) ν = U ν k (z).

10 Page 10 of 15 IOR. PRITSKR Using Theorem 1.1 and Fubini s theorem, we obtain that U ν k = m U ν k (c k) = g (c k, ζ) dν k (ζ) g (z, ζ) dν(ζ) z = g (z, ζ) dσ(z) dν(ζ) = U(z) ν dσ(z). We now apply Lemma 3.1 to estimate U ν in, which gives ( ) U ν k U ν + U µ (z) V dσ(z) ( ) = σ() U ν + U µ (z) V dσ(z). The latter inequality proves (1.6) with constant C given in (1.8). We bring C to the form of (1.7) by using Fubini s theorem and Theorem 1.1 again: ( ) U µ (z) V dσ(z) = U σ (ζ) dµ (ζ) σ() V = g (z, ζ) dµ (ζ) V σ(). z It remains to prove the sharpness of C. We use the nth Fekete points {ξ k,n} n for this purpose. Let so that ν k,n = 1 n δ ξ k,n, n ν k,n = τ n is a positive unit measure supported on. The left hand side of (1.6) may be written as follows: n U ν k,n = 1 n n g (z, ξ k,n ) = g (z, ζ) dτ n (ζ). z z Since the function log d (ζ) = z g (z, ζ) is continuous in, we obtain from (3.3) that n lim U ν k,n = g (z, ζ) dµ (ζ). z On the other hand, we have for the right hand side of (1.6) that lim U τn = lim 1 n g (z, ξ k,n ) = V, z n by (3.2). Hence (1.6) turns into equality when we pass to the limit as n. Proof of Corollary 1.3. Hence we have Recall that g D (z, ζ) = log 1 ζz, z, ζ D. z ζ ζ D r g D (z, ζ) = log 1 + r z z + r, z D,

11 RN POTNTIALS AND BLASCHK PRODUCTS Page 11 of 15 because the above imum is attained at ζ m = rz/ z. Theorem 1.22 of [6, p. 104] and Theorem 1.1 now give that dσd D r (z) = 1 ( 2π log 1 + r z ) dxdy, z = x + iy D. z + r Computing Laplacian, we obtain that dσd D r (z) = 1 ( ) r 2π z ( z + r) 2 r z (r z + 1) 2 dxdy r(1 r 2 )(1 z 2 ) = 2π z (r z + 1) 2 dxdy, z = x + iy D. (r + z ) 2 lementary integration using polar coordinates implies that σ D D r (D R ) = R(1 r 2 ) (rr + 1)(r + R), 0 R < 1, and (1.10) follows by letting R 1. It is known [13, p. 133] that V D D r = log r, and dµ (z) = dθ/(2π), z = re iθ. Hence we immediately obtain (1.11) from (1.7). Proof of Theorem 2.1. We define the measures and observe that Suppose that the Blaschke products B k have the following form B k (z) = e iθ k ν k := 1 n n k n k z z j,k, k = 1,..., m. 1 z j,k z δ zj,k, k = 1,..., m, U ν k (z) = 1 n log B k(z), k = 1,..., m. Note that the total mass of m ν k is equal to 1, so that we can apply Theorem 1.2 here. Since U ν k = 1 n log B k, k = 1,..., m, (2.1) is a direct consequence of (1.6). Using the explicit form of reen function g D (z, ζ) = log 1 ζz, z, ζ D, z ζ we obtain (2.2) from (1.7). In order to prove that C D cannot be replaced by a smaller constant, one should simply repeat the corresponding part of proof of Theorem 1.2. Proof of Corollary This result is an easy combination of Theorem 2.1 and Corollary We need the following version of Lower nvelope Theorem for reen s potentials in the sequel. Lemma 3.3. Let µ n, n N, be a sequence of positive unit Borel measures that are supported in a fixed compact subset of a bounded domain. If µ n µ then for quasi every z. lim U µn (z) = U µ (z)

12 Page 12 of 15 IOR. PRITSKR Proof. It is known that the reen potential of any measure ν supported on can be expressed via the regular logarithmic potentials of ν and its balayage ˆν from onto : U ν (z) = U ν (z) U ˆν (z), z, see Theorem 5.1 in [13, p. 124]. The properties of balayage immediately show that µ n µ implies ˆµ n ˆµ. Since supp ˆµn and supp ˆµ, we have that lim U ˆµn (z) = U ˆµ (z), z. Thus the result of this lemma follows from the Lower nvelope Theorem for logarithmic potentials (see Theorem 6.9 in [13, p. 73]), stating that for quasi every z C. lim U µn (z) = U µ (z) Proof of Theorem 2.3. Let z j,l, j = 1,..., n l, be all zeros of the product m l B k listed according to multiplicities. Define the individual Blaschke factors We first note that (2.4) implies Indeed, we have that m l lim n l b j,l (z) = z z j,l 1 z j,l z, j = 1,..., n l. n l by Theorem 2.1. On the other hand, n l nl b j,l b j,l σd (D) 1/n l n l n l B k,l b j,l e CD b j,l = e CD. (3.4) m l = B k,l, so that (3.4) follows. Passing to a subsequence, we can assume that 1/n n l l lim n l b j,l = C b j,l σ D (D) and τ nl τ as nl hold simultaneously. It is clear that τ is a unit measure with support in D. Since log d D (z) is continuous, we also have that lim log n l b j,l 1/nl = log d D (z) dτ(z) = g D(z, ζ) dτ(z). n l ζ Hence (3.4) gives that ζ g D(z, ζ) dτ(z) + σ D (D) log C = C D.

13 RN POTNTIALS AND BLASCHK PRODUCTS Page 13 of 15 Using (1.4) and Fubini s theorem, we obtain that the left hand side of the above equation may be written as gd(z, ζ) dσ D (ζ) dτ(z) + σ D (D) log C = (U(ζ) τ + log C) dσ D (ζ). Transforming the representation (2.2) for C D is a similar way, we have that ( ) (U(ζ) τ + log C) dσ D (ζ) = U µd (ζ) V D dσ D (ζ). (3.5) On the other hand, Lemma 3.1 implies that Note that U τn l (z) U τn l U τn l U µd (z) V D, z D. = log n l b j,l 1/n l Passing to lim as n l and using Lemma 3.3, we obtain that. U τ (z) + log C U µd (z) V D (3.6) holds q.e. in D. The Principle of Domination (cf. Theorem 5.8 in [13, p.130]) now implies that (3.6) holds for all z D. Furthermore, strict inequality in (3.6) for any z supp σ D would violate (3.5). Hence U τ (z) + log C = U µd (z) V D, z supp σ D. (3.7) Since supp σ D has points of accumulation on D by Theorem 1.1, and since the limiting values of reen potentials on D are zero, we obtain by letting z 1 in (3.7) that log C = V D. Thus (2.5) is proved, and (3.6)-(3.7) take the following form: U τ (z) U µd (z), z D, and U τ (z) = U µd (z), z supp σd. (3.8) Let Ω be the connected component of D \ whose boundary contains the unit circumference. The function u(z) := U τ (z) U µd (z) is superharmonic in Ω, because U µd (z) is harmonic in D \. We have that u(z) 0, z D, and u(z) = 0, z supp σ D, by (3.8). It follows that u attains its minimum in Ω, as supp σ D Ω by Theorem 1.1, so that u(z) = 0, z Ω, by the Minimum Principle. We conclude that U τ (z) = U µd (z), z Ω, and that this equality also holds on Ω by the continuity of potentials in the fine topology, see Corollary 5.6 in [13, p. 61]. If satisfies conditions of Theorem 2.3(i), then Ω = and U(z) τ = U µd (z), z D, because D = Ω. Consequently, supp τ supp µ. Integrating the above equation with respect to τ, we obtain by (1.2) that g D (z, ζ) dτ(ζ) dτ(z) = U(z) τ dτ(z) V D, which means that the reen energy of τ attains the smallest possible value among all positive unit Borel measures supported on. Hence τ = µ D by the uniqueness of reen equilibrium measure, see Theorems 5.10 and 5.11 in [13, pp ].

14 Page 14 of 15 IOR. PRITSKR To prove that τ = µ D when = H, where H is a Jordan domain, we follow the same argument to show that u(z) = 0, z H, because u is a superharmonic function that attains its minimum in H for z supp σ D H, cf. Theorem 1.1. We also use here that supp µd = = Ω = H. Consequently, U τ (z) = U µd (z) in H and everywhere in D = Ω H. Using the same energy argument as before, we obtain that τ = µ D under the assumption of Theorem 2.3(ii). Proof of Theorem 2.4. We use (1.4) again to find that log B k,n = ζ g D(z k,n, ζ) = log d D (z k,n ), k = 1,..., n. Since log d D is continuous and τ n ( lim log n µ D, we obtain that ) 1/n B k,n = lim log d D (z) dτ n (z) = On the other hand, we have that n 1/n log B k,n = U τn. g D(z, ζ) dµ D (z). (3.9) ζ Using lower semicontinuity of U τn, we conclude that the imum is attained at a point z n. We can assume that lim z n = z 0, after passing to a subsequence. The Principle of Descent (see Theorem 6.8 of [13, p. 70] and Theorem 1.3 of [6, p. 62]) implies that lim U τn (z n) U µd (z 0) = V D, where the last equality holds by (1.2) and regularity of. Hence n 1/n lim sup log B k,n V D, and we obtain from (3.9) that But lim ( n B k,n ) 1/n e CD n B k,n σd (D). lim sup by (2.1), so that (2.6) follows. ) 1/n e CD n B k,n σd (D) ( n B k,n References 1. D. H. Armitage and S. J. ardiner, Classical Potential Theory (Springer-Verlag, New York, 2001). 2. D. W. Boyd, Sharp inequalities for the product of polynomials, Bull. London Math. Soc. 26 (1994), Choquet, Diamètre transfini et comparaison de diverses capacités, Technical report, Faculté des Sciences de Paris (1958). 4. S. J. ardiner and I. Netuka, Potential theory of the farthest-point distance function, J. Anal. Math. 101 (2006), A. Kroó and I.. Pritsker, A sharp version of Mahler s inequality for products of polynomials, Bull. London Math. Soc. 31 (1999), N. S. Landkof, Foundations of Modern Potential Theory (Springer-Verlag, New York-Heidelberg, 1972).

15 RN POTNTIALS AND BLASCHK PRODUCTS Page 15 of R. S. Laugesen and I.. Pritsker, Potential theory of the farthest-point distance function, Can. Math. Bull. 46 (2003), I.. Pritsker, Products of polynomials in uniform norms, Trans. Amer. Math. Soc. 353 (2001), I.. Pritsker, Norms of products and factors of polynomials, in Number Theory for the Millennium III, M. A. Bennett, B. C. Berndt, N. Boston, H. Diamond, A. J. Hildebrand and W. Philipp (eds.), pp , A K Peters, Ltd., Natick, I.. Pritsker and S. Ruscheweyh, Inequalities for products of polynomials I, Math. Scand. 104 (2009), I.. Pritsker and. B. Saff, Reverse triangle inequalities for potentials, J. Approx. Theory 159 (2009), T. Ransford, Potential Theory in the Complex Plane (Cambridge University Press, Cambridge, 1995) B. Saff and V. Totik, Logarithmic Potentials with xternal Fields (Springer-Verlag, Berlin, 1997). Department of Mathematics Oklahoma State University Stillwater, OK U.S.A. igor@math.okstate.edu