Abstract
We study chordal Loewner families in the upper half-plane and show that they have a parametric representation. We show one, that to every chordal Loewner family there corresponds a unique measurable family of probability measures on the real line, and two, that to every measurable family of probability measures on the real line there corresponds a unique chordal Loewner family. In both cases the correspondence is being given by solving the chordal Loewner equation. We use this to show that any p
Results & Lemmas (23)
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Theorem 2.1
Theorem 2.1 (Nevanlinna Representation, [6]). Every analytic func- tion F such that ℑ(F(z)) ≥0 for z ∈H has a representation F(z) = b + cz…
Theorem 2.1 (Nevanlinna Representation, [6]). Every analytic func- tion F such that ℑ(F(z)) ≥0 for z ∈H has a representation F(z) = b + cz + Z R 1 + tz t −z ν(dt), z ∈H, where b = ¯b, c ≥0 and ν is a finite nonnegative Borel measure on R. The triple (b, c, ν) is unique and satisfies b = ℜ(F(i)), c = lim 0<y→∞ ℑ(F(iy)) y
Theorem 2.2.
Theorem 2.2. [10] For an analytic function F: H →H the following are equivalent: (i) F is the reciprocal Cauchy transform of a probability…
Theorem 2.2. [10] For an analytic function F : H →H the following are equivalent: (i) F is the reciprocal Cauchy transform of a probability measure µ on R. (ii) There exist a real number b ∈R and a finite nonnegative Borel mea- sure ν on R such that F(z) = b + z + Z R 1 + tz t −z ν(dt), z ∈H. (iii) F satisfies equation 2. For probability measures with finite variance and zero mean this result can be specified to
Proposition 2.1.
Proposition 2.1. [10] For an analytic function F: H →H the following are equivalent: (i) F is the reciprocal Cauchy transform of a…
Proposition 2.1. [10] For an analytic function F : H →H the following are equivalent: (i) F is the reciprocal Cauchy transform of a probability measure on R with finite variance and mean zero. (ii) There exists a finite positive measure ρ on R such that for all z ∈H, F(z) = z − Z R ρ(dx) z −x. 5
Proposition 3.1
Proposition 3.1 ([4]). Let µ be a probability measure on R, and let 0 < ǫ < α. There exists a β > 0 such that (i) F = 1/G is univalent in…
Proposition 3.1 ([4]). Let µ be a probability measure on R, and let 0 < ǫ < α. There exists a β > 0 such that (i) F = 1/G is univalent in Γα,β, and (ii) F(Γα,β) ⊃Γα−ǫ,β(1+ǫ). For completeness we reproduce the proof of [4].
Proposition 3.2.
Proposition 3.2. Let µ be a probability measure on R with finite variance σ2 and reciprocal Cauchy transform F. Then the restriction of F to…
Proposition 3.2. Let µ be a probability measure on R with finite variance σ2 and reciprocal Cauchy transform F. Then the restriction of F to Hσ takes every value in H2σ precisely once.
Theorem 3.1.
Theorem 3.1. Suppose that µ is a probability measure on the real line with variance σ2 and mean zero. The reciprocal Cauchy transform F of…
Theorem 3.1. Suppose that µ is a probability measure on the real line with variance σ2 and mean zero. The reciprocal Cauchy transform F of µ is univalent in H if and only if there is a chordal Loewner family {f(t; ·), t ∈ [0, ∞)} such that F(z) = f(σ2; z), z ∈H. Using the relation between chordal Loewner families and the chordal Loewner equation that we develop in Section 5 we get
Theorem 3.2.
Theorem 3.2. Suppose that µ is a probability measure on the real line with variance σ2 and mean zero. The reciprocal Cauchy transform F of…
Theorem 3.2. Suppose that µ is a probability measure on the real line with variance σ2 and mean zero. The reciprocal Cauchy transform F of µ is univalent in H if and only if there is a measurable family {µt, t ∈[0, ∞)} of probability measures on R such that if we define the family {f(t; z), t ∈ [0, ∞)} as the unique solution to the initial value problem ∂ ∂tf(t; z) = − Z R µt(dx) z −x · ∂ ∂z f(t; z), f(0; z) = z, then F(z) = f(σ2; z), z ∈H.
Theorem 3.3.
Theorem 3.3. Suppose that f is meromorphic in a domain D whose comple- ment E is compact and that f maps D into a domain D′ whose…
Theorem 3.3. Suppose that f is meromorphic in a domain D whose comple- ment E is compact and that f maps D into a domain D′ whose complement is E′. Further suppose that f ′(∞) = 1 which means that f(z) = z + a0 + a1 z + · · · for large z. 7
Corollary 3.1.
Corollary 3.1. With the notation from above, µ has Cauchy transform uni- valent in the upper half-plane if and only if the transfinite…
Corollary 3.1. With the notation from above, µ has Cauchy transform uni- valent in the upper half-plane if and only if the transfinite diameter of the complement of F(C\[Aµ, Bµ]) equals Bµ −Aµ. Next we consider the characterization by moments. To simplify notation we will assume that the support of µ is contained in the interval [−2, 2]. Let an = Z R xn µ(dx), n = 0, 1, 2, . . ., and note that G(z) = ∞ X n=0
Theorem 3.4.
Theorem 3.4. Suppose that µ is a probability measure on R such that its support is contained in [−2, 2]. Then the reciprocal Cauchy…
Theorem 3.4. Suppose that µ is a probability measure on R such that its support is contained in [−2, 2]. Then the reciprocal Cauchy transform F of µ is univalent in H if and only if for each N ∈Z+ the real symmetric matrix [cnk]N n,k=1 has all its eigenvalues in [−1, 1].
Lemma 4.1.
Lemma 4.1. If f, g ∈R, then f ≺g if and only if f(H) ⊆g(H). In this case lim y→∞iy [iy −f(iy)] ≤lim y→∞iy [iy −g(iy)] (4) with equality if…
Lemma 4.1. If f, g ∈R, then f ≺g if and only if f(H) ⊆g(H). In this case lim y→∞iy [iy −f(iy)] ≤lim y→∞iy [iy −g(iy)] (4) with equality if and only if f ≡g. 11
Theorem 4.1.
Theorem 4.1. Every f ∈R belongs to some chordal Loewner family. More generally, every chain C in R is contained in a chordal Loewner family.
Theorem 4.1. Every f ∈R belongs to some chordal Loewner family. More generally, every chain C in R is contained in a chordal Loewner family.
Theorem 4.2.
Theorem 4.2. Let fn ∞ n=1 be a sequence in R and for every n ∈Z+ let an = limy→∞iy[iy −fn(iy)]. If a = limn→∞an exists and is finite, then…
Theorem 4.2. Let {fn}∞ n=1 be a sequence in R and for every n ∈Z+ let an = limy→∞iy[iy −fn(iy)]. If a = limn→∞an exists and is finite, then there exists a function f ∈R and a subsequence n1, n2, . . . , such that for every m ∈Z+ sup z∈H1/m |f(z) −fnk(z)| →0, as k →∞. (10) Furthermore, if Ao denotes the interior of a set A, then f(H) = ∞ [ m=1
Lemma 4.2.
Lemma 4.2. Let f, f1, f2,... and g, g1, g2,... belong to R. For each n ∈Z+ let an = limy→∞iy[iy −fn(iy)], bn = limy→∞iy[iy −gn(iy)]. Assume…
Lemma 4.2. Let f, f1, f2, . . . and g, g1, g2, . . . belong to R. For each n ∈Z+ let an = limy→∞iy[iy −fn(iy)], bn = limy→∞iy[iy −gn(iy)]. Assume that supn an < ∞, supn bn < ∞, and fn →f, gn →g, uniformly on compact subsets of H, as n →∞. If fn ≺gn for each n ∈Z+, then f ≺g.
Lemma 4.3.
Lemma 4.3. Let γ: [0, 1) →H be a Jordan arc such that γ(0) ∈∂H and lim tր1 ℑ(γ(t)) = ∞. For each t ∈[0, 1) let f(t; ·) ∈R be the unique…
Lemma 4.3. Let γ : [0, 1) →H be a Jordan arc such that γ(0) ∈∂H and lim tր1 ℑ(γ(t)) = ∞. For each t ∈[0, 1) let f(t; ·) ∈R be the unique function whose range is the complement of γ([0, t]) in H, and set a(t) = limy→∞iy[iy −f(t; iy)]. Then t ∈[0, 1) 7→a(t) ∈[0, ∞) is nondecreasing, continuous, and onto. 16
Lemma 4.4.
Lemma 4.4. Let f ≺g where f, g ∈R, a = limy→∞iy[iy −f(iy)], b = limy→∞iy[iy −g(iy)], and let c be a positive number. (i) If c < b, there is…
Lemma 4.4. Let f ≺g where f, g ∈R, a = limy→∞iy[iy −f(iy)], b = limy→∞iy[iy −g(iy)], and let c be a positive number. (i) If c < b, there is an h ∈R such that lim y→∞iy[iy −h(iy)] = c and g ≺h. (ii) If b < c < a, there is an h ∈R such that lim y→∞iy[iy −h(iy)] = c and f ≺h ≺g. (iii) If a < c, there is an h ∈R such that lim y→∞iy[iy −h(iy)] = c
Theorem 4.3.
Theorem 4.3. If L is any chordal Loewner family, then f ∈L 7→lim y→∞iy[iy −f(iy)] ∈[0, ∞) is one-to-one and onto. Thus the family L has a…
Theorem 4.3. If L is any chordal Loewner family, then f ∈L 7→lim y→∞iy[iy −f(iy)] ∈[0, ∞) is one-to-one and onto. Thus the family L has a parametric representation L = {f(t; ·)}t∈[0,∞), where each f(t; ·) satisfies f(t; iy) = i y + t y + o 1 |y| ,
Theorem 5.1.
Theorem 5.1. Let L be a chordal Loewner family with associated semigroup B(a, b; ·), 0 ≤a ≤b < ∞. Then for all z ∈H, |B(a, c; z) −B(b, c;…
Theorem 5.1. Let L be a chordal Loewner family with associated semigroup {B(a, b; ·), 0 ≤a ≤b < ∞}. Then for all z ∈H, |B(a, c; z) −B(b, c; z)| ≤b −a ℑ(z) , (14) |B(a, b; z) −B(a, c; z)| ≤ 1 + b −a ℑ(z)2 c −b ℑ(z), (15) whenever 0 ≤a ≤b ≤c < ∞. Thus, for each z ∈H, (i) the function t ∈[0, ∞) 7→f(t; z) ∈H is absolutely continuous, (ii) if b > 0, a ∈[0, b] 7→B(a, b; z) ∈H is absolutely continuous,
Theorem 5.2.
Theorem 5.2. Assume the same situation as in Theorem 5.1. (i) There is a subset N of [0, ∞) of Lebesgue measure zero such that if t ∈[0,…
Theorem 5.2. Assume the same situation as in Theorem 5.1. (i) There is a subset N of [0, ∞) of Lebesgue measure zero such that if t ∈[0, ∞)\N, then ∂ ∂tf(t; z) = lim h→0 f(t + h; z) −f(t; z) h 19
Theorem 5.3.
Theorem 5.3. If f(t; ·), t ∈[0, ∞) is any chordal Loewner family, then there is a unique measurable family µt, t ∈[0, ∞) of probability…
Theorem 5.3. If {f(t; ·), t ∈[0, ∞)} is any chordal Loewner family, then there is a unique measurable family {µt, t ∈[0, ∞)} of probability measures on the real line and a subset N ⊂[0, ∞) of Lebesgue measure zero such that ∂ ∂tf(t; z) = − Z R µt(dx) z −x · ∂ ∂z f(t; z) (18) for all t ∈[0, ∞)\N, and z ∈H.
Theorem 5.4.
Theorem 5.4. In Theorem 5.3, let B(a, b; ·), 0 ≤a ≤b < ∞ be the semigroup associated to the chordal Loewner family. (i) On [a, ∞), ∂ ∂sB(a,…
Theorem 5.4. In Theorem 5.3, let {B(a, b; ·), 0 ≤a ≤b < ∞} be the semigroup associated to the chordal Loewner family. (i) On [a, ∞), ∂ ∂sB(a, s; z) = − Z R µs(dx) z −x · ∂ ∂z B(a, s; z), B(a, a; z) = z. (ii) If b > 0, then on [0, b] ∂ ∂tB(t, b; z) = Z
Theorem 5.5.
Theorem 5.5. Let µt, t ∈[0, ∞) be a measurable family of probabil- ity measures on the real line. There exists a unique family of functions…
Theorem 5.5. Let {µt, t ∈[0, ∞)} be a measurable family of probabil- ity measures on the real line. There exists a unique family of functions {B(a, b; ·), 0 ≤a ≤b < ∞} with these properties: (i) For fixed a, b, B(a, b; ·) is in R, limy→∞iy[iy −B(a, b; iy)] = b −a, and B(a, c; z) = B(a, b; B(b, c; z)) (20) whenever 0 ≤a ≤b ≤c. (ii) For fixed b > 0 and z ∈H, a ∈[0, b] 7→B(a, b; z) ∈H is absolutely continuous such that ∂ ∂aB(a, b; z) = Z R
Theorem 5.6.
Theorem 5.6. If µt, t ∈[0, ∞) is any measurable family of probability measures on the real line, then there exists a unique chordal Loewner…
Theorem 5.6. If {µt, t ∈[0, ∞)} is any measurable family of probability measures on the real line, then there exists a unique chordal Loewner family {f(t; ·), t ∈[0, ∞)} such that (26) holds.