P. L. Duren, Univalent Functions, Springer-Verlag, 1983, pp. 40---51, 95---98, 202--- 209; MR0708494 (85j:30034). Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
On Support Points Of Univalent Functions And A Disproof Of A.. - Greiner, Roth (2001) (Correct)
.... Gamma sin(m ) sin for all m;n 2; 1.6) in the case n = 2 and m = 3, because for n = 2 and m = 3 the right hand side of (1.6) equals 1=4 which is strictly larger than the bound (e Gamma 1) 4e) in Corollary 1.2. 2. Proof of Theorem 1. 1 The standard method of boundary variation (cf. SSp50] [Dur83]) shows that every support point F (z) z A 2 z 2 A 3 z 3 Delta Delta Delta of the functional L , 2 C fixed, is a solution of the Schiffer differential equation zF 0 (z) F (z) 2 1 AF (z) F (z) 2 = 1 z 2 A z B 0 Az z 2 ; 2.1) ON SUPPORT POINTS OF ....
P. L. Duren, Univalent Functions, Springer-Verlag, 1983. MR 85j:30034
Divergence Groups Have The Bowen Property - Bishop (1999) (Correct)
....tangents almost nowhere. Before beginning the proof we recall that the Schwarzian derivative of a locally univalent function on the disk is de ned by S(f) f 00 =f 0 ) 0 1 2 (f 00 =f 0 ) 2 : We shall need only two well known facts about the Schwarzian (e.g. see Section 8. 5 of [14] or Section 1.2 of [24] First, that it vanishes identically i f is a M obius transformation and second that its composition law S(f g) z) S(f) g(z) g 0 (z) 2 S(g) z) DIVERGENCE GROUPS HAVE THE BOWEN PROPERTY 7 implies that if f is a deformation of a Fuchsian group then the ....
P. Duren. Univalent functions. Springer-Verlag, 1983.
Critical exponents, conformal invariance and planar Brownian motion - Werner (Correct)
....T ) where T is the first time at which Z hits [M; 1] 5. Schramm s processes 5.1. In the plane It is possible to contruct a CCI measure P using Loewner s differential equation (that encodes a certain class of growing families of compact sets) driven by a Brownian motion. See, for instance, [17] for a general introduction to Loewner s equation. For any simply connected compact K such that the complement of K in the plane is conformally equivalent to the complement of a disc, Riemann s mapping theorem shows that there exists a unique ff K 2 R and a unique conformal map fK that maps the ....
P. Duren, Univalent functions, Springer, 1983.
Univalent Functions of Fast Growth With Gap Power Series - Eremenko, Hayman (Correct)
....point of a coe#cient region. 1991 Mathematics Subject Classification: 30C50, 30C70. We consider the class S of normalized univalent functions f(z) z a 2 z 2 . in the unit disc U. It is known [2, p. 15] that, when f # S, there always exists a limit #(f) lim n## a n n # [0, 1]. Thus if there is an infinite subsequence of zero coe#cients, we have #(f) 0. We will show in this paper that the last conclusion is best possible: the sequence (a n ) may have infinitely many long gaps and simultaneously long intervals where a n n# n . Here (# n ) is any prescribed ....
P. L. Duren, Univalent Functions, Springer, NY, 1983.
Compact Deformations of Fuchsian Groups - Bishop, Jones (2000) (Correct)
.... es a Lavrentiev type estimate (just as recti able curves do) See [12] Because of the composition properties of the Schwarzian (see Section 4) it turns out that if a deformation Fuchsian group G, then the hyperbolic Schwarzian SH ( jS( z) j 2 (1 jzj 2 ) 2 ; is group invariant (e.g. [18], 32] Thus we can rewrite I 3 as Z D SH ( z) 2 log 1 jzj dA; COMPACT DEFORMATIONS 5 where dA denotes hyperbolic area. Since log 1 jzj is the Greens function for the disk, this integral is just the same as Z R SH ( z) 2 GR (z; w)dA; 1) where R = D =G is the quotient Riemann ....
....CHRISTOPHER J. BISHOP AND PETER W. JONES If we write F 0 = e , then this can be rewritten as S(F ) z) 00 1 2 ( 0 ) 2 : 5) Recall that S(F ) 0 i F is a M obius transformation and that S satis es the composition law S(F G) S(F ) G 0 ) 2 S(G) e.g. see Section 8. 5 of [18] or Section 1.2 of [32] In particular, if G is M obius then S(F G) S(F ) G 0 ) 2 and S(G F ) S(F ) In addition, given an 0, hyperbolic disk D and a compact neighborhood K of D, there is a 0 so that jS(F )j on D implies F uniformly approximates a M obius transformation on K ....
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P. Duren. Univalent functions. Springer-Verlag, 1983.
A Survey Of Applications Of The Julia Variation - Barnard, Pearce, Campbell (Correct)
....Re L(f) max f#Sm Re L(f)forn # m. Let n be a fixed integer greater than m. Consider any function f in Sn where #f(D) has strictly more than m sides . By the assumption that #f(D) is a (bounded) Jordan curve (where f(D) is a Jordan domain) we have, by applying Caratheodory s Extension Theorem [19], that f has a continuous extension to D = D # #. Let# = f(D)and#=f(#) # Then # has m 1 or more sides. Call these sides, # 1 , # 2 , and call the preimages # 1 ,# 2 , wheref(# j ) # j . We will locally vary # by moving one side in the following manner (see Figure 1) ....
....circular symmetrization. However, Steiner symmetrization (cf. Hayman [25] can still be used in certain cases such as sectors. Another di#culty is the introduction of distinctly di#erent extremal domains for different ranges of r. Since every function in K covers a disk of radius 1 2 (cf. Duren [19]) r needs only to be considered in the interval (1 2, 1) Waniurski has obtained some partial results in [43] He defined r 1 and r 2 to be the unique solutions to certain transcendental equations where r 1 # 0.594 and r 2 # 0.673. If F # 2 is the map of D onto the half plane w :Rew 1 2 ....
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Duren, P. Univalent Functions. Spring-Verlag, Vol. 259, New York, (1980).
A Coefficient Problem for Univalent Functions Related to.. - Greiner, Roth (2000) (Correct)
....d dt a 3 (t) 4e t e iu(t) a 2 (t) 2e 2t e 2iu(t) a 3 (0) 0; 4) and denote the corresponding solution by a 2 (t; u( a 3 (t; u( then the socalled entire reachable set R : f(a 2 (1; u( a 3 (1; u( u : 0; 1) R measurableg of (4) is dense in V 3 . See for instance [3], Chapter 3. Fix p 2 R and let F (z) z A 2 z 2 A 3 z 3 2 S be an extremal function for the functional J p in S, i.e. max f2S J p (f) J p (F ) 5) Then C n F (D ) consists of one or two piecewise analytic Jordan arcs (cf. 3] p. 304) and it follows from Loewner s theory ....
....of (4) is dense in V 3 . See for instance [3] Chapter 3. Fix p 2 R and let F (z) z A 2 z 2 A 3 z 3 2 S be an extremal function for the functional J p in S, i.e. max f2S J p (f) J p (F ) 5) Then C n F (D ) consists of one or two piecewise analytic Jordan arcs (cf. [3], p. 304) and it follows from Loewner s theory that there exists a function u 0 : 0; 1) R with at most one point of discontinuity (which will be a discontinuity of rst kind) such that A 2 = a 2 (1; u 0 ( and A 3 = a 3 (1; u 0 ( In particular, max (a 2 ;a 3 )2R J p (a 2 ; a 3 ) J p ....
[Article contains additional citation context not shown here]
Duren, P. L., Univalent Functions, Springer-Verlag, 1983.
Pontryagin's Maximum Principle in Geometric Function Theory - Roth (Correct)
....degree n. Thus the function G(i) H(t; i z i Gammaz ) is a rational function of degree n (without poles on jij = 1) We first show that i 7 ReG(i) is not constant on jij = 1, i.e. h 7 ReH(t; h(z) is not constant on extr P , the set of extreme points of P . By the Toeplitz representation [5] for linear functionals on H(D ) H(t; h(z) P 1 n=0 dn (t)a n for all h(z) P 1 n=0 an z n 2 H(D ) where fdn (t)g n2N is some sequence of complex numbers. Now observe that the function z 7 1 2xz n belongs to P for all jxj 1 and all n 2 N. Therefore, if h 7 ReH(t; h(z) is constant ....
Duren, P. L., Univalent Functions, Springer (1983).
The Scenery Flow For Hyperbolic Julia Sets - Bedford, Fisher, Urbanski (1999) (1 citation) (Correct)
....and we are done. For the second proof we need Lemmas 2.1, 2.10 above plus the following lemma. A proof follows immediately from the two facts that jDf j is close to 1 (Koebe s Distortion Theorem) and that arg(DF ) is close to zero (Bieberbach s Rotation Theorem) See e.g. 3) and (6) of x2.3 in [Du]. Lemma 2.12. Assume f : B 1 (0) CI is holomorphic and 1 Gamma 1, and let it satisfy f(0) 0 and Df(0) 1. Then given 0, 9r 0 such that (for all such f) jDf(z) Gamma 1j for all jzj r: Second Proof of Theorem 2.11. We fix ffi less than the delta in Lemma 2.7. As in the ....
P. L. Duren, Univalent functions, Springer-Verlag 1983.
On the Approximation of Univalent Functions by Subordinate.. - Greiner, Ruscheweyh (Correct)
....in square brackets equals 1 2(1 Gamma cos OE) Gamma cos(n 1)OE 2(1 Gamma cos OE) 0: Comparing coefficients and using i n ff Gamma2 n j 0, ff 1, we deduce Re i oe ff n (e iOE ) Gamma 1 2 j 0. The next lemma is the well known former Wilf s conjecture ( 10] p. 86 or [5], p.254) Lemma 2 Let f; h be convex univalent functions in D, and let g OE h be holomorphic in D. Then f g OE f h: The first assertion f oe ff n OE f = f h of Theorem 2 follows readily from Lemma 1 and Lemma 2. Now we provide a technical lemma which, as a matter of fact, already implies ....
P. L. Duren, Univalent Functions, Springer-Verlag, New York, Berlin, 1983.
Scaling Limits of Loop-Erased Random Walks and Uniform Spanning.. - Schramm (1999) (5 citations) (Correct)
....y has law 0 . This is obtained by taking an appropriate P = 2 M( 0 ) Omega : X Theta X, and letting x and y be the projections on the first and second factors, respectively. Conformal maps. We review some elementary facts about conformal (aka univalent) mappings, as may be found in [Dur83], for example. Let D ae C be some domain. A continuous map f : D C , which is injective and complex differentiable is conformal. If f is conformal, then f Gamma1 : f(D) C is also conformal. Let D C be simply connected. Then Riemann s Mapping Theorem states that there is a conformal ....
P. L. Duren. Univalent functions. Springer-Verlag, New York, 1983.
On the Taylor coefficients for powers of the derivative of.. - Roth (Correct)
....i.e. has a power series expansion at z = 0: Gamma f 0 (z) Delta p = 1 X n=0 c n;p [f ] z n ; c 0;p [f ] 1: The coefficients c n;p [f ] are polynomials in the Taylor coefficients a 2 ; an 1 of f and in the parameter p. The well known distortion theorem in the class S (cf. [3], p. 32) asserts that for all f 2 S and all p 0 fi fi f 0 (z) fi fi p 1 Gamma jzj (1 jzj) 3 p = Gamma K 0 0 (jzj) Delta p = 1 X k=0 C k;p jzj k ; z 2 D ; 1) where K 0 (z) GammaK ( Gammaz) is a rotation of the Koebe function K(z) z= 1 Gamma z) 2 . From ....
Duren, P. L., Univalent Functions, Springer-Verlag, New York, 1983.
Estimates of Derivatives in alpha-Invariant Set of Analytic.. - Avhadiev, Wirths (Correct)
....functions, and ff is a parameter. Hardy spaces E p( Omega Gamma4 Bergmann spaces A p;fl and 1=p Bloch spaces B p( Omega Gamma will be considered. If Omega is the unit disc D = fz : jzj 1g, then there are several explicit estimates of M n and there is a lot of open problems (see [8] 5] [6], 10] 13] 1] 12] 2] 4] If f(z) is a bounded analytic function in an arbitrary domain Omega Gamma then M 1 is given by principle of hyperbolic metric. To find M n we introduce the functions k = k (a; Omega Gamma = 1 (k Gamma 1) Gammak Omega (a) k log 2 Omega (a) ....
....w n ; U ff ) for any 2 R, v) for w 2 C n , the function u(r) K n (rw 1 ; r 2 w 2 ; r n w n ; U ff ) is nondecreasing on 0 r 1 and log u(r) is convex with respect to log r. 4 Avhadiev Wirths Consider now Hardy, Bergmann and Bloch spaces. We use the following notations [8] [6], 1] kfk p = sup 0 r 1 Z j (z)j=r jf(z)j p jdzj 1=p for f 2 E p( Omega Gamma (p 0) where is the conformal mapping of Omega onto D, a) 0, 0 (a) 0; kfk p;fl = fl Gamma 1 ZZ Omega jf(z)j p fl Gamma2 Omega (z)dx dy 1=p (p 0; fl 1) for f 2 A ....
Duren, P. L.: Univalent Functions. Springer-Verlag, 1983.
Radii in Geometric Function Theory - Finch (Correct)
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P. L. Duren, Univalent Functions, Springer-Verlag, 1983, pp. 40---51, 95---98, 202--- 209; MR0708494 (85j:30034).
Compact Deformations of Fuchsian Groups - Bishop, Jones (1998) (Correct)
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P. Duren. Univalent functions. Springer-Verlag, 1983.
Hardy Spaces That Support - No Compact Composition (Correct)
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P. L. Duren, Univalent Functions, Springer-Verlag 1983.
Level Sets Of Neumann Eigenfunctions - Rodrigo Ba Nuelos (Correct)
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P. Duren, Univalent Functions, Springer Verlag, New York, 1983.
The Scenery Flow For Hyperbolic Julia Sets - Bedford, Fisher, Urbanski (1 citation) (Correct)
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P. L. Duren, Univalent functions, Springer-Verlag 1983.
Majorant Series - Boas (2000) (Correct)
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Peter L. Duren, Univalent functions, Springer-Verlag, 1983.
Meromorphic Functions Whose Julia Sets Contain A Free Jordan Arc - Stallard (1993) (Correct)
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Duren, P.: Univalent functions. - Springer-Verlag, New York, 1953.
The Scenery Flow For Hyperbolic Julia Sets - Bedford, Fisher, Urbanski (2000) (1 citation) (Correct)
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P. L. Duren, Univalent functions, Springer-Verlag 1983.
On a theorem of Nehari and quasidiscs - Chuaqui (1993) (Correct)
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Duren, P.: Univalent functions. - Springer-Verlag, New York, 1983.
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