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| In [[mathematics]], the '''Loewner differential equation''', or '''Loewner equation''', is an [[ordinary differential equation]] discovered by [[Charles Loewner]] in 1923 in [[complex analysis]] and [[geometric function theory]]. Originally introduced for studying slit mappings ([[conformal mapping]]s of the [[open disk]] onto the [[complex plane]] with a curve joining 0 to ∞ removed), Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of [[Carathéodory]] to the whole plane leads to a one parameter family of conformal mappings, called a '''Loewner chain''', as well as a two parameter family of [[holomorphic]] [[univalent function|univalent self-mappings]] of the [[unit disk]], called a '''Loewner semigroup'''. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part. The Loewner semigroup generalizes the notion of a [[Koenigs function#Structure of univalent semigroups|univalent semigroup]].
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| The Loewner differential equation has led to inequalities for univalent functions that played an important role in the solution of the [[Bieberbach conjecture]] by [[Louis de Branges]] in 1985. Loewner himself used his techniques in 1923 for proving the conjecture for the third coefficient. The [[Schramm-Loewner equation]], a stochastic generalization of the Loewner differential equation discovered by [[Oded Schramm]] in the late 1990s, has been extensively developed in [[probability theory]] and [[conformal field theory]].
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| ==Subordinate univalent functions==
| | If you open your web browser or turn television on are generally sure to see teeth whitening products being advertised. These adverts normally boast the player are most effective new whitener that has hit business. The question is, do these products actually work?<br><br>Speaking of local listings, another choice is to watch fireworks on the telly or online. You may want test this in readiness for a live show as part of desensitization (or as theatre folk call it, rehearsal). Rehearsal is the best way to prepare for a considerable event, and also places can be fun, so. Role play a try to a dentist or hair salon several times with your child, and be sure to modify roles here and there!<br><br>If you are not sure how much to feed, you'll love the interactive Dog Food Calculator on PetsMart's webpage! It's the result of considerable research that recently been published in scientific journals and accepted by skillfully developed. You can determine the correct quantity of food to feed your dog and discover how long that 40-lb bag will last!<br><br>Members can opt from over 87,000 access points across the country, and will eventually change dentists at once. There is no paperwork to finish or waiting periods for choosing a new DenteMax dentist. Each comparable can select his or her own dentist.<br><br>Prior to developing a routine for whitening your teeth, rendering it whiteners do not change colour of artificial teeth. Any artificial surfaces, including crowns, bonding, implants and such like dentist will cease altered in color. Getting whiter teeth probably will make artificial surfaces more likely.<br><br>Custom made teeth whitening trays is fit for home teeth whitening, dentist and yes, it is much better than whitening tapes. Actually for the best results, may possibly even need to use custom trys after your a couple of hours whitening date.<br><br>The second step involves sealing the tooth cavity. If there would be a severe infection, your dentist may use medication to eliminate it to be able to sealing. Then, a paste and rubber mixture can be as the sealant inside cavity, and a filling lies on most effective. The final step involves the placing of a crown defend and restore the teeth.<br><br>When you adored this information and also you wish to get more information relating to [https://www.youtube.com/watch?v=4To0cBm6IsM Dentists in lakeland fl] kindly visit the webpage. Be bound to select a dental clinic and get excellent treatment frequently at a fantastic price. It's a wise idea to choose general dentistry clinic located very close to your office or house so that you do not have down the sink a considerable amount of time getting going without. Regular visits to a dentist will run you some money but you can save added money actually. Your teeth tend to be in top shape if seem after them well and your smile will be very interesting. |
| Let ''f'' and ''g'' be [[holomorphic]] [[univalent function]]s on the unit disk ''D'', |''z''| < 1, with ''f''(0) = 0 = ''g''(0).
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| ''f'' is said to be '''subordinate''' to ''g'' if and only if there is a univalent mapping φ of ''D'' into itself fixing 0 such that
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| :<math>\displaystyle{f(z)=g(\varphi(z))}</math>
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| for |''z''| < 1.
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| A necessary and sufficient condition for the existence of such a mapping φ is that
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| :<math> f(D)\subseteq g(D).</math>
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| Necessity is immediate.
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| Conversely φ must be defined by
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| :<math> \displaystyle{\varphi(z)=g^{-1}(f(z)).} </math>
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| By definition φ is a univalent holomorphic self-mapping of ''D'' with φ(0) = 0.
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| Since such a map satisfies 0 < |φ'(0)| ≤ 1 and takes each disk ''D''<sub>''r''</sub>, |''z''| < r with 0 < ''r'' < 1, into itself, it follows that
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| :<math>\displaystyle{ |f^\prime(0)| \le |g^\prime(0)|}</math>
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| and | |
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| :<math>\displaystyle{f(D_r) \subseteq g(D_r).}</math>
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| ==Loewner chain==
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| For 0 ≤ ''t'' ≤ ∞ let ''U''(''t'') be a family of open connected and simply connected subsets of '''C''' containing 0, such that
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| :<math> U(s) \subsetneq U(t) </math>
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| if ''s'' < ''t'',
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| :<math> U(t)=\bigcup_{s<t} U(s)</math>
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| and | |
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| :<math> U(\infty)={\mathbb C}.</math>
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| Thus if <math> s_n\uparrow t</math>,
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| :<math> U(s_n) \rightarrow U(t)</math>
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| in the sense of the [[Carathéodory kernel theorem]].
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| If ''D'' denotes the unit disk in '''C''', this theorem implies that the unique univalent maps ''f''<sub>''t''</sub>(''z'')
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| :<math> f_t(D)=U(t), \,\,\, f_t(0)=0, \,\,\, \partial_z f_t(0)=1</math>
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| given by the [[Riemann mapping theorem]] are uniformly continuous on compact subsets
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| of [0,∞) X ''D''.
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| Moreover the function <math>a(t)=f^\prime_t(0)</math> is positive, continuous, strictly increasing and continuous.
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| By a reparametrization it can be assumed that
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| :<math> f^\prime_t(0)=e^t.</math>
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| Hence
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| :<math>f_t(z)=e^tz + a_2(t) z^2 + \cdots </math>
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| The univalent mappings ''f''<sub>''t''</sub>(''z'') are called a '''Loewner chain'''.
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| The [[Koebe distortion theorem]] shows that knowledge of the chain is equivalent to the properties of the open sets ''U''(''t'').
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| ==Loewner semigroup==
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| If ''f''<sub>''t''</sub>(''z'') is a Loewner chain, then
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| :<math> \displaystyle{f_s(D) \subsetneq f_t(D)}</math>
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| for ''s'' < ''t'' so that there is a unique univalent self mapping of the disk φ<sub>''s,t''</sub>(''z'') fixing 0 such that
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| :<math>\displaystyle{ f_s(z)=f_t(\varphi_{s,t}(z)).}</math>
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| By uniqueness the mappings φ<sub>''s,t''</sub> have the following semigroup property:
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| :<math>\displaystyle{\varphi_{s,t}\circ \varphi_{t,r}=\varphi_{s,r}}</math>
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| for ''s'' ≤ ''t'' ≤ ''r''.
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| They constitute a '''Loewner semigroup'''.
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| The self-mappings depend continuously on ''s'' and ''t'' and satisfy
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| :<math>\displaystyle{\varphi_{t,t}(z)=z.}</math>
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| ==Loewner differential equation==
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| The '''Loewner differential equation''' can be derived either for the Loewner semigroup or equivalently for the Loewner chain.
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| For the semigroup, let
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| :<math>\displaystyle{ w_s(z)=\partial_t\varphi_{s,t}(z)|_{t=s}}</math>
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| then
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| :<math>\displaystyle{ w_s(z)=-zp_s(z)}</math>
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| with
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| :<math>\displaystyle{\Re\, p_s(z) > 0}</math>
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| for |''z''| < 1. | |
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| Then ''w''(t)=φ<sub>''s,t''</sub>(''z'') satisfies the [[ordinary differential equation]]
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| :<math> \displaystyle{{dw\over dt} = -w p_t(w)}</math>
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| with initial condition ''w''(''s'') = ''z''.
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| To obtain the differential equation satisfied by the Loewner chain ''f''<sub>''t''</sub>(''z'') note that
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| :<math> \displaystyle{f_t(z)=f_s(\varphi_{s,t}(z))}</math>
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|
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| so that ''f''<sub>''t''</sub>(''z'') satisfies the differential equation
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| :<math>\displaystyle{\partial_t f_t(z)= zp_t(z) \partial_zf_t(z)}</math>
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| with initial condition
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| :<math>\displaystyle{f_t(z)|_{t=0} =f_0(z).}</math>
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| The [[Picard–Lindelöf theorem]] for ordinary differential equations guarantees that these
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| equations can be solved and that the solutions are holomorphic in ''z''.
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| The Loewner chain can be recovered from the Loewner semigroup by passing to the limit:
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| :<math>\displaystyle{ f_s(z) = \lim_{t\rightarrow \infty} e^t \phi_{s,t}(z).}</math>
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| Finally given any univalent self-mapping ψ(''z'') of ''D'', fixing 0, it is possible to construct a Loewner semigroup
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| φ<sub>''s,t''</sub>(''z'') such that
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| :<math>\displaystyle{\varphi_{0,1}(z)=\psi(z).}</math>
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| Similarly given a univalent function ''g'' on ''D'' with ''g''(0) =0, such that ''g''(''D'') contains the closed unit disk, there is a Loewner chain ''f''<sub>''t''</sub>(''z'') such that
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| :<math> \displaystyle{f_0(z)=z,\,\,\, f_1(z)=g(z).}</math>
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| Results of this type are immediate if ψ or ''g'' extend continuously to ∂''D''. They follow in general by replacing mappings ''f''(''z'') by approzimations
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| ''f''(''rz'')/''r'' and then using a standard compactness argument.<ref>{{harvnb|Pommerenke|1975|pp=158–159}}</ref>
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| ==Slit mappings==
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| Holomorphic functions ''p''(''z'') on ''D'' with positive real part and normalized so that ''p''(0) = 1 are described by the [[Herglotz representation theorem]]:
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| :<math>\displaystyle{ p(z) =\int_0^{2\pi} {1 + e^{-i\theta}z\over 1 -e^{-i\theta}z} \, d\mu(\theta),}</math>
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| where μ is probability measure on the circle. Taking a point measure singles out functions
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| :<math> \displaystyle{p_t(z)= {1+\kappa(t) z\over 1-\kappa(t) z}}</math>
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| with |κ(''t'')| = 1, which were the first to be considered by {{harvtxt|Loewner|1923}}.
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| Inequalities for univalent functions on the unit disk can be proved by using the density for uniform convergence on compact subsets of '''slit mappings'''. These are conformal maps of the unit disk onto the complex plane with a Jordan arc connecting a finite point to ∞ omitted. Density follows by applying the [[Carathéodory kernel theorem]]. In fact any univalent function ''f''(''z'') is approximated by functions
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| :<math> \displaystyle{g(z)=f(rz)/r}</math>
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| which take the unit circle onto an analytic curve. A point on that curve can be connected to infinity by a Jordan arc. The regions obtained by omitting a small segment of the analytic curve to one side of the chosen point converge to ''g''(''D'') so the corresponding univalent maps of ''D'' onto these regions converge to ''g'' uniformly on compact sets.<ref>{{harvnb|Duren|1983|pp=80–81}}</ref>
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| To apply the Loewner differential equation to a slit function ''f'', the omitted Jordan arc ''c''(''t'') from a finite point to ∞ can be parametrized by [0,∞) so that the map univalent map ''f''<sub>''t''</sub> of ''D'' onto '''C''' less ''c''([''t'',∞)) has the form
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| :<math>\displaystyle{ f_t(z)=e^t(z+b_2(t)z^2 + b_3(t) z^3 + \cdots)} </math> | |
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| with ''b''<sub>''n''</sub> continuous. In particular
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| :<math>\displaystyle{f_0(z) = f(z).}</math>
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| For ''s'' ≤ ''t'', let
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| :<math>\displaystyle{\varphi_{s,t}(z)= f_t^{-1} \circ f_s(z)= e^{s-t} (z+a_2(s,t)z^2 + a_3(s,t) z^3 + \cdots)}</math>
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| with ''a''<sub>''n''</sub> continuous.
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| This gives a Loewner chain and Loewner semigroup with
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| :<math>\displaystyle{p_t(z)={1+\kappa(t) z\over 1-\kappa(t) z}}</math>
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| where κ is a continuous map from [0,∞) to the unit circle.<ref>{{harvnb|Duren|1983|pp=83–87}}</ref>
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| To determine κ, note that φ<sub>''s,t''</sub> maps the unit disk into the unit disk with a Jordan arc from an interior point to the boundary removed.
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| The point where it touches the boundary is independent of ''s'' and defines a continuous function λ(''t'') from [0,∞) to the unit circle. κ(''t'') is the complex conjugate (or inverse) of λ(''t''):
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| :<math>\displaystyle{\kappa(t)=\lambda(t)^{-1}.}</math>
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| Equivalently, by [[Carathéodory's theorem (conformal mapping)|Carathéodory's theorem]] ''f''<sub>''t''</sub> admits a continuous extension to the closed unit disk and λ(''t''), sometimes called the '''driving function''', is specified by
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| :<math>\displaystyle{f_t(\lambda(t))=c(t).}</math>
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| Not every continuous function κ comes from a slit mapping, but Kufarev showed this was true when κ has a continuous derivative.
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| ==Application to Bieberbach conjecture==
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| {{harvtxt|Loewner|1923}} used his differential equation for slit mappings to prove the [[Bieberbach conjecture]]
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| :<math> \displaystyle{|a_3|\le 3}</math>
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| for the third coefficient of a univalent function
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| :<math>\displaystyle{f(z)=z + a_2 z^2 + a_3z^3 +\cdots}</math>
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| In this case, rotating if necessary, it can be assumed that ''a''<sub>3</sub> is non-negative.
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| Then
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| :<math>\displaystyle{\varphi_{0,t}(z)=e^{-t}(z+a_2(t)z^2 + a_3(t) z^3 +\cdots)}</math>
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| with ''a''<sub>''n''</sub> continuous. They satisfy
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| :<math>\displaystyle{a_n(0)=0,\,\, a_n(\infty)=a_n.}</math>
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| If
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| :<math>\displaystyle{\alpha(t)=e^{-t}\kappa(t),}</math>
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| the Loewner differential equation implies
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| :<math>\displaystyle{\dot{a_2}=-2\alpha} </math>
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| and
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| :<math>\displaystyle{\dot{a_3} =-2\alpha^2 -4\alpha\, a_2.}</math>
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| So
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| :<math>\displaystyle{ a_2 =-2\int_{0}^\infty \alpha(t) \, dt}</math>
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| which immediately implies Bieberbach's inequality
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| :<math>\displaystyle{|a_2|\le 2.}</math>
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| Similarly
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| :<math> \displaystyle{a_3=-2\int_0^\infty \alpha^2\, dt +4\left(\int_0^\infty \alpha\, dt\right)^2}</math>
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| Since ''a''<sub>3</sub> is non-negative and |κ(''t'')| = 1,
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| :<math> \displaystyle{|a_3|=2\int_0^\infty |\Re \alpha^2|\, dt +4\left(\int_0^\infty \Re \alpha\, dt\right)^2}
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| \le 2\int_0^\infty |\Re \alpha^2|\, dt +4\left(\int_0^\infty e^{-t}\,dt\right)\left(\int_0^\infty e^t(\Re \alpha)^2\, dt\right)
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| =1 +4\int_0^\infty (e^{-t}-e^{-2t}) (\Re \kappa)^2\, dt \le 3, </math>
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| using the [[Cauchy-Schwarz inequality]].
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{citation|last=Duren|first=P. L.|title=
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| Univalent functions|series=Grundlehren der Mathematischen Wissenschaften|volume= 259|publisher= Springer-Verlag|year= 1983|isbn= 0-387-90795-5}}
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| *{{citation|last=Kufarev|first= P. P.|title=On one-parameter families of analytic functions|journal= Mat. Sbornik|volume= 13 |year=1943|pages= 87–118}}
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| *{{citation|last=Lawler|first= G. F.|authorlink=Greg Lawler|title=Conformally invariant processes in the plane|series=Mathematical Surveys and Monographs|volume= 114|publisher= American Mathematical Society|year= 2005|isbn= 0-8218-3677-3}}
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| *{{citation|first=C.|last=Loewner|authorlink=Charles Loewner|title=Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I|journal= Math. Ann.|volume=
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| 89|year=1923|pages= 103–121|doi=10.1007/BF01448091}}
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| *{{citation|last=Pommerenke|first= C.|authorlink=Christian Pommerenke|title=Univalent functions, with a chapter on quadratic differentials by Gerd Jensen|series= Studia Mathematica/Mathematische Lehrbücher|volume=15|publisher= Vandenhoeck & Ruprecht|year= 1975}}
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| [[Category:Complex analysis]]
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