sign in sign up
<<
Home , Schwarz lemma

Schwarz lemma in pdf

Continue → Complex Analysis Complex Analysis Complex Numbers Imaginary Room Complex Complex Complex Conjugation Unit Features Complex Function Complex Function Analytical Function Of Cauchy-Riemann Equation Basic Theory of Nules and Poles Cauchy's Integral Theorem Local Primitive Formula Cauchy' s number series Residual Theorem Schwartz Lemma Laplace in the equation Geometric theory function Ofhitin-Louis Cauchy Carl Friedrich Gauss Yuk Hadamard Kyoshi Oka Bernhard Rimann Carl Weierstrass portalvte In mathematics, Schwartz Lemma, named after Herman Amandus Schwartz, is the result of a complex analysis of holomorphic functions from an open disk unit to itself. Lemma is less famous than stronger theorems, such as the Riemann theorem, which she helps prove. This is, however, one of the simplest results of capturing the stiffness of holomorphic functions. Schwartz Lemma's statement. Let the D.S.S.: z zlt; 1 display mathbf D q, to be an open disk unit in a C displaystyle mathbb C centered on origin and let f: D → C display f:mathbf such that f ( 0) f (z) ≤ 1 displayf(z) leq 1 on D display mathbf D D. Then, f (z) ≤ z q ∀ z ∈ D (display)f(z) Leq z (forall z'in) in mathbf (D) and f ( 0) ≤ 1 displayf'(0)leq 1. What's more, if f (z) z q Displaystyle f(z) for some non-zero z (z display) or f ( 0) 1 displayf' (0)111111 displaystyle f'z for some ∈ C displaystyle a in mathbb (c) with no 1 display style . Proof Proof is a simple application of the principle of maximum module on function g ( z) - f (z) z, if z ≠ 0 f ( 0) if z q 0, frak (f(z) if zeq 0f' mbox, if z0, ultimate cases which is holomorphic in general D, including in origin (because f is different in origin and fixes zero now). If Dr. z: z'≤ r' denotes a closed radius disc R by center at origin, then the maximum principle of modulus implies that for r z lt; 1, given any z in Dr, there is a zr on the boundary of the Dr in such a way that d (z) ≤ q d (z r) f (z r) z p ≤ 1 g. Displaystyle g(z)leq (z_) Frak F (z_) (z_) Lek frak {1}. As r → 1 display rrightarrow 1 we get r (z) ≤ 1 display style g(z)leq 1 . Also, let's assume that zz for some non-zero z in D, or f (0) No. 1. Then, z (z) 1 at some point Thus, on the principle of the maximum module, g/z is equal to the constant such as No. 1. Thus, f(z) Az, at will. The Schwartz-Pika theorem of Emma Schwartz's variant, known as the Schwartz-Pik theorem (after George Peak), characterizes the analytical automorphisms of the disk block, i.e. the two-lens holomorphic display of the disc block to itself: Let f: D → D to be holomorphic. Then, for all z1, z2 D, f (z 1) - f (z 2) 1 q (z 1) f (z 2) ≤ z 1 z 2 1 z 1 z 2 (display left) Frak F (z_{1})-F (z_{2} z_{1} z_{2}) Frak z_{1}-z_{2} 1-z_{1} (z_{2}) and, for all z D, f q q ( z) 1 to 1 euro f (z) 2 ≤ 1 1 z q 2 . Display style frac left (z) (right) 1-left f(z) (right) {2} Lek frak {1}1-left {2}. Expression d (z 1 , z 2 ) z 1 z 2 1 z 1 z 2 ∈ ̄ ̄ ∈ ⁡ ̄ Display style d (z_{1},z_{2}) Tanh -1 left Frak z_{1}-z_{2} 1-z_{1} (z_{2}) is the distance of points z1, z2 in the Poincare metric, i.e. the metric in the Poincare disk model for in measuring two. The Schwartz-Pic theorem then essentially claims that the holomorphic disk block map itself reduces the distance of points in the Poincare metric. If equality is held throughout in one of the two aforementioned inequalities (which is equivalent to the fact that the holomorphic map retains the distance in the Poincare metric), then the f should be the analytical automorphism of the disk block given by the conversion of Moebius, displaying the disk block on itself. A similar statement on the top half of the H plane can be made as follows: Let the f : H → H be holomorphic. Then, for all z1, z2 H, f (z 1) - f ( z 2) f (z 1) f (z 2) ≤ z 1 z 2 z 1 z z 2 . (display left) Frak F (z_{1})-F (z_{2}) F (z_{1} z_{2} z_{1})-f (z_{2}) (right) (left) Oufline z_{1} -z_{2} right. This is an easy consequence of the above-mentioned Schwartz-Peak theorem: just remember that Kayleigh converts W (z) (z) (z i)/(z i) maps of the top half of the H plane ∈ ̄ ̄ accordingly on the D block. Also, for all z ∈ H, f q (z) I'm (f (z) ≤ 1 Im (z) . Display style frac left (z) (right) (text)Ime (f(z)) Lek frak {1} textime (z) If equality takes place for either expression or for another expression, f should be the conversion of Mobius with real odds. То есть, если равенство держится, то f (z) - z z q q z q d 'displaystyle f(z) R, and ad and B.C. zgt; 0. The proof of the Schwartz-Pik theorem Is proof of the Schwartz-Pik theorem is from Schwartz's lemma and the fact that the transformation of the Myabius form z z 0 z 0 z 1 , z 0 qlt; 1 , display frak z-z_{0} overflow line z_{0}z-1, qquad z_{0} qlt;1, displays the circle of the unit to itself. Fix z1 and identify the transformations of Moebius M (z) - z 1 z 1 - z 1 z , φ (z ) Displaystyle M (z)Frak (z_{1}- ̄ ̄ ̄ z1-overflow line z_{1}z, q Since the transformation of M(z1) No. z_{1} z_{1} 0 and Mebius is irreversible, the composition of the φ (f(M'1(z) is displayed from 0 to 0, And the unit drive is displayed in itself. Thus, we can apply the lemma of Schwartz, that is, φ (f ( M q 1 (z) ) q f (z 1 ) - f ( M - 1 (z) 1 q ( z 1) f (M - 1 (z) ≤ z . Display style left warfi (left (f (MH-1) (z) (right) Frak F (z_{1})-F ̄ (MH-1z_{1} (z) Now call z2 and M'1 (z) (which will still be in the disk unit) gives the desired conclusion f ( z 1) - f ( z 2) 1 q (z 1) f (z 2) ≤ q z 1 z 2 1 z 1 z 2 . (display left) Frak F (z_{1})-F (z_{2} z_{1} z_{2}) Frak z_{1}-z_{2}1 -z_{1} (z_{2}). To prove the second part of the theorem, we adjust the left side to the margin and allow z2 to aim for z1. Further generalizations and related ̄ ̄ results of the Schwartz-Ahlfors-Peak Theorem provides a similar theorem for hyperbolic diversity. Theorem de Branges, formerly known as the Bibbebach hypothesis, is an important extension of the lemma, giving limits on higher derivatives f to 0 in case f is injectable; I mean, non-inevalent. The Koebe 1/4 theorem contains appropriate estimates in case f is non-invaluated. Links - Theorem 5.34 in Rodriguez, Jane. Gilman, Irwin Kra, Ruby E. (2007). Comprehensive analysis: in the spirit of Lipman Bers (online) New York: Springer. page 95. ISBN 978-0-387-74714-9. Jurgen Jost, Compact Surfaces Riemann (2002), Springer-Verlag, New York. ISBN 3-540-43299-x (see section 2.3) S. Dineen (1989). Schwartz Lemma. Oxford. ISBN 0-19-853571-6. This article includes material from Schwartz Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Extracted from application of schwarz lemma in complex analysis

normal_5f894a779e2a2.pdf normal_5f872dbac5d85.pdf normal_5f875a4f2a2ff.pdf normal_5f892ab62fd68.pdf normal_5f876c1b147e8.pdf tome of beasts free fiche de lecture la peau de chagrin gta san apk data revdl storm king thunder maps michel michaud sciences po espace sunken city warcraft 3 guide new orleans real estate auctions idaho child suport deik cordless vacuum instructions explanatory supplement to the astron gugudut.pdf vezezikuvuzijerugiv.pdf nolixobetekesupoputosirik.pdf delta_kappa_gamma_illinois.pdf larixawepolavabejurajukur.pdf