Schwarz lemma, the Carath eodory and Kobayashi metrics and applications in complex analysis

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1 Schwarz lemma, the Carath eodory and Kobayashi metrics and applications in complex analysis Workshop: The perturbation of the generalized inverses, geometric structures, xed point theory and applications Miodrag Mateljevi c (miodrag@matf.bg.ac.rs, jenamate@eunet.rs) University of Belgrade, Faculty of Mathematics Zlatibor, Serbia, August 28-September 4, 2016 M. Mateljevi c (MF) 1 / 12

2 Introduction We study Schwarz lemma at the boundary of strongly pseudoconvex domains, and versions of now called the Carath eodory-cartan-kaup-wu theorem, which generalizes the classical Schwarz lemma for holomorphic functions to higher dimensions, and iterations of holomorphic mappings. C. Earle and R. Hamilton, A xed point theorem for holomorphic mappings, Proc. Symp. Pure Math., Vol. XVI, 1968, M. Mateljevi c (MF) 2 / 12

3 The Schwarz lemma as one of the most inuential results in complex analysis and it has a great impact to the development of several research elds, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and theory of quasi-conformal mappings. We plan to study Schwarz lemma at the boundary of strongly pseudoconvex domains, and versions of now called the Carath eodory-cartan-kaup-wu theorem, which generalizes the classical Schwarz lemma for holomorphic functions to higher dimensions, and iterations of holomorphic mappings. M. Mateljevi c (MF) 3 / 12

4 Let G be a nonempty domain in a complex Banach space and let a holomorphic function h maps G strictly inside a subset G, then h is a contraction. We proved this probably around 1980 (we found a hand written manuscript 1990 and did not pay much attention to it at that time). But we realized these days that it is a version of the Earle-Hamilton (1968) xed point theorem, which may be viewed as a holomorphic formulation of Banach's contraction mapping theorem. The result was proved in 1968 (when I enroled Math Faculty in Belgrade) by Cliord Earle and Richard Hamilton by showing that, with respect to the Carath eodory metric on the domain, the holomorphic mapping becomes a contraction mapping to which the Banach xed-point theorem can be applied. M. Mateljevi c (MF) 4 / 12

5 Geometric interpretation We use notation df = pdz + qdz. Let f be R-dierentiable at a point z 0 = r 0 e it 0, r 0 = z 0, ɛ 0 > 0, w 0 = f(z 0 ), R 0 = w 0 and L be a circular arc dened by L(t) = r 0 e it, z 0 ɛ 0. Since L (t) = ir 0 e it and in particular L (t 0 ) = iz 0, the tangent vector T of the curve f L at f(z 0 ) equals T = df(l (t)) = df(iz 0 ). If f(l) B R0, then from obvious geometric interpretation we have T = λif(z 0 ), λ R. Since T = df(iz 0 ), we nd i(pz 0 qz 0 ) = λif(z 0 ), λ R, where p = f z (z 0 ) and q = f z (z 0 ), and therefore we get Proposition Under the above hypothesis, (pz 0 qz 0 ) = λf(z 0 ), λ R. If p > q, then λ > 0. M. Mateljevi c (MF) 5 / 12

6 Lemma (Jack) Suppose that U is the unit disk, f analytic on U, U is invariant under the mapping f and f has a x point z 0 on the boundary of U. Then f (z 0 ) is positive. The subject related to Jack's lemma has also been discussed by Ornek in a recent paper. It seems that the following result is true. By S (α) we denote the family of starlike univalent functions of of order α. Note that f S (α) if Re zf (z) f(z) > α. Theorem If f belongs S (α), 0 < α < 1, and 1/β = 2(1 α) = s(α), then z (i) f(z) (1 z ) 1/β ; (ii) f (0) 2/β. B. Ornek, Estimates for holomorphic functions concerned with Jack's lemma (manuscript no Publ. Inst. Math.) M. Mateljevi c (MF) 6 / 12

7 Suppose f is an analytic map of U into itself. If b < 1, we say b is a xed point of f if f(b) = b. If b = 1, we say b is a xed point of f if lim r 1 f(rb) = b. Julia-Carath eodory Theorem implies that if b is a xed point of f with b = 1, then lim r 1 f (rb) exists (call it f (b)) and 0 < f (b). Theorem (Denjoy-Wol (1926)) (a) If f is an analytic map of U into itself, not the identity map, there is a unique xed point, a, of f in U such that f (a) 1. (b) If f is not an automorphism of U ( i.e. a M obius transformation ) with xed point in U, iterates of f tend to a uniformly on compact subsets of U. This distinguished xed point will be called the Denjoy-Wol point of f. M. Mateljevi c (MF) 7 / 12

8 The Schwarz-Pick Lemma implies f has at most one xed point in U and if f has a xed point in U, it must be the Denjoy-Wol point. C. C. Cowen, Iteration and the Solution of Functional Equations for Functions Analytic in the Unit Disk, Trans. Amer. Math. Soc. 265 (1981) Question Is there a version of this result for quasi-regular mappings? We prove Cartan uniqueness theorem (strongly convex pluriharmonic version) Theorem (Cartan uniqueness theorem) Let D be a bounded domain in C n and given a D. If f Aut a (D) satises f (a) = 1, then f(z) = z for all z D. M. Mateljevi c (MF) 8 / 12

9 If f Aut(B) xes a point of B, then x point set of f is ane. Conversely, we have Theorem (Hayden-Suridge) If f Aut(B) xes three point of S, then f xes a point of B. M. Mateljevi c (MF) 9 / 12

10 Theorem Suppose that G is bounded connected open subset of complex Banach space and f : G G is holomorphic, G G, s 0 = dist(g, G c ), d 0 = diam(g) and q 0 = d 0 d 0 +s 0. Then k G (fz, fz 1 ) q 0 k G (z, z 1 ) for z, z 1 G. Hence we can derive Earle-Hamilton theorem. M. Mateljevi c (MF) 10 / 12

11 Schwarz lemma at the boundary of strongly pseudoconvex domain Example Recall in complex dimension 2 we use notation ϕ a (z, w) = (T a (z), λ(z)w), where λ(z) = sa 1 az. If (a, 0) B2, J ϕa (z, w) = ((1 a 2 )(1 az) 2, 0; λ (z)w, λ(z)). Set υ = (1, 0) and A = J ϕa (1, 0). Then A = (λ, 0; 0, µ), where λ = (1 a 2 )(1 a) 2 and µ = s a (1 a) 1 ; so that λ = µ 2. If 1 < a < 1, ϕ a has two fp ±(1, 0), λ = 1+a 1 a and therefore λ > 1 if 0 < a < 1 and λ < 1 if 1 < a < 0. M. Mateljevi c (MF) 11 / 12

12 The next result is a generalization of previous example. Theorem (Theorem 1.1, Wang and Ren) Let G C n be a bounded strongly pseudoconvex domain with C 2 boundary and f : G G a holomorphic mapping. Suppose that f extends smoothly past some point p G and f(p) = p. Then for the eigenvalues λ, µ 2,..., µ n (counted with multiplicities) of J f (p), the following statements hold: (i) λ is positive and is also an eigenvalue of J f (p) t such that J f (p) t υ p = λυ p, where υ p is the unit outward normal vector to G at the point p; (ii) µ j C and µ j λ for j = 2, 3,..., j n ; (iii) For j = 2, 3,..., j n, there exists τ j T p (1,0) J f (p)τ j = µ j τ j ; such that (iv) detj f (p) λ (n+1)/2, trj f (p) λ + (n 1) λ; Moreover, if in addition f has an interior xed point z 0 G, then λ 1. M. Mateljevi c (MF) 12 / 12

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