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Schwarz Lemma and Kobayashi Metrics for Holomorphic And SCHWARZ LEMMA AND KOBAYASHI METRICS FOR HOLOMORPHIC AND PLURIHARMONIC FUNCTIONS M. MATELJEVIC´ This is a working version. We will polish the text as soon as possible. 1. Introduction The Schwarz lemma as one of the most influential results in complex analysis and it has a great impact to the development of several research fields, such as geometric function theory, hyperbolic geometry, complex dynamical systems, and theory of quasi-conformal mappings. In this note we mainly consider various ver- sion of Schwarz lemma and its relatives related to holomorphic functions including several variables. We use Ahlfors-Schwarz lemma to give a simple approach to Kalaj-Vuorinen results [7] (shortly KV-results) and to put it into a broader perspective. But, it turns out that our methods (results) unify very recent approaches by D. Kalaj- M. Vuorinen, H. Chen, K. Dyakanov, D. Kalaj, M. Markovi´c, A. Khalfallah and P.Melentijevi´c.1 We use several dimension version of Schwarz lemma (we call it the geometric form of Kobayashi-Schwarz lemma, Theorem 4) to generalize these results to several variables. In particular our considerations include domains on which we can compute Kobayashi distance, as the unit ball, the polydisc, the punctured disk and the strip. There is a huge literature related to Schwarz lemma (see for example H. Boas [21], R. Osserman [22], D.M. Burns and S.G. Krantz [2], S.G. Krantz [14] and the literature cited there) and we apology if we did not mention some important papers. 2. Schwarz lemma,background Concerning material exposed in this section, we refer the interested reader to well written Krantz’s paper [12] that can provide greater detail. Let D be a domain in arXiv:1704.06720v1 [math.CV] 21 Apr 2017 z = x + iy-plane and a Riemannian metric be given by the fundamental form ds2 = σ dz 2 = σ(dx2 + dy2) | | which is conformal with euclidian metric. Often in the literature a Riemannian metric is given by ds = ρ dz , ρ> 0, that is by the fundamental form | | ds2 = ρ2(dx2 + dy2) . Date: 28 Dec,2016. 1991 Mathematics Subject Classification. Primary 30F45; Secondary 32G15. Key words and phrases. harmonic and holomorphic functions,hyperbolic distance and do- mains,the unit ball,the polydisc. 1 After writing a version of manuscript [16], Petar Melentijevi´c[17] sent me his preprint (at 18 Jan 2017). We learned from his preprint about a few related recent results [20, 5, 8, 15]; we also found [23] which may be related to our paper. 1 2 M. MATELJEVIC´ In some situations it is convenient to call ρ = ρD shortly metric density. For v TzC vector, we define v ρ = ρ(z) v . If γ is a piecewise smooth path in D, we∈ define γ = ρ(z) dz |and| d (z ,z| |) = inf γ , where the infimum is | |ρ γ | | D 1 2 | |ρ taken over all paths γ in D joining the points z and z . R 1 2 For a hyperbolic plane domain D, we denote respectively by λ = λD (or if we wish to be more specific by HypD) and δD (in some papers we use also notation σD) the hyperbolic and pseudo-hyperbolic metric on D respectively. By HypD(z) we also denote the hyperbolic density at z D. For planar domains G and D we denote by∈ Hol(G, D) the class of all holomorphic mapping from G into D. For complex Banach manifold X and Y we denote by (X, Y ) the class of all holomorphic mapping from X into Y . O If f is a function on a set X and x X sometimes we write fx instead of n ∈ n f(x). We write z = (z1,z2, ..., zn) C . On C we define the standard Hermitian n ∈ Cn inner product by <z,w >= k=1 zkwk for z, w and by z = √<z,z> we denote the norm of vector z. We also use notation∈ (z, w) instead| | of < z,w > on P n some places. By B = Bn we denote the unit ball in C . In particular we use also notation U and D for the unit disk in complex plane. Proposition 2.1 (classical Schwarz lemma 1-the unit disk). Suppose that f : D D is an analytic map and f(0) = 0. The classic Schwarz lemma states : f(z) →z and f ′(0) 1. | | ≤ | | | |≤ It is interesting that this (at a first glance) simple result has far reaching appli- cations and forms. Define z z1 T 1 (z)= − , z 1 z z − 1 ϕ = T and z1 − z1 z z1 δ(z ,z )= T 1 (z ) = − . 1 2 | z 2 | |1 z z | − 1 The classical Schwarz lema yields motivation to introduce hyperbolic distance: If f Hol(U, U), then δ(fz1,fz2) δ(z1,z2). Consider∈ F = ϕ f ϕ , w≤ = f(z ). Then F (0) = 0 and ϕ (w ) w1 ◦ ◦ z1 k k | w1 2 | ≤ ϕz1 (z2) . | Hence| 1 fz 2 (2.1) f ′(z) − | | . | |≤ 1 z 2 − | | By the notation w = f(z) and dw = f ′(z)dz, we can rewrite (2.1) in the form dw dz (2.2) | | | | . 1 w 2 ≤ 1 z 2 − | | − | | We can rewrite this inequality in vector form. Namely, define the density λ(z) = 1 ∗ 2 . For v T C vector we define v = λ(z) v and set v = df (v). Hence, 1−|z| ∈ z | |λ | | z we can rewrite (2.2) in the form: v∗ v . Thus we have | |λ ≤ | |λ Proposition 2.2 (classical Schwarz lemma 2-the unit disk). Suppose that f : D D is an analytic map. → (a) Then (2.1). (b) If v T C and v∗ = df (v), then v∗ v . ∈ z z | |λ ≤ | |λ SCHWARZ LEMMA 3 If we choose that the hyperbolic density (metric) is given by 2 (2.3) HypD(z)= , z D . 1 z 2 ∈ − | | then the Gaussian curvature of this metric is 1. We summarize − 1+ δU 1+ δH (2.4) λU = ln , λH = ln . 1 δU 1 δH − − Let G be a simply connected domain different from C and let φ : G U be a G → conformal isomorphism. Define ϕa (z)= ϕb(φ(z)), where b = φ(a), and the pseudo hyperbolic distance on G by G δG(a,z)= ϕa (z) = δU(φ(a), φ(z)). One can verify that the pseudo hyperbolic distance on G|is independent| of conformal mapping φ. In particular, using confor- w−w0 H U mal isomorphism A(w)= Aw0 (w)= w−w0 of onto , we find ϕH,w0 (w)= A(w) and therefore δH (w, w0)= A(w) . C | ′ | ′ ′ For a domain G in and z,z G we define δG(z,z ) = sup δU(φ(z ), φ(z)), and ∈′ ′ the Carath´eodory distance cG(z,z ) = sup δU(φ(z ), φ(z)), where the suprimum is taken over all φ Hol(G, U). Of course the Caratheodory distance can be trivial-for instance if G is∈ the entire plane. On simply connected domains, the pseudo-hyperbolic distance and the hyper- bolic distance are related by δ = tanh(λ/2) and we have useful relation: (I-1) If G and D are simply connected domains domains and f conformal mapping ′ of D onto G, then HypG(fz) f (z) = HypD(z). The uniformization theorem| says| that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature. Every Riemann surface is the quotient of a free, proper and holomorphic action of a discrete group on its universal covering and this universal covering is holomorphically isomorphic (one also says: ”conformally equivalent” or ”biholomorphic”) to one of the following:the Riemann sphere,the complex plane and the unit disk in the complex plane.If the universal covering of a Riemann surface S is the unit disk we say that S is hyper- bolic.Using holomorphic covering π : U S, one can define the pseudo-hyperbolic and the hyperbolic metric on S. In particular,→ if S = G is hyperbolic planar domain we can use ′ (I-2) HypG(πz) π (z) = HypD(z) and (I-3) If G and D| are| hyperbolic domains and f conformal mapping of D onto G, then Hyp (fz) f ′(z) = Hyp (z). G | | D Proposition 2.3 (Schwarz lemma 1- planar hyperbolic domains). (a) If G and D are conformally isomorphic to U and f Hol(G, D), then ∈ δ (fz,fz′) δ (z,z′), z,z′ G. D ≤ G ∈ (b) The result holds more generally: if G and D are hyperbolic domains and f Hol(G, D), then ∈ Hyp (fz,fz′) Hyp (z,z′), z,z′ G. D ≤ G ∈ 4 M. MATELJEVIC´ (c) If z G, v T C and v∗ = df (v), then ∈ ∈ z z v∗ v . | |Hyp ≤ | |Hyp For a hyperbolic planar domain G the Carath´eodory distance CG λG with equality if and only if G is a simply connected domain. ≤ (A) holomorphic functions do not increase the corresponding hyperbolic dis- tances between the corresponding hyperbolic domains. The Caratheodory and Kobayashi metrics have proved to be important tools in the function theory of several complex variables. We can express Kobayashi- Schwarz lemma in the geometric form (see Theorem 4, which is our main tool): In particular, we have: n (B) If G1 and G2 are domains in C and f : G1 G2 holomorphic function, then f does not increase the corresponding Caratheodory(Kobayashi)→ distances. But they are less familiar in the context of one complex variable.
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