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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
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. Krantz [13] gathers in one place the basic ideas about these important invariant metrics for domains in the plane and provides some illuminating examples and applications. We consider various generalization of this result including several variables. There is an interesting connection between hyperbolic geometry and complex geometry. (C) Pseudo-distances defined by pluriharmonic functions. In [23] the author constructs αM,P , a new holomorphically invariant pseudo- distance on a complex Banach manifold M using the set of real pluriharmonic functions on M with values in P , a proper open interval of R. It is well known that the Kobayashi pseudo-distance is the largest and the Caratheodory pseudo- distance is the smallest one which can be assigned to complex Banach manifolds by a Schwarz-Pick system. Therefore C α Kob . M ≤ M,P ≤ M 3. Schwarz lemma for real harmonic functions
Let S = S0 = w : Rew < 1 and S1 = w : Rew < π/4 . tan maps S1 onto D π{ | | } { | |π } S . Let B(w)= 4 w and f0 = tan B, ie. f0(w) = tan( 4 w). Then f0 maps 0 onto D. ◦ 2 Let r < 1, A (z) = 1+z , and let φ = i lnA ;. that is φ = φ A , where 0 1−z π 0 0 ◦ 0 φ = i 2 ln. Let φˆ be defined by φˆ(z) = φ(iz). Note that φˆ = 4 arctan is the 0 π − π ˆ′ 4 D S inverse of f0. Hence φ (0) = π and if f is a conformal map of onto 0 with ′ 4 f(0) = 0, then f (0) = π . By the subordination| principle, D S ′ 4 (II) If f is an analytic map of into 0 with f(0) = 0, then f (0) π . There is analogy of classical Schwarz lemma for harmonic| maps.≤
Lemma 1 ([4]). Let h : D S0 be a harmonic mapping with h(0) = 0 . Then Reh(z) 4 tan−1 z and this→ inequality is sharp for each point z D . | |≤ π | | ∈
Example 1. Let fa(z)=Reφˆ(z) + iay and ga(z)=Reφˆ(z) + iaImφˆ(z). For a R, D S ′ ∈ fa and ga are harmonic maps of into 0 and fa(0) = ga(0) = 0. Since fa(0) a 4 | | ≥ | | and Lga (0) = a π , there is no reasonable estimate for the distortion of harmonic functions which| | maps the unit disk into the strip. It is interesting to note that Lemma 1 shows that we can control the growth of the real part of a harmonic mapping which maps D into S0 and keeps the origin fixed. SCHWARZ LEMMA 5
Let λ0 be a hyperbolic density on S0. Then π 1 (3.1) λ0(w) = HypS0 (z)= π . 2 cos( 2 u) If F is holomorphic map from D into S0, then by Ahlfors-Schwarz lemma
(3.2) λ (F (z)) F ′(z) 2(1 z 2)−1, z D. 0 | |≤ − | | ∈ Using that 1 cos( π x) = 2 sin2( π x) and the inequality 4 t √2 sin t, 0 t π , − 2 4 π ≤ ≤ ≤ 4 we prove cos( π x) 1 x2, x 1, and therefore we get 2 ≤ − | |≤ (A1) π (1 u2)−1 λ (w). Hence, we get Kalaj-Vuorinen [7], see also [20]: 2 − ≤ 0 Proposition 3.1. If f is harmonic map from D into ( 1, 1) with f(0) = 0, then − 4 π (3.3) f(z) cos( f(z))(1 z 2)−1, z D. |∇ |≤ π 2 − | | ∈ 4 1 f(z) 2 (3.4) f(z) − | | , z < 1. |∇ |≤ π 1 z 2 | | − | | Outline. Let F be analytic such that f = F on D with F (0) = 0. Then f(z) F ′(z) , z D. By Ahlfors-Schwarzℜ estimateℜ (3.2), |∇ | ≤ | | ∈ λ (f(z)) f(z) 2(1 z 2)−1, z D. 0 |∇ |≤ − | | ∈ Now an application of the formula (3.1) yields (3.3). By (A1), (3.4) follows from (3.3). 4 By (A1), HypD(x1, x2) π HypS0 (x1, x2), x1, x2 ( 1, 1). If F is holomorphic map from D into S and ≤u = ReF , then Hyp (uz∈,uz− ) Hyp (Fz ,Fz ) 0 S0 1 2 ≤ S0 1 2 ≤ HypD(z1,z2). Theorem 1. Suppose that D is a hyperbolic plane domain v : D ( 1, 1) is real 4 → − harmonic on hyperbolic domain D. Then HypD(v(z ), v(z ) Hyp (z ,z ). 1 2 ≤ π D 1 2 The author discussed the results of these types with SH. Chen, S. Ponnusamy and X. Wang, see also [19]. Proof. If D is the unit disk D this result follows from (3.4) and it has been proved by Kalaj and Vuorinen [7]. In general case one can use a cover : D D and definev ˆ = v . For z, w D, let z′ −1(z), w′ P −1(w→). Then ◦P ∈′ ′ 4 ∈P ′ ′ ∈P Hyp (ˆvz , vwˆ ) HypD(z , w ). Hence we get Theorem 1. D ≤ π 1 If Π= w : Rew> 0 , then HypΠ(w)= Rew . There is{ tightly connection} between harmonic and holomorphic functions. A few year ago I had in mind the following result: Theorem 2. Suppose that D is a hyperbolic plane domain and G = S(a,b) = (a,b) R,
In the case D = D, (II) is proved for G = S( 1, 1) in [7], and for G = S(0, ) in [15]. − ∞
Proof. First suppose that D = D. In general case one can use a cover : D D as in the proof of Theorem 1. Let f = u+iv, F +U +iV be analytic such thatP f→= F ℜ ℜ on D. Let h TzC, h = 1, and dfz(h)= λv, λ> 0 and v TfzX. If the measure 2 ∈ | | ∈ 4 1−|f(z)| of the angle between v and e1 TpC,p = f(z), is α, then λ cos α π 1−|z|2 . ∈ ′ ≤ Note that dfz(h)= duz(h) and duz(h) F (z) . ℜ ′ | | ≤ | | ′ ′ If we choose h such that f (z) = dfz(h) , we get cos α f (z) F (z) . Since ′ | | | | | | ≤ | | HypG(Fz) F (z) HypD(z) and HypG(f(z)) = HypG(F (z)) = HypG(Ref(z)) Hence, we| have (I).| ≤ An application of (I) with α = 0 yields (II). In particular, if G = S( 1, 1) we have − 4 1 Ref(z) 2 f ′(z) cos α − | | , | | ≤ π 1 z 2 − | | and if G = S(0, ) we have ∞ 2Ref(z) (3.5) f ′(z) cos α . | | ≤ 1 z 2 − | |
4. Kobayashi-Schwarz lemma -Several variables Definition 3. Let G be bounded connected open subset of complex Banach space, p G and v T G. We define k (p, v) = inf h , where infimum is taking over ∈ ∈ p G {| |} all h T0C for which there exists a holomorphic function such that φ : U G such that∈ φ(0) = p and dφ(h)= v. →
We also use the notation KobG instead of kG. We call KobG Kobayashi-Finsler norm on tangent bandle. For some particular domains, we can explicitly compute Kobayashi norm of a tangent vector by the corresponding angle. We define the distance function on G by integrating the pseudometric kG: for z,z G 1 ∈ 1 (4.1) KobG(z,z1) = inf kG(γ(t), γ˙ (t)) dt γ Z0 where the infimum is taken over all piecewise paths γ : [0, 1] G with γ(0) = z → and γ(1) = z1. It is convenient to introduce kG = 2kG. By (2.3), Kobayashi pseudometric kD and the Poncare metric coincide on D. For complex Banach manifold X and Y we denote by (X, Y ) the class of all holomorphic mapping from X into Y . If φ (D,X) andO f (X, Y ), then φ f (D, Y ). ∈ O ∈ O ◦We∈ can O express Kobayashi-Schwarz lemma in geometric form: Theorem 4. If a X and b = f(a), u T X and u = f ′(a)u, then ∈ ∈ p ∗ (4.2) Kob(b,u ) u Kob(a,u) u . ∗ | ∗|e ≤ | |e Hence SCHWARZ LEMMA 7
Theorem 5 (Kobayashi-Schwarz lemma). Suppose that G and G1 are bounded connected open subset of complex Banach space and f : G G1 is holomorphic. Then → (4.3) Kob (fz,fz ) Kob (z,z ) G1 1 ≤ G 1 for all z,z G. 1 ∈ If φ is a holomorphic map of D into G, we define L u(p, v) = sup λ : φ(0) = G { p, dφ0(1) = λv , and LG(p, v) = sup LGu(p, v), where the supremum is taken over all maps φ} : D G which are analytic in D with φ(0) = p. Note that → LG(p, v)kG(p, v) = 1. By Definition 3, 1 (4.4) KobG(p, v)= . LG(p, v)
If G is the unit ball, we write Lφ(p, v) instead of LGφ(p, v). For u T Cn we denote by u euclidean norm. ∈ p | |e 4.1. A new version of Schwarz lemma for the unit ball. Using classical Schwarz lemma for the unit disk in C, one can derive:
Proposition 4.1 (Schwarz lemma 1-the unit ball). Suppose that (i) f (Bn, Bm) and f(0) = 0. Then (a) f ′(0) 1. ∈ O (b) If u T Cn and u =| f ′(0)|≤u, then u u . ∈ 0 ∗ | ∗|e ≤ | |e Proof. Take an arbitrary a Bn and set b = f(a). For z U define g(z) =< f(za∗),b∗ >. Since g Hol(U∈, U), then by the unit disk version∈ of Schwarz lemma, ∈ ∗ we find (i) g(z) z , z U. Choose z0 such that a = z0a An application of (i) to z , yields| f(|a ≤) | | a .∈ It is straightforward that we get (a) and (b). 0 | | ≤ | | We need some properties of bi -holomorphic automorphisms of unit ball (see [18] for more details). For a fixed z, B = w : (w z,z)=0, w 2 < 1 and z { − | | } denote by R(z) radius of ball Bz. Denote by Pa(z) the orthogonal projection onto the subspace [a] generated by a and let Qa = I Pa be the projection on the orthogonal complement. For z,a Bn we define − ∈ a Pz s Qz (4.5)z ˜ = ϕ (z)= − − a , a 1 (z,a) −
1 2 1 2 (4.7) M(p,u)= MB(p,u)= 4 cos α + 2 sin α. ssp sp 8 M. MATELJEVIC´
k k p p Proof. Set A = dϕp and u = u1 + u2, where u1 TpU and u2 TpQ , and ′ k ∈ ∈ uk = A (uk), k =1, 2. By the classical Schwarz lemma 2-the unit disk, Proposition ′ 2 ′ 4.1 (Schwarz lemma 1-the unit ball) and (A2), u1 e = u1 e/sp and u2 e = u2 e/sp. ′ ′ ′ ′ ′ | | | | | | | | Then u = A(u)= u1 + u2 and u1 and u2 are orthogonal. ′ ′ 2 ′ 2 Hence, since u1 e = cos α u e, u2 e = sin α u e and u e = u1 e + u2 e, we find (4.7). | | | | | | | | | | | | | | p It is clear that 1 1 (4.8) M(p,u) 2 . sp ≤ ≤ sp Suppose that (i) f (B , B ), a B and b = f(a), u T Cn and u = f ′(a)u. ∈ O n m ∈ n ∈ p ∗ If A = dϕa, B = dϕb then by Proposition 4.2, we find Au e = M(a,u) u e and Bu = M(b,u ) u . By Schwarz 1-unit ball, Bu |Au| . Hence | | | ∗|e ∗ | ∗|e | ∗|e ≤ | |e n Theorem 6. Suppose that f (Bn, Bm), a Bn and b = f(a), u TpC and ′ ∈ O ∈ ∈ u∗ = f (a)u. Then (4.9) M(b,u ) u M(a,u) u . ∗ | ∗|e ≤ | |e In particular, we have
Theorem 7 (Schwarz lemma 2-unit ball,[8, 16]). Suppose that f (Bn, Bm), a B and b = f(a). ∈ O ∈ n Then s2 f ′(a) s , i.e. (1 a 2) f ′(a) 1 f(a) 2. a| |≤ b − | | | |≤ − | | Theorem 8. Let a B and v T Cn. For Bp, Kob(a, v)= M(a, v) v . ∈ n ∈ p n | |e n Proof. Let φ be a holomorphic map of D into Bn, φ(0) = a, v TaC , v e = 1, ′ ∈ | | dφ0(1) = λv = v . 1◦. Consider first} the case a = 0. Let p be the projection on [v]. Then φ1 = p φ is a holomorphic map of D into v ′ ◦ 0 U . By classical Schwarz lemma φ1(0) 1 and therefore φ (ζ)= vζ is extremal. Hence Kob(0, v) = 1. | |≤ 2◦. If a = 0, in general the part of the projection of φ(D) on [v] can be out of 6 ◦ Bn.So we can not use the procedure in 1 directly and we consider φa = ϕa φ and ′ ′ ◦ set v∗ = (ϕa φ) (0) = (dϕa)a(v ). ◦ ′ ◦ ′ a Hence v∗ e = M(a, v ) and by 1 , λ M(a, v ) 1. Hence the mapping φ = ϕ φ0 is| extremal.| | | ≤ a ◦ 4.2. polydisk. For the polydisk, see [9], p.47,
KobUn (z, w) = max Kob(z , w ): k =1, ,n . { k k ··· } 2 2 Let p = (c, d) U , T = (ϕc, ϕd), A = dTp, and u TpC . ∈ 2 ∈ If the measure of the angle between u TpC and z1-plane is α = αu = α(p,u) and u′ = A(u), one can check that(see below)∈
′ ′ ′ 1 1 2 2 2 (4.10) u e = M (p,u)= MU (p,u)= 4 cos α + 4 sin α . | | ssc sd If c d , then | | ≥ | | 1 ′ 1 2 M (p,u) 2 . sd ≤ ≤ sc SCHWARZ LEMMA 9
2 (III) Now let φ(0) = p, u TpC , u e = 1, and dφ0(1) = λu and consider T φ. ∈ | | ′ ◦ If v = dTp(u), then d(T φ)0(1) = λv and v e = M (p,u) u e. Hence ′ ◦ | | | |′ (A3) if u e = 1, M (p,u)L(p,u) √2 and KobU2 (p,u) M (p,u)U2 u e/√2. It turns| | out that we need an improvement≤ of (A3) in order≥ to compu|te| Kobayashi- Finsler norm. Namely, our computation of Kobayashi-Finsler norm on U2 is based on: (A4) If v = (v , v ) T C2 and v v , then Kob(p, v)= v = cos α v . 1 2 ∈ 0 | 1| ≥ | 2| | 1| | |e 2 Proof. Let φ be analytic from U into U and φ(0) = (0, 0). Then φ1 and φ2 map U intoself. Hence dφ (1) √2. If v = (v , v ) T C2 and v v , the map | 0 | ≤ 1 2 ∈ 0 | 1| ≥ | 2| z (zv1, zv2)/ v1 shows that L(0, v)=1/ v1 (note that L(0, v) = v / v1 √2 if 7→v = 1). By| equation| (4.4), Kob(p, v)= |v | = cos α v . | | | | ≤ | |e | 1| | |e Now we check (4.10) and that 2 2 (B1) v1 e = cos α u e/sc, v2 e = sin α u e/sc , and | | | | | | | | 2 2 2 (B2) k(p,u)= M(p,u) u e, where M(p,u)= MU (p,u) = max cos α/sc, sin α/sd . 2 2 | | { 2 } In particular, if sd/sc tan α, then k(p,u)= v1 e = cos α u e/sc. 1 ≥ 2 | | | | Proof. Set A = (A , A ) = dTp and u = u1 + u2, where u1 Tp[(c, 0)] and ′ k ∈ u2 Tp[(0, d)], and uk = A (uk), k =1, 2. ∈ ′ 2 ′ By the classical Schwarz lemma 2-the unit disk, u1 e = u1 e/sp and u2 e = ′ ′ ′ ′ ′ | | | | | | u2 e/sp. Then u = A(u)= u1 + u2 and u1 and u2 are orthogonal. | | ′ ′ 2 ′ 2 Hence, since u1 e = cos α u e, u2 e = sin α u e and u e = u1 e + u2 e, we find (4.10), (B1)| and| (B2). | | | | | | | | | | | | p Thus we get Proposition 4.3. Let p = (c, d) U2. If the measure of the angle between u 2 ∈ 2 2 ∈ T C and z -plane is α = α = α(p,u), then kU2 (p,u) = max cos α/s , sin α/s u . p 1 u { c d}| |e Using a similar procedure as in the proof of Proposition 4.3 one can derive: Proposition 4.4. Let D and G be planar hyperbolic domains, Ω = D G, p = (c, d) Ω and u T C2. Then × ∈ ∈ p Kob(p,u)= M (p,u) u , Ω | |e where M(p,u)= M (p,u) = max Hyp (c)cos α, Hyp (d) sin α . Ω { D G } We can restate this result in the form: (a) If Hyp2 (c)cos2 α Hyp2 (d) sin2 α, then Kob(p,u)= v := cos α u Hyp (c). D ≥ G | 1| | |e D (b) If Hyp2 (c)cos2 α Hyp2 (d) sin2 α, then Kob(p,u)= v := sin α u Hyp (c). D ≤ G | 2| | |e D Proof. Let ψc and ψd be conformal mappings of D and G onto U such that ψc(c)= d c d ψ (d) = 0 respectively. If T = (ψ , ψ ), A = dTp and v = A(u), one can check that k (p,u)= kU2 (0, v), 2 v = λ (c) u and 2 v = λ (c) u . Hence Ω | 1|e D | 1|e | 2|e G | 2|e (4.11) v = M ′(p,u)= M ′ (p,u)= Hyp2 (c)cos2 α + Hyp2 (d) sin2 α . | |e Ω D G q If v v , then k (p,u)= λ (c) u . | 1| ≥ | 2| Ω D | 1|e Using Theorem 4, Propositions 4.2 and 4.4, we have 10 M. MATELJEVIC´
2 Theorem 9. Suppose that f (B2, Ω), a B2 and b = f(a), u TpC and ′ ∈ O ∈ ∈ u∗ = f (a)u. Then
(4.12) Kob (b,u )= M (b,u ) u MB (a,u) u , Ω ∗ Ω ∗ | ∗|e ≤ 2 | |e where KobΩ is described in Proposition 4.4. Set G = S(a,b) = (a,b) R, (4.13) Kob 2 (u(z),u(w)) KobB (z, w), z,w B . S(a,b) ≤ 2 ∈ 2 B S2 Proof. Under the hypothesis, there is an analytic function f : 2 0 such that 2 → Re f = Re u on B2 and f : B2 (a,b) . By Theorem 5 (Kobayashi-Schwarz→ lemma) and (C2) we have (4.13). 2 Let p B2 and v TpB . Note that df(v)= dRe f(v)+ idIm f(v). ∈ ∈ 2 If the measure of the angle between v TpC and x1x2-plane β = β(p,u), ′ ′∈ ′ ′ (dRe f)p(v) = cos β dfp(v) , where β = β(p , v∗), v∗ = dfp(v) and p = f(p). | B | S2 | | ′ Let h : 2 0 be a pluriharmonic function. Let v = (dRe h)p(v), v∗ = dhp(v) and p′ = h(p).→ We leave the reader to check that ′ ′ ′ ′ (D1) k 2 (p , v ) tan β kB2 (p, v)λ(p )/λ(p ). S(a,b) ∗ ≤ 2 1 Outline. Then there is analytic function f : B S2 such that Re f = Re h on 2 → 0 B . Note that HypS (w) = HypS (Re w), w S . 2 0 0 ∈ 0 If I = ( 1, 1) and J = (0, ) we can consider I2, J 2 and I J. In a similar way, we can− extend this result∞ to pluriharmonic functions u : B × (a,b)m. n → 4.3. Invariant gradient and Schwarz lemma. The mapping w = eiz maps H onto the punctured disk. The Poincare metric on the upper half-plane induces a metric on the punctured disk U′ 4 ds2 = dq dq, q U′ . q 2(log q 2)2 ∈ | | | | Hence 2 1 ′ HypU′ (z)= − dz = − dz , z U . z (log z 2)| | z (log z )| | ∈ | | | | | | | | Ahlfors [1] proved a stronger version of Schwarz’s lemma and Ahlfors lemma 1. Theorem 11. ( Ahlfors lemma 2 ). Let f be an analytic mapping of D into a region on which there is given ultrahyperbolic metric ρ. Then ρ[f(z)] f ′(z) λ . | | ≤ SCHWARZ LEMMA 11 Let G be open subset of B, f C2(G) and a G. Set ( ˜ )f(a)= (f ϕ )(0). ∈ ∈ △ △ ◦ a Operator ˜ comutate with automorphisms of ball and we call it invariant Laplacian. For λ △C, we denote by X the space of all functions f C2(B), which satisfy ∈ λ ∈ ˜ f = λf. Elements of X0 we call -harmonic functions. △ Recall: Suppose that D and G areM hyperbolic planar domains and f is an analytic ′ mapping of D into G. Then HypG(fz) f (z) HypD(z), z D. n | | ≤ ∈ If G C and f : G C, define Df(z) = (Dj f(z), ..., Dnf(z)) and Df˜ (a) = D(f ϕ⊂)(0). For h T →Cn, set u = (dϕ ) (h). Then s2 h u = (Dϕ ) (h) ◦ a ∈ 0 a 0 a| | ≤ | | | a 0 |≤ sa h . Note that (d(f ϕa))0(h)= dfa(u). | | n◦ Set Df(z) = ( D f(z) 2)1/2. If f(z)=z, z Bn, then Df(z) = √n. Let | | j=1 | j | ∈ | | f be complex-valued function defined on Bn. Cn P ′ n For a , define g(z)= ga(z)= f(a1z, ..., anz). Then g (z)= k=1 Dkf(za)ak = df(a). ∈ Bn ′ P If f is an analytic mapping of into G, then HypG(fz) g (z) a e. ′ | | ≤ | | Set ak = Dkf(0)/ Df(0) , we find g (0) = Df(0) = df0(a) . ′ | | | | | | | | ′ Hence f (0) Df(0) . Since there is a, a = 1, such that f (0) = df0(a) , f ′(0) = |Df(0)| ≤. | | | | | | | | | There| | is u |T Cn such that Df(z) = df (u) . If v = (dϕ) (u), then (df ∈ z | | | z | z ◦ ϕz)0(v) = dfz(u). Hence Df(z) = dfz(u) = (df ϕz)0(v) Df˜ (z) v e. Since 2 | | | | | ◦ | ≤ | || | v e 1/sz, we find | | ≤2 ˜ (A) sz Df(z) Df(z) . Set |F = f| ≤ϕ | . There| is v T Cn such that DF (z) = dF (v ) and set ◦ z 0 ∈ 0 | | | 0 0 | u0 = (dϕ)0(v0). Since u s , dF (v )= df (u ), we find df (u ) df u s df . Thus | 0|e ≤ z 0 0 0 0 | 0 0 | ≤ | 0|| 0|≤ z| 0| (B) Df˜ (z) sz Df(z) . Hence| |≤ | | (C) s2 Df(z) Df˜ (z) s Df(z) . z| | ≤ | |≤ z| | Theorem 12. Let G be a hyperbolic plane domain and let f be an analytic mapping of Bn into G, a Bn, b = f(a), u T Cn and u = f ′(a)u. Then ∈ ∈ p ∗ (4.14) Hyp (b) u MB (a,u) u . G | ∗|e ≤ 2 | |e In particular, (i) Hyp (fz) f ′(z) 1/s2, G | |≤ z (ii) Hyp (fz) Df˜ (z) 1. G | | ≤ By a version of Schwarz lemma, (i) holds. For z = 0, since Df˜ (0) = f ′(0) , (ii) holds. An application of (ii) to the function F = f ϕ at z =| 0, and| the| definition| ◦ z of Df˜ (z), show that (ii) holds in general. If G is the punctured disk, we get a Dyakonov result [5]: ′ Proposition 4.5. Suppose that f (Bn, U ), a Bn, b = f(a) and ρ = HypU′ . Then ρ(b) f ′(a) 2/s2, i.e. (1 a∈2) Of ′(a) 2 b ∈ln 1 . | |≤ a − | | | |≤ | | |b| Acklowedgment. We have discussed the subject at Belgrade Analysis seminar (in 2016) and in particular in connection with minimal surfaces with F. Forstneric and get useful information about the subject via Forstneric [6]. We are indebted to the members of the seminar and to professor F. Forstneric for useful discussions. References [1] L. Ahlfors, Conformal invariants, McGraw-Hill Book Company, 1973. 12 M. MATELJEVIC´ [2] D.M. Burns and S.G. Krantz, Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary, J. Amer. Math. Soc. 7 (1994), 661-676. [3] D. Chelts, A generalized Schwarz lemma at the boundary, Proceedings of the American Math. Society 129 (2001), No.11, 3275-3278. [4] P. Duren, Harmonic mappings in the plane, Cambridge Univ. Press, 2004. [5] K. M. Dyakonov, Functions in Bloch-type spaces and their moduli, Ann. Acad. Sci. Fenn. Math., 41(2):705712, 2016, arXiv:1603.08140 [math.CV]. [6] Communication with F. Forstneric, 2016. [7] D. Kalaj and M. Vuorinen, On harmonic functions and the Schwarz lemma, Proc. Amer. Math. Soc. 140 (2012), no. 1, 161-165. [8] D. Kalaj, Schwarz lemma for holomorphic mappings in the unit ball, arXiv:1504.04823v2 [math.CV] 27 Apr 2015 [9] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel-Dekker, New York, 1970. [10] S. Kobayashi, Hyperbolic Complex Spaces, Berlin: Springer Nature, (1998), ISBN 3-540- 63534-3, MR 1635983 [11] S. G. Krantz, The Kobayashi metric, extremal discs, and biholomorphic mappings, Complex Variables and Elliptic Equations, Volume 57, 2012 - Issue 1 [12] S. G. Krantz, Pseudoconvexity, Analytic Discs, and Invariant Metrics,http://www.math.wustl.edu/∼sk/indian.pdf [13] S. G. Krantz, The Carath´eodory and Kobayashi Metrics and Applications in Complex Analysis, arXiv:math/0608772v1 [math.CV] 31 Aug 2006 [14] S. G. Krantz, The Schwarz Lemma at the Boundary, September 16, 2010 [15] M. Markovic, On harmonic functions and the hyperbolic metric, Indag. Math., 26(1):19-23, 2015. [16] M. Mateljevi´c, Schwarz lemma, Kobayashi Metrics and FPT, preprint November 2016, to appear in Filomat [17] P. Melentijevic, Invariant gradient in refinements of Schwarz lemma and Harnack inequal- ities, manuscript. [18] W. Rudin, Function Theory in the Unit Ball of Cn, Springer-Verlag, Berlin Heidelberg New York, 1980. [19] S. H. Chen, M. Mateljevi´c, S. Ponnusamy and X. Wang, Schwarz-Pick lemma, equivalent modulus, integral means and Bloch constant for real harmonic functions, to appear. [20] H. Chen, The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings, Sci. China Math. November 2013, Volume 56, Issue 11, pp 2327-2334 [21] H. Boas, Julius and Julia: Mastering the Art of the Schwarz Lemma - ESIA, http://www.esi.ac.at [22] R. Osserman, From Schwarz to Pick to Ahlfors and beyond, Notices Amer. Math. Soc.46 (1999) 868-873. [23] A. Khalfallah, Old and new invariant pseudo-distances defined by pluri- harmonic functions, Jan 2014, Complex Analysis and Operator Theory, https://www.researchgate.net/profile/Adel− Khalfallah2 Faculty of mathematics, University of Belgrade, Studentski Trg 16, Belgrade, Yu- goslavia E-mail address: [email protected]