ON HEINZ S INEQUALITY

Transcription

1 ON HEINZ S INEQUALITY DARIUSZ PARTYKA AND KEN-ICHI SAKAN In memory of Professor Zygmunt Charzyński Abstract. In 1958 E. Heinz obtained a lower bound for xf + yf, where F isaone-to-oneharmonicmappingoftheunitdiscontoitselfkeeping the origin fixed. We improve Heinz s inequality in the case where F is the Poisson integral of a sense-preserving homeomorphic self-mapping f of the unit circle. As an application we infer a version of Heinz s inequality for harmonic and quasiconformal self- mappings of the unit disc. Introduction Write Hom + (T) for the class of all sense-preserving homeomorphic self-mappings of the unit circle T := {z C : z =1}. Given a function f : T C integrable on T we denote by P[f](z) thepoissonintegraloff, i.e. P[f](z) := 1 (.1) f(u)re u + z π T u z du, z D, where D := {z C : z < 1} is the unit disc. It is well known that the Jacobian J[P[f]] is positive on D for every f Hom + (T); see e.g. [1] or [4, p. 43]. Modifying considerations in [4, pp. 4-43] we obtain a stronger result given by Theorem 1. in Section 1. This implies [6, Remark.3] and thereby completes consideration in [6]. In 1958 E. Heinz proved that the inequality (.) x F (z) + y F (z) π holds for every z = x + iy D, provided F is a one-to-one harmonic mapping of D onto itself and F () = ; cf. []. Applying Theorem 1. and [6, Lemma.1], we are able to improve Heinz s inequality (.) in two cases. The first one, discussed in Section, deals with the case where F =P[f] forsomef Hom + (T); see Theorem.. The second one, discussed in Section 3, deals with the case where F is a quasiconformal (qc. in abbreviation) mapping; see Theorem 3.. The results were presented on Seminar: Generalized Cauchy-Riemann Structures and Surface Properties of Crystals, 3-3 July, 1, Bȩdlewo-Czȩstochowa, Poland. Date: November 5, Mathematics Subject Classification. Primary 3C6. Key words and phrases. Harmonic mappings, Poisson integral, Jacobian, quasiconformal mappings. The research of the second named author was supported by Grant-in-Aid for Scientific Research No , Japan, Society for the Promotion of Science. 1

2 DARIUSZ PARTYKA AND KEN-ICHI SAKAN 1. A lower estimate for the Jacobian Given f Hom + (T) andz T set (1.1) d f := ess inf f (z), z T where f f(u) f(z) (1.) (z) := u z u z provided the it exists and f (z) :=otherwise. Lemma 1.1. If f Hom + (T), then d f 1 and for every Borel subset I T, (1.3) si n f(i) 1 where I 1 is the arc-length measure of I. d f si n I 1, Proof. Let m := d f.obviouslym. If V is a Borel subset of T, then (1.4) f(v ) 1 f (z) dz m dz = m V 1. V In particular T 1 = f(t) 1 m T 1, and hence m 1. Applying (1.4) we obtain (1.5) f(i) 1 m I 1 and f(t \ I) 1 m T \ I 1 = m(π I 1 ). Since f(t \ I) =T \ f(i) weconcludefrom(1.5)that (1.6) f(i) 1 = T \ (T \ f(i) 1 =π f(t \ I) 1 π m(π I 1 ), and hence (1.7) m I 1 f(i) 1 π m(π I 1 ). Since f(i) 1 / π we conclude from (1.7) that (1.8) sin( f(i) 1 /) min{sin(m I 1 /), si n(π m(π I 1 /))} =min{si n(m I 1 /), sin(m(π I 1 /))}. Since R t si n t is a concave function on [; π], we have sin(mt) m si n t for t π. Thus (1.9) sin( f(i) 1 /) m min{sin( I 1 /), sin(π I 1 /))} = m si n( I 1 /) follows from (1.8), which yields (1.3). Theorem 1.. If f Hom + (T), then (1.1) inf J[P[f]](z) z D d3 f. Proof. Given h Hom + (T), t R and s [; π] define (1.11) h(t, s) 1 := h(i(e it,e i(t+s) )) 1, h(t, s) := h(i(e i(t+s),e i(t+π) )) 1, h(t, s) 3 := h(i(e i(t+π),e i(t+s+π) )) 1, h(t, s) 4 := h(i(e i(t+s+π),e i(t+π) )) 1, V

3 ON HEINZ S INEQUALITY 3 where I(z, w) is a closed arc directed counterclockwise from z T to w T. Following Douady and Earle [1] the Jacobian J[P[h]]() of h is equal to J[P[h]]() = 1 π π (1.1) (sin s R π h (t, s)dt)ds, where R h (t, s) :=sin h(t, s) 1 + h(t, s) si n h(t, s) + h(t, s) 3 see also [4, pp. 4-43]. For a D and z D write h a (z) := z a 1 az. Fix z D and set (1.13) h(u) :=f h z (u), u T. From (1.1) and (1.3) it follows that J[P[h]]() = 1 π π (sin s R π f h z (t, s)dt)ds d3 f π π (sin s π si n h(t, s) 1 + h(t, s) 3 R h z (t, s)dt)ds = d 3 f J[P[h z ]](). Hence J[P[f]](z) =J[P[h h z ]](z) =J[P[h] h z ](z) =J[P[h]](h z (z)) J[h z ](z) =J[P[h]]() J[h z ](z) d 3 f J[h z ]() J[h z ](z) =d 3 f J[h z ](h z (z)) J[h z ](z) = d 3 f J[h z h z ](z) =d 3 f J[id](z) =d3 f, which proves (1.1).. The case where F isgivenbythepoissonintegral Recall that the formal derivative operators and are defined by the usual real partial derivatives x and y as below ; (.1) := 1 ( x i y ) and := 1 ( x + i y ). Let f Hom + (T). From [6, Lemma.1] it follows that for a.e. z T both the functions P[f] and P[f] haveradialitingvaluesatz and the following equalities hold [ f(z) P[f](rz) (.) z P[f](rz) = r 1 z P[f](rz) = ] + zf (z) [ ] f(z) P[f](rz) zf (z) Thus we may define (.3) d f := ess inf z T P[f](rz). Following Heinz [], we will prove the following lemma..

4 4 DARIUSZ PARTYKA AND KEN-ICHI SAKAN Lemma.1. If f Hom + (T) and if F =P[f], then (.4) inf F(z) z D d f. Proof. From [5, (1.1)] it follows that F(z) > 1 ( ) 1 a π >, z D, 1+ a where a D is a unique point satisfying F ( a) =. Hence the holomorphic function 1/ F on D belongs to the Hardy class H,andso sup z D Then(.4)followsfrom(.3),asclaimed. F(z) 1 ess sup z T r 1 F(rz) 1. Theorem.. If f Hom + (T) and if F := P[f] satisfies F () =, then (.5) inf z D F(z) 1 π d f max{d f, d 3 f} and (.6) inf ( xf(z) + y F (z) ) z D π + 1 d f + 1 max{d f, d 3 f } Proof. From(.)itfollowsthatfora.e.z T the its exist and the following equalities hold: (.7) and J[F ](rz) ( F(rz) + F(rz) )= f (z) + as well as (.8) ( F(rz) F(rz) )= J[F ](rz). Combining (.7) with (.8) we see that the equality (.9) r 1 F(rz) = 1 4 f (z) J[F ](rz) r 1 holds for a.e. z T. Since F is harmonic on D, F(D) =D and F() =, we conclude from [, Lemma] that (.1) F (z) 4 π arctan z, z D. Actually, this is a version of Schwarz s lemma for harmonic self-mappings of D. From (.1) we see that for every z T and r [; 1), f(z) F (rz) 1 4 π (.11) arctan r as r 1. π By [6, Theorem.] and by Theorem 1. we have J[F ](rz) 1 max{d f, d 3 f} for a.e. z T.

5 ON HEINZ S INEQUALITY 5 Combining this with (.9) and (.11) we obtain (d f ) 1 π d f max{d f, d 3 f } Thus Lemma.1 yields (.5). Applying (.1) we get (.1) x F(z) + y F(z) =( F(z) + F(z) ), z D. Combining (.5) with (.1) we obtain (.6), which completes the proof. 3. The case where F is a quasiconformal mapping It is well known that a quasiconformal self-mapping F of D has a homeomorphic extension F to the closure D; cf. [3]. We call the restriction f := F T the boundary iting valued function of F. Suppose that F is additionally a harmonic mapping. Then F = P [f] ond, as a unique solution to the Dirichlet problem with the boundary function f. Lemma 3.1. Given K 1 let F be a K-quasiconformal and harmonic self-mapping of D satisfying F() =. Iff is the boundary iting valued function of F, then (3.1) d f πk. Proof. From(.)itfollowsthatfora.e.z T, [z F(rz)+z F(rz)] = (3.) [z F(rz) z F(rz)] = zf (z). Since F is a K-quasiconformal mapping, we see from (3.) that for a.e. z T, f (z) = z F(rz) z F(rz) ( F(rz) F(rz) ) r 1 1 K ( F(rz) + F(rz) ) 1 K ( z F(rz)+z F(rz) ) = 1 K. Hence by (.11) we deduce (3.1). Theorem 3.. Given K 1 let F be a K-quasiconformal and harmonic selfmapping of D satisfying F() =. Iff is the boundary iting valued function of F,thentheinequalities (3.3) F(z) K +1 Kπ and x F (z) + y F(z) ( π 1+ 1 ) (3.4) K hold for every z D. Proof. Since F is a K-quasiconformal mapping, we have (K +1) F(w) (K 1) F(w), w D,

6 6 DARIUSZ PARTYKA AND KEN-ICHI SAKAN and hence (3.5) (K +1) F(w) (K +1) ( F(w) + F(w) ), w D. Combining (.7) with (3.1) and (.11) we see that for a.e. z T, + F(rz) ) ( F(rz) π K + π = ( 1+ 1 ) (3.6). π K From this and (3.5) it follows that for a.e. z T, (3.7) K +1 F(rz) r 1 πk. Applying now (.4) we deduce (3.3). Then (3.4) follows directly from (3.3) and (.1). References 1. A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), E. Heinz, On one-to-one harmonic mappings, PacificJ.Math.9 (1959), O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, nd ed., Grundlehren 16, Springer, Berlin, D. Partyka, The generalized Neumann-Poincaré operator and its spectrum, Dissertationes Math., vol. 366, Institute of Mathematics, Polish Academy of Sciences, Warszawa, D. Partyka and K. Sakan, A note on non-quasiconformal harmonic extensions, Bull. Soc. Sci. Lettres Lódź 47 (1997), 51 63, Série: Recherches sur les déformations 3. 6., Quasiconformality of harmonic extensions, J. of Comp. and Appl. Math. 15 (1999), Faculty of Mathematics and Natural Sciences, Catholic University of Lublin, Al. Rac lawickie 14, P.O. Box 19, -95 Lublin, Poland address: partyka@kul.lublin.pl Department of Mathematics, Graduate School of Science, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka, 558, Japan address: ksakan@sci.osaka-cu.ac.jp