From: "Volker Milbrandt" Subject: Re: Lowner Lemma Date: 10 May 1999 10:00:02 -0500 Newsgroups: sci.math.research nowhere!nobody@uunet.uu.net wrote on 7 May 1999 14:30:03 -0500: : Where can I find Lowner lemma. It is something about holomorphic functions on : the unit disk. Probably proved in the 20's or 30's. : : Pawel Gora : pgora@vax2.concordia.ca I only know the following theorem by K. Loewner: Let A be a bounded set (with non-empty interior) in R^d. There exists one and only one ellispoid E of minimal volume containing A (in the interior and on the border), the so-called Loewner ellipsoid. You may find this theorem and the prrof in Detlef Laugwitz: Differentialgeometrie, Teubner, Stuttgart, 1960. As far as I can see, there is no direct connection to holomorphic functions. Volker.Milbrandt@mathematik.uni-stuttgart.de ============================================================================== From: Lukas Geyer Subject: Re: Lowner Lemma Date: 10 May 1999 10:00:08 -0500 Newsgroups: sci.math.research nowhere!nobody@uunet.uu.net wrote: > > Where can I find Lowner lemma. It is something about holomorphic functions on > the unit disk. Probably proved in the 20's or 30's. > > Pawel Gora > pgora@vax2.concordia.ca You can find it in e.g. Pommerenke, Ch.: Boundary behaviour of conformal maps, Springer-Verlag, 1992, p. 72. The statement is the following: Let f be an analytic map of the unit disc into itself and f(0)=0. Let A be a subset of the unit circle such that f(A) (as radial limits) also is a subset of the unit circle. Then L(A)<=L(f(A)) where L is (outer) linear measure on the circle. Lukas Geyer ============================================================================== Mathematical Reviews on the Web Selected Matches for: Anywhere=(Lowner Lemma) 7,288e 30.0X Kametani, Shunji; Ugaheri, Tadasi A remark on Kawakami's extension of Löwner's lemma. Proc. Imp. Acad. Tokyo 18, (1942). 14--15. Soit dans le domaine circulaire unite $D$ de frontiere $\Gamma f(z)$ holomorphe telle que $\vert f(z)\vert <1$, $f(0)=0$. Soit sur $\Gamma$ un ensemble $E$ ou $f$ admet une limite radiale $\rho$ de module 1. Les points $\rho$ forment un ensemble $E'$ sur $\Gamma$. Les auteurs montrent, par usage des resultats de Fatou [Acta Math. 30, 335--400 (1906)] sur l'integrale de Poisson, que la mesure interieure de $E$ est majoree par la mesure exterieure de $E'$. Reviewed by M. Brelot _________________________________________________________________ 4,9a 30.0X Paatero, V. Über beshränkte Funktionen, welche gegebene Paare von Randbogen ineinander überführen. (German) Math. Z. 47, (1941). 175--186. Lowner's lemma is concerned with analytic functions $w=f(z)$ which map the unit circle $\vert z\vert <1$ on a portion of the unit circle $\vert w\vert <1$, with $f(0)=0$, and which remain continuous on $\vert z\vert =1$ and map an $\text{arc}\,\alpha$ of $\vert z\vert =1$ on an $\text{arc}\,\beta$ of $\vert w\vert =1$; the conclusion is that $\text{meas}\,\beta\geq\text{meas}\,\alpha$. This lemma gives rise to an existence problem when there are more than one pair of arcs involved. The author previously has solved the problem for two pairs of arcs; in the present paper the result is extended to three pairs. Necessary and sufficient conditions for the existence of mapping functions are given in terms of cross ratios of sets of the points involved. [For a direct extension of Lowner's lemma, also concerned with more than one pair of arcs, see the papers by Yosiro Kawakami, Jap. J. Math. 17, 569--572 (1941); these Rev. 3, 202.] Reviewed by E. F. Beckenbach _________________________________________________________________ 3,202a 30.0X Kawakami, Yosiro On an extension of Löwner's lemma. Jap. J. Math. 17, (1941). 569--572. The author proves the following theorem: A domain $D$ with boundary $C$ included in the unit circle $\vert z\vert <1$ in the $z$-plane is represented on the unit circle $\vert w\vert <1$ in the $w$-plane by a simple function $w=F(z)$ such that $F(0)=0$. If a closed set $B$ on $C$ on $\vert z\vert =1$ is represented on a closed set $B'$ on $\vert w\vert =1$, then we have $\text{meas}(B)\geq\text{meas}(B')$, the equality holding only when $D$ coincides with the unit circle if $B$ is of positive measure. This theorem extends one of Lowner, which proves the same result in the case $C$ is a simple curve and $B$ a segment thereof. For the proof the author makes use of a "hypofunction" $\underline H\sb D{}\sp f$ defined as follows: Let $f$ be a bounded function on $C$ and $F\sb i$ the class of subharmonic functions in $D$, none of whose limiting values on $C$ exceed $f$. Then $\underline H\sb D{}\sp f$ is the superior envelope of the functions of $F\sb i$. It is harmonic in $D$ and is equal to $\int\sb Cf(Q)\,dµ(e\sb Q,P)$, where $µ$ is the mass distribution obtained from sweeping out unit mass from $P$ onto $C$. In case $f$ is the characteristic function of a closed set $B$, this hypofunction is a generalization of harmonic measure. Using the integral representation for $H$ and the invariance of the sub-harmonic property under conformal transformation, the author obtains the stated result. Reviewed by J. W. Green _________________________________________________________________ 1,308c 30.0X Joh, Kenzo Theorems on "schlicht" functions. IV. Proc. Phys.-Math. Soc. Japan (3) 22, (1940). 329--343. Employing the principle of harmonic measure due to R. Nevanlinna, the author gives new proofs of two recent theorems by H. Unkelbach [Jber. Deutsch. Math. Verein. 49, 38--49 (1939)]. The first of these theorems is to the effect that if $W = F(z)$ be regular for $\vert z\vert < R$, $F(0) = 0$, $F(z) \ne 1$ for $\vert z\vert < R$, $F(z)$ univalent and convex for $\vert z\vert < R$, then the Lebesgue measure $\alpha $ of the set of points on $\vert z\vert = R$ for which $\vert F(z)\vert \geqq R$ satisfies the inequality $\alpha < 8$. And for any arbitrary given small positive number $\straightepsilon$, there exists a value of $R$ and a function such that $\alpha > 8 - \straightepsilon$. If instead of being convex $F(z)$ is univalently star-like for $\vert z\vert < R$ in addition to the normalization mentioned above, the second theorem establishes the fact that the Lebesgue measure $\beta $ of the set of points on $\vert z\vert = R$ for which the radial limiting value $\lim \sb {\vert z\vert \to R} \vert F(z)\vert \geqq R\sp 2$ is such that $\beta < 8$ and again the number 8 cannot be replaced by a smaller one. Unkelbach gave a first proof of these theorems using as a basis Lowner's lemma which is that, if $W = \straightphi (z)$ be a function regular for $\vert z\vert < R$, $\straightphi (0) = 0$, $\vert \straightphi (z)\vert \leqq R$ and $\vert \straightphi (z)\vert = R$ continuously on an arc $A$ on $\vert z\vert = R$ of length $\alpha \sb {(z)}$, and if $\alpha \sb {(W)}$ be the length of the arc corresponding to the arc $A$, then $\alpha \sb {(W)} \geqq \alpha \sb {(z)}$. Unkelbach conjectured that the second theorem is correct if the normalized $F(z)$ is merely univalent (instead of univalently star-like). The difficulties of this conjecture were overcome by the author through the powerful principle of harmonic measure which appears to be natural and essential to the proof. A further extension is obtained for multivalent functions $F(z)$. Reviewed by M. S. Robertson © Copyright American Mathematical Society 1999