Comment. Math. Helvetici 69 (1994) 659-668 0010-2571/94/040659-1051.50 + 0.20/0 9 1994 BirkMuser Verlag, Basel A h l f o r s - W e i l l e x t e n s i o n s o f c o n f o r m a l m a p p i n g s and critical p o ints o f the P o i n c a r 6 metric M. CHUAQUIAND B. OSGOOD I. Introduction Nehari showed in [10] that if f is analytic in the unit disk D, and if its Schwarzian derivative S f = ( f " / f ' ) - ( 1 / 2 ) ( f,, / f, ) 2 satisfies ' 2 Isf(z)l (1 -tzl ) ( l. l ) then f is univalent in the disk. Ahlfors and Weill showed in [1] that if the Schwarzian satisfies the stronger inequality 2t Isf(z)l ~ (1 -Iz12) 2 (1.2) for some 0 -< t < 1 then, in addition, f has a quasiconformal extension to the sphere. They gave an explicit formula for the extension. The class of analytic functions satisfying either of these conditions is quite large. It was shown by Paatero in [13] that any convex univalent function satisfies (1.1). This was later established in a different way by Nehari in [11], and he went on to prove that a bounded convex function satisfies the Ahlfors-Weill condition. In [6], Gehring and Pommerenke made a careful study of Nehari's original univalence criterion and showed, among other things, that the condition (1.1) implies that f ( D ) is a Jordan domain except when f is a M6bius conjugation of the logarithm, 1 1 + z (1.3) Fo(z) = ~ log 1---~z " By this we mean t h a t f = T o Fo o T, where T and z are M6bius transformations and z(d) = D. The function Fo has SFo(z) = 2/( 1 - z2) 2, and Fo(D) is an infinite parallel 659
660 M. CHUAQUI AND B. OSGOOD strip. For topological reasons, it then follows from the Gehring-Pommerenke theorem that other than in the exceptional case f h a s a homeomorphic extension to the sphere. See also [4]. The main result in this paper is that the same Ahlfors- Weill formula defines a homeomorphic extension off, though it will not in general be a quasiconformal extension. We discuss this phenomenon via a relationship between the Ahlfors-Weill extension and the Poincar6 metric of the image of f This may be of independent interest. For economy of notation, though at the risk of sinking a crowded ship, we introduce explicitly several subclasses of univalent functions associated with Nehari type bounds. Thus we let N denote the set of analytic functions in the disk satisfying (1.1), N* the elements of N other than M6bius conjugations of the function F0, and N t those functions satisfying (1.2). We use the notation No, No*, N6 to indicate that a function f in any of the classes has the normalization f ( O) = O, f ' ( O) = 1, f " ( O) = 0. If f (z) = z + a2z 2 +. 9 9 is in any of the classes, then f/(1 + a z f ) is in the corresponding class of normalized functions, the point being that the normalized function is still analytic, [2]. The function F0 is normalized in this way. Furthermore, according to [3], Lemma 4, functions in N* are bounded. The family of normalized extremals for the Ahlfors-Weill condition (1.2) is At(z) = 1 (1 + z ) ~ - (1 - z ) ~ x/1 t. (1.4) 9 (1 + z)" + (1 - z)"' ~ = -- We thank the referee for his thoughtful and helpful remarks. 2. Preliminary estimates Several distortion theorems for the classes No and N~ were proved in [2] using comparison theorems for the second order, ordinary differential equation associated with the Schwarzian. We continue somewhat in the same vein here for a few basic estimates. We refer to our earlier paper for further background. LEMMA 1. I f f e No then 1'~' 21z[ (2.1) (z) I - I z l =" Equality holds at a single z v~ O if and only if f is a rotation o f Fo(z). I f f e N6 then f" 2tlz[ T (z) 1 - I z l 2 (2.2)
Ahlfors-Weill extensions of eonformal mapping and critical points of the Poincar6 metric 661 The inequality (2.2) is not sharp. The proof will show how one may obtain a sharp estimate, but it is not as convenient and explicit as the one given here. Proof Let y =f"/f'. Then y ' = l y 2 + 2p, y(0) = O, with 2p(z) = Sf(z). We consider the real equation w' lw2 2 = 2 + ( ~ l - - x' 2 ) w(o) =0, on (--1,1), whose solution is w ( x ) = 2 x / ( 1 - x 2 ). ly(z) I < w(izl). Fix Zo, Izol = 1, and let We want to show that go(t) = Jy(~zo)J, 0 < ~ < 1. Unless f(z) = z identically the zeros of go are isolated. Away from these zeros go is differentiable and go'(z) < [y'(zzo)[. Since I p(tzo)l< 1/(1 - z z) 2 we obtain d 1 (go(v) - w(t)) ~ Jy'(vzo)J- w'(t) ~ 5 ([y(vzo)j = - w2(t)) 1 = ~ (go(v) - w(t)) (go(r) + w(t)). This, together with go(0)- w(0)= 0, implies that go(z)- w(z) can never become positive. Now suppose ihat equality holds in (2.1) at zl # 0. Let z0 = z,/izll and let go(z) be defined as above. Then go(lzl[)= w(iz~l) which, by the previous analysis, can happen only if go(z) = w(z), first on [0, Izl I], and then for all z e [0, 1) since both functions are analytic. Hence y(tzo) is of the form eio(*)w(t). Since all inequalities above must be equalities, it follows easily that 0(0 must be constant. From this, it follows in turn that y(z)= cw(~oz) for all Izl < l, with Icl = 1. Integrating this equation and appealing to the normalizations on f shows that f(z) = e-~~176 This proves the first part of the lemma. Next, suppose t h a t f e N~. The proof that tf"/f'j has the bound in (2.2) proceeds exactly as above with the single difference that the comparison equation is, l 2 2t w = ~ w + ( l ~ _ x 2 ' ) w(o)=o.
662 M. CHUAQUI AND B. OSGOOD The solution is given by 2 x 2~t 2 w(x) - 1 -- x 2 1 -- X 2 A t ( x ) ' where A t ( X ) is defined in (1.4). It can be checked that At(x) is convex on [0, 1], and hence max A,(x) - A~(0) = 1. 0 ~ x ~ l X Therefore 1 - - X 2 - - w ( x ) < 1 - - ~2 = t, 2x which proves (2.2). 3. Bounds for the Poincar~ metric The Poincar6 metric 2a ]dw I of a simply connected domain f2 is defined by 1 2 a ( f ( z ) ) l f ' ( z ) [ = 2D(Z) -- 1 --izl = ' where f : D ~ t2 is a conformal mapping of the unit disk onto f2. From Schwarz's lemma and the Koebe 1/4-theorem one has the sharp inequalities 1 1 1 ~ a ( z ) ~ - - 4 d(z, Or2) d(z, t3t2)' where d(z, 0~2) denotes the Euclidean distance from z to the boundary. Writing w = f ( z ) and taking the dz = O/Oz derivative of the logarithm of (3.1) gives Ow2a(f(z)).... ~, 1 f " j t z ) - 1 - - I z t 2 2 f ' (z). (3.2) Observe for a normalized function f e No that the Poincar6 metric 2a of the image
Ahlfors-Weill extensions of conformal mapping and critical points of the Poincar6 metric 663 12 has a critical point at w = 0, and, by L e m m a 1, that this must be the unique critical point if f is not a rotation of the logarithm F0. (In the latter case f2 is a parallel strip and the critical points of 20 are all the points of the axis of symmetry of f2.) Assuming that f e N is bounded, we can drop the normalization and reach the same conclusion: L E M M A 2. lf f ~ N is bounded, then 2~ has a unique critical point. Proof Since f2 = f ( D ) is bounded and 2a(w) ~ oo as w ~ 0f2, 2a must have at least one critical point. By replacing f by f o T~ where T~ is a M6bius transformation of the disk to itself, and then by T2 o f o T,, where Tz is a complex affine transformation, we may assume that one such critical point is w = 0 = f ( 0 ), and furthermore that f ( 0 ) = 0 a n d f ' ( 0 ) = 1. The identity (3.2) then forces f " ( 0 ) = 0, i.e. t h a t f e N*. Hence, as above, w = 0 is the unique critical point for )-a s i n c e f c a n n o t be a rotation of the log. R E M A R K. These are sharp results in the sense that for any 0 < E < 2 there is a bounded, univalent function f with S f ( z ) = - 2 ( l + Q / ( l - z ~ ) 2 such that 20, f2 = f ( D ), has more than one critical point. In fact, consider the (normalized) functions A_,(z) for 1 < t < 3, where A,(z) is defined in (1.4). For each t the function A_ t has S A _, ( z ) = - 2 t / ( 1 - z 2 ) 2 and maps D onto the quasidisk f2, consisting of the interior of the union of the circles through the points l/a, - 1/a and + / ( I / a ) tan (na/4), where a = x/1 + t. One can check directly that when t > 1 the Poincar6 metric for I2, has exactly three critical points, one at 0 and two on the imaginary axis which are conjugate. In [7] Kim and Minda showed that log 2a is a convex function if and only if t2 is a convex domain. See also the papers [9] and [ 14]. Using the fact that N contains the convex conformal mappings we can add: C O R O L L A R Y 1. If f2 is a bounded, convex domain, then 2~ has a unique critical point. In [12] it was shown that IV log 2al < 42a (3.3) as a consequence of (3.2) and the classical bound for [f"/f'[ that holds for any univalent function in the disk. The inequality (3.3) is equivalent to the coefficient inequality [a21 ~ 2. We now give some lower bounds for IV log 2o[.
664 M. CHUAQUI AND B. O ~ D LEMMA 3. I f f ~ N*, then there exists a constant c > 0 such that IV log 2a(W)l > clwl2a(w ) I/2 (3.4) If f e N~, then Iv log ;~,~(w) I ~ 2(1 - t)3/2lwl2~(w). (3.5) Recall that a functionf~ N~' is bounded. The constant in (3.4) depends on the bound forf. In an appendix we will give an example to show that the exponent 1/2 is essentially best possible in (3.4). Proof. The estimate (3.4) is implicit in [6]. We show how it can be deduced, adopting the notation used there. Let h be the inverse of F0 and let g = f o h. For z e R we have 2lg'(v)l -- (1 --Ih(T)12) If'(h(T))l -- 2a(g(z))-1. It was shown in [6] that v = I g ' l - 1 / 2 is convex, with v(0) = 1, v'(0) = 0. It is not constant whenfis not equal to Fo. Now, ' d 2 ~- (z) = ~ log 2a(g(z)) < IV log 2u(g(z))[ [g'(~)l = I v log 2,~(g(~))lv(~) -2, hence 117 log 2u(g(z))[ > 2v(z)v'(z) = 23/2v'(T)Aa(g('0)1/2. Since v is not constant and f is bounded, it follows that there exists a constant a such that v'(z) > alg(z) t for z > 0. The estimates can be made uniformly on different rays from the origin by considering f(ei~ This proves (3.4). Now suppose that f e Nh and write w =f(z). Using (3.2),,f f " (z) 1117 log 2a(w)[ = I(Ow2")(f(z))[ -~(1 -Izl2)T 2a(f(z)) = (1 -Izl2)tf'(z)t From Lemma l, (2.2) we then obtain 2( 1 - t)izl = 2(1 - t)izl2n(w), 117 log 3. a(w) l ~ ( 1 -- Iz 12) If'(z) l with w = f ( z ). But from [2], a function in N~ is subject to the sharp bound
Ahlfors-Weill extensions of conformal mapping and critical points of the Poincar6 metric 665 If(z)l ~ A,(H), where A, was defined in (1.4). This can be rearranged to (1 + ~lwl)~/~-(1- ~lwl) TM Izl ~ (1 - ~lwl)l/~t "3 L (1 0 ~ [ w l ) 1 / ~ - - ~,(~lwl), 9 = x / l - t. The function (s) is concave on [0, 1] with ~k(0) = 0 and ~(1) = 1. Hence ~k(s) > s and (3.5) follows. 4. H o m e o m o r p h i c e x t e n s i o n s L e t f e N withf(z) = z + a2 z2 +. 9 9. It was shown in [2] that - l/a 2 Sf(D), and it follows from Lemma 4 in [3] that unlessfis conjugate to Fo the point - l/a2 will actually lie outside (f(-d-). For a fixed ( e D renormalize in the usual way to.(z +~'x - s ( ~ 1 2) z 2 ( l - - l ~ 1 2 ) f ' ( 0 = z + ( I - I ~ l ( 0 - +..., which is again in N, and which has g ( 0 ) = 0-2 / g " ( 0 ) 6 g(d) is equivalent to saying that Ef(O =f(c) + ( 1 - J(I2)f'(C) e f ( D ), 1 2) ' f ~--~(i-1r (o and g ' ( 0 ) = 1. To say that (4.1) and, again, i f f is not conjugate to F0 then E A O e f(d). (4.2) In terms of the Poincar6 metric, Ef has the expression 1 E:(z) =f(z) + dw(log ;to) ( f ( z ) ) ' (4.3) by (3.2). T H E O R E M 1. l f f ~ N* then F(z) = ~f(z) for Izl ~ 1, [E/(1/2) for Izl > I, (4.4) is a homeomorphic extension o f f to the sphere.
666 M. CHUAQUI AND B. OSGOOD The extension Ey has the important property that it commutes with M6bius transformations o f f. If T is a M6bius transformation, then Erof = T(Ef ). (4.5) This can be checked directly from the definition, first for complex affine transformations and then, less obviously, for an inversion. It is also true that Efo ~(z) = Ef(z(z)) for all M6bius transformations r of the disk onto itself, but we will not make any use of this fact. Proof of Theorem 1. We first show that F is continuous at all points of the sphere. If [z[ < 1 this is obvious. Next, using (4.5) we may normalize further and assume that f e N*. It is then clear from fe(z) =f(z) + (I - [z()f'(z) 1 1-1 z l 2) (z) that F is continuous outside D; from Lemma 1 the denominator vanishes only at z = 0, which corresponds to oo under the reflection in Izl = 1. Finally, recall by the Gehring-Pommerenke theorem that f has a homeomorphic extension to D. Thus sincef(d) = f2 is a Jordan domain, to show that F is continuous on Izl = 1 we must see that Ey matches with f there. Because we have normalized to get f e N* we know that t2 is bounded, and so it suffices to show that Es(z ) - f ( z ) ~ 0 as Izl-, 1. This is equivalent to iv log Xo(w) I --, ~ as w ~ a o, which follows from the first part of Lemma 3. We also now conclude that the range of F is all of (2. Since f is a homeomorphism of D, it remains to show that E i is injective. Suppose that Ef(zl) = Ef(z2). Appealing again to (4.5) we may c h a n g e f t o T o f by an appropriate M6bius transformation T and assume that this common value is oe. But (4.2) now implies t h a t f m u s t be bounded, while on the other hand (4.3) shows that an infinite value of E s must be a critical point of log 2o. By Lemma 2 such a critical point is unique, hence Zl = z2 because f is univalent. We have proved that the mapping F is continuous and injective, and is therefore a homeomorphism onto its range, (2. This completes the proof of the theorem. The function E s is precisely the Ahlfors-Weill extension. For f satisfying [Sf(z)[ ~ 2t(1 -Iz12) -2 the function F defined by (4.4) is a (1 + t / 1 - t)-quasiconformal mapping which extends f. In [5] Epstein made an enlightening differentialgeometric study of this extension. Independent of the Gehring-Pommerenke results, a function in N6 is already x / 1 - t-h61der continuous in D, and so, in particular, it can be extended to!); see [2].
Ahlfors-Weill extensions of conformal mapping and critical points of the Poincar6 metric 667 The complex dilatation #E = t3ef/o~f of the Ahlfors-Weill extension at a point in the exterior of the disk is 1 ~ ( ~) = - ~ ( ~ _ i~]:) ~ Sf(z), where z = 1/(. It will therefore not define a quasiconformal mapping at points where [Sf(z)[ is at least 2/(1 -Iz12) 2. There are, however, functions in N * \ U, < ~N' which do have quasiconformal extensions. For example, take f to be a solution of S f = - 2 in D. The function has ISf(z)l ~ t~2/2 for t = 4/n 2 < 1, and so by [10] and [6] its image is a quasidisk. But the formula (4.4) will not provide a global quasiconformal extension. Also, recall from the remark following Lemma 2 that the functions A_,(z), 1 < t <3, with Sf(z) = - 2 t / ( 1 - z 2 ) 2 (too big to be in N when t > 1) all have quasiconformal extensions, but again not via the Ahlfors-Weill extension. Appendix: An example We return to the first part of Lemma 3. We want to construct a f u n c t i o n f e NO* showing that the exponent 1/2 in the bound IV log ).a(w)] > alwl (w) "=, t~ = f < m, is, in general, best possible. As the proof of Lemma 3 shows, this will be the case provided the convex function v, introduced in the proof, has bounded derivative. The extremal F0 maps the disk onto the strip - n / 4 < Im w < n/4. We want to construct g, analytic in this strip, so that f = g ofo will be in No*, and v ( r ) = ]g'(r)l-'/2 will be convex with bounded derivative for z on the real axis. Let a > 0, to be chosen, and let g ' ( O = a a + ~ 2" If a > x/~/2 then g' will be regular in the strip and v(r), r ~ R, will be a convex function with bounded derivative. We compute the Schwarzian of g to be - 2 a s g ( o (a + ~ ~)~" 2{ 4a} Sf(z) -- Sg(Fo(z))(F'o(Z)) 2 + SFo(z) (1 - zz) 2 1 (a ~-~2)2, =F0(z). Then f = g o Fo is normalized and
668 M. CHUAQUI AND a. OSGOOD It is n o t hard to s h o w that if a is sufficiently large then 1 4a - (a ~---(2)2 < 1, so that f e N~'. REFERENCES [ 1] AHLFORS, L. and WEILL, G., A uniqueness theorem for beltrami equations. Proc. Amer. Math. Soc., 13: 975-978, 1962. [2] CHUAQUI, M. and OSGOOD, B., Sharp distortion theorems associated with the Schwarzian derivative. Jour. London Math. Soc. (2), 48: 289-298, 1993. [3] CHUAQUI, M. and OSGOOD, B., An extension of a theorem of Gehring and Pommerenke. Israel Jour. Math., to appear. [4] DUREN, P. L. and LENTO, O., Schwarzian derivatives and homeomorphic extensions. Ann. Acad. Sci. Fenn. Ser. A 1 Math., 477: 3-11, 1970. [5] EPSTEIN, C., The hyperbolic Gauss map and quasiconformal reflections. J. reine u. angew. Math., 372: 96-135, 1986. [6] GEHRING, F. W. and POMMERENKE, CH., On the Nehari univalence criterion and quasicircles. Comment. Math. Helv., 59: 226-242, 1984. [7] K1M, S.-A. and MINDA, D., The hyperbolic and quasihyperbolic metrics in convex regions. J. Analysis, 1: 109-118, 1993. [8] MINDA, D. and OVERHOLT, M., The minimum points of the Poincar~ metric. Complex Variables, 21: 265-277, 1993. [9] NEHARI, Z., The Schwarzian derivative and schlicht functions. Bull. Amer. Math. Soc., 55: 545-551, 1949. [10] NEHARL Z., A property of convex conformal maps. Jour. d'analyse Math., 30: 390-393, 1976. [11] OSGOOD, B., Some properties off"if' and the Poincarb metric. Indiana Univ. Math. Jour., 31: 449-461, 1982. [12] PAATERO, V., Ober die konforme Abbildung yon Gebieten daren Rfinder yon beschr~inkter Drehung sind. Ann. Acad. Sci. Fenn. Ser. A XXXIII, 9: 1-79, 1931. [13] YAMASHITA, S., The Schwarzian derivative and local maxima of the Bloch derivative. Math. Japonica, 37: 1117-1128, 1992. P. Universidad Cat61ica de Chile, Chile and Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA Received March 21, 1994