Harmonic Mappings onto Convex Domains

LEMMA 2.1. Let gn and g belong to L [0, 2TT], lim gn = g a.e., and r- J o ' J \gn(x) \dx ^ M for all n = 1, 2, 3, . . . . Then there is a subsequence {gn } of {gn } such that gn dx converges in the weak * topology as k —> oo to gdx -f ds, where ds is a singular measure on [0, 277] with respect to the Lebesgue measure dx.

LEMMA 2.2. Assume that / 2TT A 0 P(z, t)fn(elt)dt and that https://doi.org/10.4153/CJM-1987-071-4 Published online by Cambridge University Press

THEOREM 2.3. Let D be a convex domain. Fix w0 e D, and let a e H(U) satisfy a(U) c U. Then there exists a univalent, harmonic, orientation- preserving mapping f with the following properties. (a) f(U) c D, /(0) = w0, andfM > 0; (b) / is a solution of fj = afz\ (c) f/*£ //ra/to lim f(relt) exist and belong to dD for a.e. t.

THEOREM 2.4. Let f be a univalent, harmonic, orientation-preserving mapping from U onto an unbounded convex domain D which is neither a strip nor a half-plane. Then ( a ) / £ / » ' ; (b) there is only one point el that corresponds to oo; (c) f(z) = fj P(z, t)f(e")dt + AP(z, X) for some constant A G C; (d) there is a countable set E c dU\{el } such that (i) the unrestricted limit lim f(z) exists as z —» e , z <E U, and is continuous for all points el e dU\[E U {el } ],

THEOREM 2.6. Let f be a univalent, harmonic, orientation-preserving mapping from U onto an unbounded convex domain D. Choose X and a so that there are points zk Œ U for which zk -> e'\ \f(zk) | -> oo, and Rzk)/\f{zk)\-*em ask-* cxi. Then for each r > 0 / + re'aP( •, A) is a univalent, harmonic, orientation-preserving mapping of U onto an unbounded convex domain contained in D.

THEOREM 3.1. Iff G SW, then (a) f <= h\ f(U) c W, f is not constant, andf{elt) G dWfor almost allt; _ (b)fj = afz for some analytic function a with a(U) c U; (c) // \a\ 3É 1, then f is univalent and orientation preserving, f(U) is convex, and f has the representation (3); (d) if\a\ == 1, thenf(U) is either a line segment or half line through the point 1 with endpoint(s) on dW.

Theorem 2.3. Hence fis not constant. A • In order to show that fie1 ) G 8 W for almost all t, consider the functions <p o fn where is a conformai mapping from U onto W. Then <p o /w maps 3f/ into dU, is orientation-preserving, and, by Helly's selection theorem as in the proof of Theorem 2.3, part (c), has a subsequence that converges almost everywhere to a function 77 with |TJ| = 1. Since <p is continuous from U into the Riemann sphere, a subsequence of {fn} converges to cp o 17 almost everywher

THEOREM 3.2. For a ^ / ? ^ y ^ É a + 277 the functions T^a^y) belong to Sw.

THEOREM 3.3. HSW = X

COROLLARY 3.4. / / % ( * , A) = ^(/c, A). We shall add the prefix E to denote the set of extreme points. https://doi.org/10.4153/CJM-1987-071-4 Published online by Cambridge University Press

THEOREM 3.5. EHSW = {T(aAy) : 0 ^ a < 2TT, a ^ fi ^ y ^ a + 2TT) and EHSW(K, X) = {TMy) : X ^ P ^ K ^ y ^ \ + 2TT}.

LEMMA 3.6. The function 1 2 1 2 G(x,y) = -s(x) + -s(y) - s(x)s(y) sin(x + y) satisfies 0 ^ G(x, y) ^ 1 for all x, y ^ 0, #«d G(x, j ) = 1 ow/y w/iefl JC = y = 0.

LEMMA 3.7. For x, _y = 0, the function H{x,y) = ^s(x)2 + ^(^y) 2 + *(x>(j) sin(x + y) is nonnegative and assumes its maximum at a unique point (x0, x0). In particular, sup H(x, y) = max H(x, x) = H(x0, x0) ~ 1.7114 JC,V^0 (77/8)^X^(77/4) where x0 « 0.5875.

THEOREM 3.8. Let f belong to Sw, and suppose that f(z)=\ + 2 a„z" + 2 b„z". n = I « = 1 https://doi.org/10.4153/CJM-1987-071-4 Published online by Cambridge University Press

THEOREM 3.9. Iff = h + g belongs to Sw, then \h\0) | ^ 2/77. Equality occurs only when f = 2(«,a + ir,a + ,r)> 0 ^ « < 2<77.

THEOREM 3.10. Iff = h + g belongs to S^, then 4 4 — — ^ |A'(0) | ^ - . 77 + 2 77 The lower bound is sharp only for the functions 77 477 / = -—-W*,«+W) + - x ^ P ( ' 'a)' ° = a < 2"> 77 + 2 77 + 2 and the upper bound is sharp only for the functions

Theorem 3.9. In particular, we may represent n - n . <WA y) *'<°> = J L > *"* I 2 2 J and, similarly, ^ ° ) = X(K,x, e_A I 2 2 J where

LEMMA 4.1. If f G SR, then Re / = 277P( • , A) for some X, 0 ^ X < 277.

Theorem 2.4, we conclude that there is exactly one point el which corresponds under / to 00. Since Re / > 0, we have Re / G h1 and / 2TT Except for the point el , the radial limits of Re / all exist and are zero. Therefore /x is equivalent to a point mass at / = À. https://doi.org/10.4153/CJM-1987-071-4 Published online by Cambridge University Press

THEOREM 4.2. 7/"/ e SR, then there are a probability measure ju and real numbers A, b > 0, and c so //ia? (13) f(z) = 2mP{z, A) https://doi.org/10.4153/CJM-1987-071-4 Published online by Cambridge University Press

THEOREM 4.3. Iff e JT with b > 0, f/zeft / is a univalent, harmonic, orientation-preserving mapping of U onto a convex domain.

Theorem 2.3 in case D = R. Part (c) of the theorem is clearly satisfied because Re /tends to zero at every point of dU\{el }. The normalizations in (a) may be achieved by composing /with elementary mappings. Using an argument similar to that in [8, Lemma 2.6 and Theorem 2.7], one obtains the following result. We omit the proof.

THEOREM 4.5 Jf = S£. In the remaining part of this section, we shall study the subclasses SR and JT° of SR and X> respectively, of functions / = h -f g that satisfy the additional condition /F(0) = gxo) = o or, equivalently, p(0) = 1. It is clear from the definitions that S% c j f ° and s | = JT°. Since b = 1 and c = 0 in (13), it is also clear that Jf ° is compact. Next, we characterize the extreme points of Jf°.

THEOREM 4.6. EJf° = {2TTP( •, A) + iK( •, t, X) : 0 ^ /, A < 2TT}.

THEOREM 4.9. 7/" /(z) = 1 + 2 *„*" + 2 V W=l « = 2 belongs to JT°, */H?« |a,| = 2, |aj ^ AI + 1, AAK/ \bn\ ^ n - \ for n ^ 2. Equality in either of the inequalities occurs only for the functions from Example 4.7.

Theorem 4.6. For these functions we have 2 nanzn x = 2TTPZ(Z, X) + iKz(z, t, X) « = i https://doi.org/10.4153/CJM-1987-071-4 Published online by Cambridge University Press

LEMMA 5.3. Let S be a nonparametric minimal surface over a simply-connected domain D ¥= C. Let f = h + g and a be as in Proposition 5.2. Then for z e U the Gaussian curvature of S at the point I Re /(z), Im /(z), 2 Im J V**'<k — i/i i x i k(z) ,a(2)|(l + Kz)|)V(z)| 2' /« addition, one has (15) |*(z)| and (16) |*(0)| 4(1 - |a(z)|) 2 (1 - |z|2)2(l + \a(z) | )4|fc^) I2

THEOREM 5.5. Let S be a nonparametric minimal surface in R that lies above a domain D in C. Let k denote the Gaussian curvature of S at a point P that lies above w0 e D. (a) If D = W = {w : |arg w\ < 77/4} and w0 = 1, then \k\ ^ IT2. In addition, assume that S has a horizontal tangent plane at P. (b) IfD = Wand w0 = 1, then \k\ ^ (TT + 2)2/4. (c) IfD = R = {w : Re w > 0} awrf w0 - 1, then \k\ ^ 1. (d) IfD = fi = {w : |Im w| < TT/4} ^«J W0 = 0, then \k\ ^ 4. The estimate (16) that we have used in