ON BLOCH'S CONSTANT MARIO BONK. The lower bound for Bloch's constant is slightly improved.

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1 proceedings of the american mathematical society Volume 110, Number 4, December 1990 ON BLOCH'S CONSTANT MARIO BONK (Communicated by Irwin Kra) Abstract. The lower bound for Bloch's constant is slightly improved. Let //(D) denote the class of holomorphic functions in D := {z G C: \z\ < 1}. Given a function F G //(D) define BF to be the least upper bound of all numbers r > 0 such that there exists a number z0ec and a region il ç D which is univalently mapped onto {z G C: \z - zj < r} by F. Now Bloch's constant B is B := inf{bf : F G //(D) A F'(0) = 1}. The purpose of this note is to give a new proof of the well-known inequality B > \/3/4 (cf. [3, 5, 8]). Indeed our method of proof enables us to show B > \/3/4 + 10"14. We need the following Theorem 1. B = inf{bf : F G AAS A ^'(0) = 1} (cf. [4]). Here AA denotes the class of all functions F g H(D) with Now we can prove F(0) = 0 and sup /7'(z) (l - z 2) < 1. z D Theorem 2. Let F G AA and F'(0) = 1. Then Proof. Define and Calculation yields ReF'(z)> l,v/i z, for \z\ < JÎJ1,. -(l-^t73 z )3 '-V f(w) := 3 1 -w/3 G(w) := \w (1 - \w) \G(w)\(l-\fi(w)\2) = 1 forwedd. Received by the editors September 14, 1988 and, in revised form, January 24, Mathematics Subject Classification (1985 Revision). Primary 30C25; Secondary 30C American Mathematical Society /90 $1.00+ $.25 per page

2 890 MARIO BONK Thus if F e (1) F'(f(w)) G(w) < 1 for w GdD. Now let If FeJ H(w) := and F'(0) = 1, then F'(f(w)) G(w) - 1 w (w-iy \F'(z)\ = l+f"(0)z +!< 1-lzl = i + ur + and therefore F"(0) = 0. Considering local developments in w = 1 we see that // is a holomorphic function in D. Now ( 1 ) gives and so Hence we get ReH(w)>0 ReH(w)>0 forweôd, forwed. Re F'(f(w)) > G(w) for we [0,1]. This is equivalent to the assertion, for without loss of generality we may assume 2 [0,y/T/3]. Corollary 1. B > \fi/4. Proof. If F G AA and F'(0) = 1, we see from Theorem 2 that ReF'(z) > 0 for \z\ < y/t/3. Hence F is injective in M := {z G C: \z\ < ^1/3}. The inequality 1/3 LV^ l-y/3t Jo Jo (i-y/tßty \F(y/T/3ei9)\> [X ReF'(te'v)dt> /' ' VJ1 dt = v^/4 shows that every boundary point F(^/\ße"p), tp g [-n, n], of F(M) has a distance to the origin which is greater than or equal to v/3/4. Because 0 = F(0) G F(M), F(M) contains a disc of radius %/3/4. This means BF > \/3/4 and so Theorem 1 shows B > y/3/4. To improve this result we need some preparation. Lemma 1. If F GAS and if F' has the Taylor series then oo F'(z) = l+272anz", n=2 N<1, kl<5

3 ON BLOCH'S CONSTANT 891 and Proof. oo EKHzI" < 10 z 4 for \z\< 1/10. n=4 \a2\ < 1 follows at once from \F'(z)\ = \\+a2z2 +!< l/(l- z 2)= l + z If 0 < r < 1, then This implies and thus Kl2 * i For «= 3 and r = \/3/5 l«_l< «'- (1-r 1 ^2~? rn-\\-r2) 2x2 we immediately obtain a3 <5. 1 for n > 3, for «> 3. For n > 4 set r = ^/(n - \)/(n + 1). For n > 2 the inequality n + 1 /i- 1 (n-d/2 = 1 + «- 1 (n-l)/2 is valid, because the function (l-l-l/x)'v in (0, 00) is monotonically increasing and tends to e for x 00. So we arrive at. ^ n + l /«+ 1\(""1)/2 fñ+3 ^ [7 n + l This yields, for \z\ < 1/10, 00 co rzr 1 ç /"*y 00 «Mzl ^EV5e^~ z Ei 111«^ v^ / / W t 1,,«~> 1 v ^» -r 1, i«<e =2V5^_1~ Z n=4 n=4 n=4 ^ ^2V5^ Z 5 /7 En 1 i3v^ «=1 1" 5 /7 z = -\/-<?- 2V5 (i_ < 10 z. Lemma 2. Let F G â with F'(0) = 1, ç»0 = 2arcsin(l/20) a«<2 z < 1/ ?^«the following inequality is valid for cp G [ n, 7t] \ f-^0, tp0] : 1 n re.', v. c-', (> V1. 1 V 3\Z\ Re [F (z) + F (e z)]>-_l, + ~\z\. 2 (1 - y/tjj\z\)3 21 Proof. Since F G AAS and F'(0) = 1 the function / "' has a Taylor series whose second coefficient vanishes: 00 F'(Z)= i+j2anz"- n=2

4 892 MARIO BONK Lemma 1 shows that, for cp G [ n, n] and \z\ < 1/10, i Re[F'(z) + F'(e'vz)] - X~^, 2 (1-/T73 z )3 El,,, in<p. n Re «(l+e )anz L«=2 > Re oo. + E oo 2, n - 1 «=2 v^" if 2, î(3/2)ç» 3 3 e a2z coscp + e a^z cos-=cp. i,2 8%/3..3 1AI.4 + z +-ñ- z - 10 z >(l- cos^ ) z 2+[^-5 cos -tp z 3-10 z 4 Now if tp G [-n, -n + <p0] or cp G [n - cp0, n] set a = (n + cp)/2 resp. a (n - cp)/2. Then a [0, cp0/2] and we obtain the following lower bound for the expression above: (*Ç. -Ssirxla) z 3 _ l0 z 4 > (* - lssina") \zf - 10 z 4 ^i -ivl3-10l-l4hn3-i0lzl4. If on the other hand cp G [-ti + cpq, -tp0] u [tp0, n - <p0], we obtain the lower bound (1 -cosç>0) z 2-4 z 3-10 z 4 = jiglzl2-4 z 3-10 z 4. Now for \z\ < 1/1000 both and Thus we get the inequality 3U 3-10 zf>i z 5iüiz 2-4 zi3-ioizr>i zr. I4^ 1U 3 l-v/3 z >i,3 ^Re[y(z) + F'(e",z)]- 2 (l-v/t/ilzl)3 2 which is valid for ç? e [-7r, 7r] \ [-<p0, cp0] and z < 1/1000. Now we can prove: Corollary 2. B > x/3/4 + 10"'4. Proof. It suffices to show that every function F G AS with F'(0) = 1 covers a disc of radius n/3/4 + yj 10" univalently. If min Re[e-ivF(^T/3e"')]>^ + r-a0-n, <p [-7t,7t] then the assertion follows from the same arguments as those in the proof of Corollary 1.

5 ON BLOCH'S CONSTANT 893 If on the other hand the above minimum is less than v/3/4 + yjlo-1, we can without loss of generality assume that it is obtained for cp 0. Letting cpq - 2arcsin(l/20) Lemma 2 shows, for tp G [ n, 7t]\[-cpQ, <p0], Re[F(y/l/3) + e ^F^lße*)} hence fv ' / 3 r V (F'(r) + F'(e"pr))dr Jo f 1/1000 i/1uuu 1/3 = / ^Re[F'(r) + F'(e"pr)]dr= +/ Jo * 'o ii1/1000 / 1/1000 ~rdr i + v^ l-y/jr. 73^1,.-12 I>0 l JO Jo (1 Tßr\ Re[e'i,pF(y/Tj3e"p)]> ^ + ^lo'12. For cp g [-<p0, <p0] we have the following inequality which follows at once from Theorem 2: -If T?, /,,t _ if Re[e-'*F(^\ße'v)]>&. Now let z0 := because for cp G [-n,.id for cp g[-ç>0, 10~12 and r0:=v/3/4 + -Li0"12. Then {zgc:\z-z0\<r0}nf({zgc:\z\ F(VW n] \ [ q>0, çp0] we have \F(^Y]3e'v) - z0\ = \r»f{y/ïjw) <p0] we have Tß}) - «T'%1 > Re[e-"pF(s/Tj3ei,p)] - \z0\ > ^ " = r i l If n /i li "P\ ~'f l z0\ = \e F(y/l/3ev)-e z0\ >Re[e-ivF(^\~ße"p)-e~"pz(A] Thus in all cases F covers a disc of radius r0 = \/3/ This means BF > \ß/4+ -L10"12. univalently. Remarks. Theorem 2 also gives the lower bound S > y/tß for the Marden constant S for Bloch functions which appears in [6]. A systematic treatment of the ideas developed in this paper is given in [2]. The main result is the determination of the variability regions for z., z, g D and w, G C. V(zx, wx ; z2) := {F'(z2): F g AS a f'(zx) = wx}

6 894 MARIO BONK References 1. L. V. Ahlfors, An extension of Schwarz 's lemma. Trans. Amer. Math. Soc. 43 (1938), M. Bonk, Extremalprobleme bei Bloch-Funktionen, Dissertation Braunschweig, M. Heins, On a class of conformai metrics, Nagoya Math. J. 21 (1962), E. Landau, Über die Bloch 'sehe Konstante und zwei verwandte Weltkonstanten, Math. Z. 30 (1929), C. D. Minda, Bloch constants, 3. Analyse Math. 41 (1982), _, Morden constants for Bloch and normal functions, 3. Analyse Math. 42 (1982/83), E. Peschl, Les invariants différentiels non holomorphes et leur rôle dans la théorie des fonctions. Rend. Sem. Math. Messina 1 (1955), Ch. Pommerenke, On Bloch functions, 3. London Math. Soc. (2) 2 (1970), Institut für Analysis, TU Braunschweig, 33 Braunschweig, Federal Republic of Germany