CHAPTER 2. Special classes: convex, starlike, real, typically real, close-to-convex, bounded boundary rotation We shall examine some special families for which we can determine the closed convex hulls and corresponding extreme points. In principle then, all linear extremal problems for these families are elementary. We shall restrict our attention to normalized analytic functions N= (£EH(U) Then f(O)=O, e(O) =l} . nN S = Hu(U) (f E 5 : £(U) is convex} S*= (f E S: f(U) is starlike with respect to the origin} S = (f E S: lR T = {f EN : lR f (n) (0) E lR K = C = (f E N: for all (Jm z) (Jm f(z)) ~ n ~ 2} O} ~e[ f' /eillq>'] > 0 for some c:pEK and IlElR} are the familiar normalized schlicht, convex, starlike, real, typically real, and close-to-convex classes, respectively. Functions in Functions in if f E TlR Tm then to the real axis. K and S* have obvious geometric properties. have real Maclaurin coefficients. f (z) = f (z) Clearly and f (U) is symmetric with respect SlR c: T lR , but nonunivalent functions (e.g., z + z3) Consequently, TlR contains also some On the other hand, close- to-convex functions turn out to be univalent, and this is the motivation for the definition of LEMMA2,l. Proof. Since D If Suppose is convex, FEH(D) C. where D F (zl) = F (z2) (l-t)zl + tZ 2 ED is convex and for some for ReF'>O, then zl' z2 ED, zl" z2 • t E [0,1 ] • Therefore 7 1 S (~eF')dt > 0 o presents a contradiction. THEOREM 2.2. Assume .f.!:Q.Q1. cp E K, and If Ct E 1R • f E C, then fEH(U) f and is univalent. F = fa f -1 E H (D) Then ReF'= ~ef'/q;/> 0, by Lemma 2.1 Since t e where D = f (U) is and F.~=f where ~ef'/~' >0 F cp, convex. are univalent. We shall need to convert the geometric definitions for S* io. K and into analytic relations. THEOREM 2.3. If f E s, then the following are equivalent: (a) fES*: (b) Dr = f(\z\ <r) 0 <r <1 : origin for (c) Proof. zf'/fEP. o (b) => for some (a) Iw(z) 1 :s; (a) oDe arg f (re i9 ) = Jm oDe log f = Re zf' If sufficiently close to (b): Izl for is a non- i.e., zf' If \ z=O = 1 • is obvious since each point of For O<t<l, 1f- l 0 < t < 1. (tw 0) So 1= Dr f (U) = Cl(tf(Z» Suppse woE Dr' Iw belongs to 1. UJ(z) by Schwarz's lemma. 1f- l (~) 1 < rand tWoE Dr e, by the minimum principle, since r => :s; arg f (re ie ) is starlike iff (b)" (c): decreasing function of Re zf If> 0 is starlike with respect to the (2 0 ) 1 $ 1201 < is starlike. satisfies Then r. IZol = Therefore Dr THEOREM 2.4. f E 5, then tbe following are equivalent: (a) f EK (b) Dr (c) 1 (d) zf' ES* • Proof. ae f(lzl <r) is convex for ~ (c): Dr is convex iff the tangent angle R.e {l + zftl If'] > 0 (b) '" (a) D r A for r sufficiently close to Suppose hence, w,. = f(zA) = g(Z2) where zl ;i z2 for zA E U. for some g(z) = Af(zz l /z2) + (l-A)f(z). wA ~~ '" ~e (1 + zftl If') • [1 + zt" If' Jz=O '" 1 • by the minimum principle since = ).f(zl) + (1-A)f(z2) E feU) g(U) Cf(U). i.e., is obvious since any two points of (a) '" (b) : w e, is a nondecreasing function of Jm oOe log some O<r<lr + zf tl If' E P r (b) Of (re i9 ) arg If By Theorem 1.1, Then feU) . 1 I z l l1OI z 2 1 < r 0<1..<1 since Then is convex; feU) We must show that gEH(U), • belong to wI.. € Dr' g(O) = D, and g(lzl <r) CD ; in particular, r EDr • (c)" (d): f satisfies (c) iff g zft satisfies part (c) of Theorem 2.3 • COROLLARY 2.5. and C (f E N: ie[zf 'le iC1 gJ > 0 for some 9 E S* a E lR} • ~. Apply Theorem 2.4(d) to the definition of COROLLARY 2.6. ~. choosing Let a K C S* =D and C. K C S* C C • is geometrically obvious. g= f in Corollary 2.5 • S* C C follows by 9 It will be useful later to know that the analytic conditions of Theorems 2.3(c) and 2.4(c) actually imply univalence. We therefore sketch the proof: Let THEOREM 2.7. (a) If (b) If Proof. fEN. l+zf"lf'Ep, then f ( I z I= r) Yr f assumes If (a) n(r/O) so that 1 2TT Yr in zf'/fEP winds around Iz z = re S2n 1 2TT Yr arg Wo and also the number (1 i9 then I + positive powers of z)d9 = 1 0 Yr curve. asa arg f(re is ) Since and 1 + zf" I f ' E P 211 2TT ~~ld9 = S~e (1 + z{' )de = ~e f o z = re i 9 I then (1 + positive powers of z)d8 = 2n o So the total variation of the tangent angle to for e dz is a Jordan curve. If (b) (z) Wo ~ Vr I < r. and S2n zf' f de = f' 2ni f(z)-w o Izl=r winds once around the origin. 2TT o Wo S _1_ o o ~ezf'lf > 0, SI :e S~ -w-w 2ni gives the number of times fEK is a closed curve and for 1 n(r,wo ) of times fES * zf'/fEP, then Yr is 2TT There- can wind around any point at most once and is a Jordan Since the total change in the tangent angle is also 2TT, the winding is in the positive direction. n(r/wo ) In both cases (a) and (b) the winding numbers identically one for in the exterior of interior of Yr w a vr of Yr. f maps Therefore exactly once in point of the exterior of mapping, in the interior of I z\ <r vr in Iz I Iz I vr and zero for are Wo assumes each point in the f <r <r and does not assume any Since f is an open in a one-to-one fashion onto the interior This is true for all r < 1. Therefore f is univalent • 10 in U. Since now f E S, the respective assertions follow from Theorems 2.3 and 2.4 • THEOREM 2. B (Ruscheweyh and Sheil-Small [R4 1 ). If f e K, then -L} ~e{......L.k.:L f(z)-f(t) _ z-C z-t f(C)-f(t) z-C Proof. > 1 - The function z, for all 2 C, t E u f(z)-f(t) f(C)-f(t) ~.k.:L F(z,C,t) - z-C z-t _ • ~ z-C _ 1 is analytic in all three variables when extended by continuity for z = c, z = t, and ~e Since F(e ia t, e f(lzl ,;r) arg whenever is • is convex for if 0 < f(el.Pt)-f(t) if 0 <: a. <: 'Q [-TT, OJ (f(e~at)-f(t)} Jm and I = I cI = I t I z,C,tEU. ~e the minimum principle implies Actually ~eF>O since F(O,O,O) { Iz and finally for I "I CI first for Iz It < 1 .) II I (; \ I z \ I = COROLLARY 2.9 (Suffridge [S23J). zf'(z) Re { f(z)-f(C) ~. Let t If we also let ~ z - -L}>l z- C '2 It I If I ~e =1. { I z I = r} 3 I for (Toapply is the individually to then for f EK for all F "= 0 <: r} 3, or else one can z,C,t apply it successively in the variables a iteF(z,C,t) "=0 is a harmonic function F(z, Ct) distinguished boundary of the po1ydisk Re F > S <: 2TT By continuity, Since < 21'1 have the same sign f(el.St)-f(t) the minimum principle one can observe that obtain ~ <a f(eiUt)_f(t) E {[ 0,1'1 ] of all three variables I all then r= Itl, ~eF(eiat, eiet,t) ~O. Iz a.1 S, 0 < a, S <: 21'1, If 1 ~ J [f(eiCtt)_f(t») t, t) _ - sin ~ (a-~) s~ mtf (ei~t) -fIt) sin J., (a-[:l) Therefore so that C= t Z, I I CI , I z I ,; \ t then CE U in Theorem 2.8 • C ~ z, the above inequality implies that \ I 11 1 + zf" If' e P. It follows then from Theorem 2.7 (b) that the condi- tions of Theorem 2.8 and corollary 2.9 are not only necessary, but also sufficient for fEN to belong to are invariant when replacing f characterize the convexity of K. In fact, the expressions Af + B. by f(U) They therefore independent of normalization. COROLLARY 2.10 (Stroh"acker [S21J, Marx[Ml]). ~e zf'/f .R!Q2f. Set (, = 0 > ~ fEK, then If Re f/z > ~ • and in Corollary 2.9, and let t = 0, C" 0 in Theorem 2.8. f is called starlike of order ~ if fore convex mappings are starlike of order Re zf' If > ~. There- (l. More importantly, the second condition of Corollary 2.10 turns out to describe precisely the closed convex hull of are due to K. This and related results that follow L. Brickman, T. H. MackGregor, and D. R. Wilken [B10J. THEOREM 2.11 ([SlOJ). coK K (fEN: Ref/z and co K are compact, >~) ( jz/(l-~z)d~; ~ is a probability measure on 1~1=1}, \'1")1=1 and ECOK'" Proof. (z/(l-T'lz) The mapping homeomorphism of H(U) IT'll "'l} • r, defined by r, (g) onto the subspace r,(P) = [f EN; Re f/z >J:i). comes from Theorem measure ~ f(z) 1.6~ namely, P A second representation for f E '£(P) ~z f[l+ (l+T'lz)/(l-"z))d~ OJ. By onto a compact convex .£(P) iff there is a probability such that 1,,1=1 is a linear (h E H(U); h(O) = Theorem A.2 it maps the compact convex set set "'!o:!z (1 + g) Sz/(l-llz)d~ I~\"'l 12 It follows from Corollary 2.10 that K K is a subset of !(P) is closed since the relation of Theorem 2.4(c) is preserved K. under locally uniform convergence of functions in K !(P) is a compact subset of the compact convex set co K C:.l: (P) Observe that the mappings C::co .s:(P) =co E!(p) Therefore Therefore • By Theorems A.2 and 1.5, and Consequently, ! (p) = co E.s:(p) = £(Ep) = (z/(l-nz): K. belong to z/(l-nz ) Hence Inl =l} • E£(p)C:K by the Krein-Milman theorem (Appendix A). K K, and the proof is complete. The extreme points z/(l-nz) whose boundaries have distance @ map U onto the half-planes from the origin. ~ /......z/(l-nz)li ~ / / / I ,/ i +'1--1'l....+--f".;:-I / L(f) = f{n) (z) Since the extreme values of the must occur at an extreme point (Theorem A.3). we have the following immediate application. TIiEOREM 2.12. If I'" z + cnz n E K, then n=2 f(z) If(z)1 "Izl/(l-Izl) and If(n)(z)1 "nl/(l_lzl)n+l for all and In particular, '..l I c n I .. 1 for zeu n:;, 1 n = 2,3, • •• . We turn to the class s*. THEOREM 2.13. iff there exists a probability measure f E S* such that f(z) =z exp[-2 Jlog(l-nz)diJ. ] 1T'l1=1 Moreover, the probability measure iJ. is unique. . . 13 Proof. If fE5*, then by Theorems 2.3(cl and 1.6 there is a unique probability measure such that ~ [zf'/f-ll!z = S[(l+"zl/(l-"zl-lJ/z d\.l = S21l/(1-"Z)d\.l • 1,,1=1 Therefore 1111=1 log f/z = -2 Slog(l-11Z)d\.l Conversely, if f E S* zf I If E P and by Theorem 2.7 (a) • COROLLARY 2.14. all by integration. Inl=l f has the given form, then f E 5*, then If lim arg f (re r-+l e. Proof. Represent is ) arg f/z = -2 Sarg(l-11z)dIJ. exists for The radial I" 1=1 limit exists by the Lebesgue bounded convergence theorem. To determine the co 5* we shall exploit the connection with convex mappings rather than Theorem 2.13. THEOREM 2.15 ([BlOJ). S* co S* = ( !z/(1-11Z)2d\.l : \.l Proof. defined by ~ homeomorphism of the space Theorems 2.4(d) and 2.7. 111 I=1} , I" 1=11 (z/ (l-11Z) 2 : The mapping are compact, is a probability measure on 1,,1=1 and co S* and = zg' ~(g) is a linear (hEH(U) :h(O) =O}, and ~(K) =5* by The results now follow from applying Theorem A.2 to Theorem 2.11. The starlike functions and map u @ are called ~ functions z/(l-11z ) 2 onto the complement of a ray from I/U// i ! .. 'j / I z/(l-Tlz) 2 "----'" " . , to i ... i_\1T'L,i I. I. '/' / ...-:-r : ... ;' 0 /' ./ ' I J By examining just these extreme points we have the following application. 14 THEOREM 2.16. If I'" anz n E S*, fez) = z + then n=2 If(n)(z)1 ,.;; nl(n+lzl)/(1_lzl)n+2 In particular, I an I ,.;; n for for all SlR and TJR. TJR is precisely the closed convex hull of n;'O. We shall see that SlR' The following are equivalent: (al fET lR : (b) (1-z2)f!zEP (cl there exists a [-l,lJ and n = 2,3, • •• • We turn now to the classes THEOREM 2.17. zEU and f(n) (0) ElR for all n"'2 (unique) probability measure ~ on such that fez) Jz/(1-2xz + z2)dlJ [-1,1] Proof. if z (a) => (b): is real so that If f E T lR , fen) (0) E lR then by continuity for all n. f is real 0 <r < 1 If and I z I = 1, then Re[(1-z 2 )rf(rz)lz] = 2 Jm(rz) • Jmf(rz) "'0 • By the minimum principle, r ot Re(1-z 2 )f(z)/zl;,0. l: then minimum principle since (b) where => Re[ (1-z 2 )rf(rz)/z];'0 (c): pEP. in U. Let Zero is not possible by the (1-z2)f/zl = 1 • z=O Since all Maclaurin coefficients of f are real, By Theorem 1.6 there is a probability measure ') such that fez) = Sz/(1-2(~e TI)z+z2)d\l = Jz/(1-2xz+z2ldl-! I TlI=l where "I-l(x) = \I(TI)+\I(ii)," Le., AC[-l,l]. [-l,lJ I-l(A) = For the uniqueness we assume '9 \I£e1.: cosBEA}) for fez) = Sz/(1-2Xz+z 2 )dl-l k • [-1,1] 15 i.e., "k(B)=J.,lJ.k([cos6: e for Bc::(lnl =l} • i9 -i8 EB,os;es;n)) 2 J(l+Tlz) / (l-llz)dvk= (l-z )f/z, and by the =1 uniqueness of the Herglotz representa tion v 1 = v 2: hence U 1 = IJ. 2 (c) '" (a): Then EB,O",Bsn})+J.,lJ.k((cos9:e \ nl It is clear that (1_\z\2)\1_2xz+z 2 \-2.9mz that fEN. Since .9m[z/(1-2xz + z2) ] has the same sign as Jm z, it follows f E TlR • THEOREM 2.18 ([BIO]). co 8lR = also convex, 8lR and TlR are compact, TlR is T lR , and (z/(1-2xz+z 2 ): xE[-I,I]). lR That SlR and TlR are closed, T]R is convex, and ET Proof. SlR c:: T lR is clear from their definitions. TlR C 2 (zp/(l-z ): Hence both By Theorem 2.17 (b) , pEP}, which is compact since SlR and co TlR are compact and P is compact. SlR c;; TlR • Since the representation in Theorem 2.l7(c) is unique, = {z/ (1-2xz + z2): x E [-1,1]). In fact, each function ]R 2 z/(1-2xz + z) belongs to SlR so that by the Krein-Milman theorem ET TlR = co ETlR C co S]R: therefore z/(1-2xz + z2) The functions real slits from (Note for 4 x =::1 co (1 + x) -1 to SlR = T]R • ®U! / appears and we have O· 2 II, 1/ ' ET These mappings are starlike, i.e., we have the estimates of Theorem 2.161 f(z) = z '" lR /./ , ....-----) / If ~(I_x)-1 j z/(1-2xz + z ) Koebe functions.) THEOREM 2.19. onto the complement of and from -'" one of the slits dis- U map , I i ' / /' / ,I ,' 1 _'1 ; -~ ( 1 + xl I C /1' ,/ 0/ I ;' (l-x) I / ! I .' ! ! ! / .", S*, so as an application + ~ anz n ET lR , then n=2 1/ /' 1/,: / / ,i i ,J I'. . '/,i" -1 i I,' I • to 16 ~ nl(n+ Izl)/(1_lzl)n+2 If(n)(z)1 In particular, I an I S n for for all zEu and n;;'O. n = 2,3, • •• • The following theorem is a fundamental tool in the further study of special families. THEOREM 2.20 (Brannan, Clunie, and Kirwan [BB]). F = [fEH(U) : f-< (l+cz)/(l-z)} Fet = [fet:fEF} let Proof. F Since GC1. = ( and probabili ty measure on for fixed I Tl I=l} ITlI=l Then . P. do not vanish so that po. Since Go. on is weakly compact. \-l Go. are compact, (l+z )/ (l-z) +J.,(l-c), F is compact and I cis: 1, the functions in The is compact since the set of probability measures Suppose now measures is a \-l is well defined and compact. convex set I Til = 1 ~(l+c) Therefore EF = «l+cTlz)/(l-Tlz) : ITlI =l} F F, Fa. , and a ., 1 For Icl sl • SC (1 + cTlz) I (l-Tlz) Jet d\-l (l+cz)/(l-z) = is an affine image of c, Let on g E EGa.. I Tli = 1 Let be the set of probabili ty Pg such that S[ (1 + cTlz) /(l-Tlz) ]ctdl-l = g(z) ITl I =l " 9 I- ¢ and is compact in the weak topology. the Krein-Milman theorem. Let some probability measures VI C1. 1 dv 2 • Since vEEr.' If 9 and v 2 ' then Therefore E" '" ¢ by '1 v = A. VI + (I-A.) v 2 for g(z) = ASC JC1. d v l + g E EGet, this is impossible unless v l ,v 2 E "g ; but the latter is impossible since v must be a point mass and v E Ef>. Therefore Ct g EGa. c [[ (1 + CTlz) I (l-Tlz) 1 : I ril =l} • The sets are actually equal since i f Agl(z) + (1-A)g2(z), then e v ery (1-A)g2(nnoz) If and f E F-E F ' [(1 + cTloz) I (l-Tloz) C(l+cTlz) / (l-Tlz)] et it = = A91(T11loz) + EGa = ¢ . then distinct f l , f2 eP and I. E (0,1) exist 17 such that Af1 + (1-A) f 2 • ~-lfk' k where maps = f U = 1.2, Now fa. = are distinct. onto a convex set and 0.:;" Afa.-1f1 + (1-)..) Since ~-lf2 loge (1 + cz) / (l-z) ] 1 , 10gf + (l/a)log fk < log[(l+cz)/(l-z)] (1-1/0.) so that Therefore By the Krein-Milman The next theorem was verified in many cases in [ BB] and in general first by D. Aharonov and S. Friedland [A3]. The following elementary proof is due to D. A. Brannan [B7]. THEOREM2.21. Then for 0.:;" 1 Suppose the coefficients of sponding coefficients of Proof. g f«l+cz)/(l-z) By g 'it< h fo. [(1 + z)/(l-z) for some ~ Fix Jo. • we mean that the coefficients of 0.:;,,1. 1>< [(l+z)/(l-z),a If Icl "1 is obvious. and employ induction. If and Iclslo are dominated by the corre- are dominated by the corresponding coefficients of O.:k':n. c, zk for for h f< (l+cz)/(l-z), then fa « [ (1 + z) 1 (l_z)]o. We assume n Icl sl, then [(l+Cz)/(l_z)]l-lIa[(l+z)/(l_z)]l/a < (l+yz)/(l-z) for some y, I yl S I . Therefore [(1+cz)/(1-z)]a-1[(1+z)/(1-Z)] «[(l+z)/(l_z)]o. n by the induction hypothesis. Since [(l+z)/(l-z)]o. and 1/(l_z 2) have all nonegative coefficients, it follows that (1 + cz)o.-l = (1 + CZ)a-1(~) __ 1_ « (l_z)(l + 1 'l-z 1-z (1_z2) n Now azo [(l+cz)/(l-z)] 0. <~ (1 + z)o. 1 1-z (l_zZ) = o.(l+c)(l+cz) a -1 l(l-z) Cl + 1 20. (1 + z) (1-1 /(l-z) 0.+1 . (l+z)o.-l (l_z)o. + 1 18 By integra tion, [(1 + CZ)/(l-z)Ja <<. n+.l [(1 + z)/(l-z)Ja If now f< (1 + cz)/(l-z), then by Theorems A.3 and 2.20 we may dominate the coefficients of fa by dominating the coefficients of the extreme 1 ri~l [(l+z)/(l-z)]a, we have fa ~h [(l+Z)/(l-z)](1, and the induction is complete. Recall that f is close-to-convex if there exist ~e[ f' /e iClcp '1 cP E K and > 0, or equivalently, ex. E lR such that <~. We first extend this definition in a way that will be useful also for functions of bounded boundary variation. DEFINITION. of order ~ (tl > 0) C(a) = {fEN: Note that f e C (5) P EH(U) iff with The class of normalized close-to-convex functions is larg[f l /e i (1cp'JI C(l) fEN Denote > for some cpEK and is the old close-to-convex class and ~ep <~8TT f' = e iCl p 8cp' for some ClElR}. C. Also a E lR , til E K, and o. f = t[ (1-~Z)/(1-T\z)]e+l_11l( (6+.1 ) (T'I-s)} if~=Tl \ z/ (l-f'\z) where lsi = iT'll = 1 . Observe that Cl E lR k(z;~'Tl'S) such that Then eC(e), for choose cp(z) = z/(l-T'lz) EK ~e[e -io/S (l-~z)/(l-Tlz) J > 0 in U. and 19 THEOREM2.22«(B8 ] l. Then C(S) COC(~) and Let B~l >l EcoC(~) c (k(z:s,n,l3) : \ e;\ = In\ = 1 , Re p > 0 • Since [\ ~\ =1} x (lnl = 1} is a probability measure on T and T . are compact, = (.fk(z:~,'I'),~)du : coC(~) and l=e ia p(O) S feN, for an appropriate choice of c with is by Theorem 2.20 a probability measure eia.p(z)~ e; 1T1} • e ia/l3 p -< (1 + cz) /(1-z) and I cl T), = 1. '1 on Since S 1 e;1 = 1 ~ 1, there such that = S[1+ce;z)/(1-e;z)]6C\11 \e;1=1 By Theorem 2.11 there is a probability measure that !p' 2 (z) = Jl/(l-n z ) C\l2 • \n1=1 £'(z) = \J2 on 1T11 = 1 such Therefore J(1+c~z)/(1-e;z)J6/(1-T1Z)2 dt.-t 1 x \J2' T Since 10g(1-z) [~/(B maps + 2)1 U onto a convex set, 10g(1-~z) + [2/(13 + 2>1 10g(1-nz) -< log (l-z) ~I'ld 1/[ 1-e;z) 13/(8+ 2 ) (l-nz) 2/(13+2) J -< l/(l-z) c=O and By Theorem 2.20 (with 0.=6+2) there is a probability measure \J3 on le i =1 such that Therefore [(1+c~zl/(1-~Z)J6/(1-T1Z)2 belongs to :z = r(1+c~z)S/(1-ez)I3+2<\.t3 I e1 =1 e;,T1dl) C\I: jC4J. = l}. B I = {J k (z: Since B' is convex T T and compact (the set of probability measures on T is cnmpact in the weak topology), f "'EB'. By integration, fEB = (jK(z: ;,n.e) : T fdtJ = 1} T 20 Clearly B CIs) fn E C (S), is also convex and compact. is a closed subset of I arg[ roC (S ) c: B B: hence compact. f~/ei(Irtp~1 1 < ~eT1', t+l n E K, and suitable subsequences S Hence t(l ~ .. t(l (Indeed, if f n " f, then for I arg[ f Ie io.t+l ' ] I E K, and I If the latter is a strict inequality, then ~s'IT. not, then by the rnaximmn or minimum principle fEC(S): if f '" «:p E C (S) . ) Just as in the proof of Theorem 2.20, extreme points of must be generated by point masses. 1; 1 = 1,, 1 = Hence l } c: CI s ) • Therefore EBC: (k(z : C",B) : By the Krein-Milman theorem, BCcoC(S) EcaC(S ) c: (k(z:t;,Tl,S): the function lsi = k(z:",,,,a) = z/(l-Tlz) ITlI = I} • However, for ~=Tl cannot be an extreme point of since it is not even an extreme point of the subset * Kc: SeC C C (8) (Observe the chain = • coCCa) = B Now 1;1 B PROBLEM. Determine PROBLEM. Show that each 1,,1 = I , 6>1. roC (6) for and 8;' I .) EeoC (6) ~=l 0 < e <1 • for k(z:~,Tl,e) E EcoC (S ) For - * coS for ~;i" , this fact is contained in the following theorem on the close-to-convex class . THEOREM 2.23 ([BIOl) • coC where 2 U[z-~(g +,,)z J /(I-Tlz) T T ={\~\=11 C 2 t\l and coC are compact, : 11 is a probability measure on T} x (In\=l}, and I, ;;iTll . 21 Proof. Since C = C (1) e=1 everything is a special case of Theorem 2.22 for assertion about extreme points. \ ~o \ For = \ T]o \ = 1, except the 1;0 1"0 ' we must show that fe\! = 1, T is possible only if (l-~oz)/(l-"oz) atives we have and let is a point mass at \.l r ... 1 . 3 f(l-~z)/(l-T]z) T 3 Then where Set To O!l:as1 ~oTio - and Set ~ 1 - ~ o~ 0 By the Lebesgue bounded convergence theorem we have S(l-~no)~ Taking deriv- (~o'''o). (l-a) = ~o S ~~ . Then a = Sdl-l • To On the one hand, 50' a, and on the other hand, ! Sono-(l-a) \ ~l-(l-a) both inequalities must be equality. is possible only if only if a = 1. is a unit point mass at \.l ~on 0 = (~ o 2 2 t h and their n - !o:i(n-1)~ - ~T] zJ/(l-T]z) n+2 Therefore nor ~C\J is possible To ,T] ) 0 By expressing the extreme points of C in the form derivatives as -!o:i(~-'T'1)z /(l-T\z) a = I!:o;:j 011, the latter Since Moreover, \nJ~<\l\ To To n!" n-2 z/(l-T]z) [~(n+ 1)T] and using elementary estimates, we have as an application the following extension of Theorem 2.16. co THEOREM 2.24. If (n) If f(z) = z (z)\ s:n!(n+\z\l/(l-\z\) In particular, \ a \ "n n for n +n~2 anz n n +2 = 2, 3, E C, then foraH zeU __ . _ and n:<:O. 22 PROBLEMS. There are many intriguing extremal problems that are not linear. One example is the conjecture of M. S . Robertson [R3] that J. A. holds for the coefficients of close-to-convex functions. Jenkins [J2] has shown that the conjecture fails in the full class S. C However, Robertson (R2J has verified the conjecture in case n-m is an even integer and in some other cases . sufficient to find a single functional la n holds in the starlike class functional 'X : K'" C -X an- 11 x : 5 * .. C in It is such that cl 5*, or equivalently, to find some such that Ina n - (n-nxa n- 1\ s 1 K holds in the convex class for all n. Perhaps the inequality of Theorem 2.8 can be used profitably to attack the last problem. ~lli. is A natural candidate for the functional since it works for ~a2 EcoS*' X : S* .. C However, verify that the in- equalities la n are violated for each if e n:;, 4 -~a2a n- 1\ s 1 by the functions is positive, but sufficiently small. z(l+z) -e (l-z) e-2 ~ and v a finite signed measure if a =~-v where are finite (nonnegative regular Borel) measures and denote its corresponding variation measure by * This observation is due to R. Barnard. We will call a ES lal . 23 The class of functions of bounded boundary rota- DEFINITION. tion is z V = (f exp[ Jlog(l-~z)dcrJdz : a is a finite signed measure on o ITll=l and is the subclass of Vk V for which By Theorems 2.13 and 2.4(d), For f E Vk the constant k {dial ITl =1 V2 S (k ::e 2) k 1 ~1=1 Jda = -2) • ITl1=1 • is just the convex class K. restricts the rotation of the boundary in the following sense. y r = f<\zl=r) tangent along at f(re iS ) Yr is then is The tangent to the level curve ~ f(re ie ) The variation of the 09 S21'Td larg.2....f(re i9 ) 1= S21'T l l e arg..Q....a f(re ie ) Ide o e Since 09 0 f E v, 0 0 the harmonic function u ~ {Re[ (l+Tlz)/(l-Tlz)]da, and it is known that lTi =1 creases to the limit ZfH = Re (1 + ft) = S2TT lu (re i9 ) IdB o in- TT Jdlo I as r -+ 1. If f E Vk ,then kTT is ITll=l a bound in a limiting sense for the variation of the tangent angle at the boundary. The characterization of domains of bounded boundary rotation has been studied by V. Paatero [Pl). In order to- show that functions of bounded boundary rotation are close-to-convex of some order, we shall use the following theorem. THEOREM 2.25. If \-1 1 and are probability measures on ITiI =1, then there exists a constant -c, Icl =1, such that exp( {lOgn-TlZ)d(\.1rIJ2)} -< (1 + cz)/(l-z) . ITl =1 24 Proof. Assume first that the probability measure generated by a continuous strictly increasing function [0,2""J onto [O,lJ, Since loge (l-e width ,.,., ia j = 1,2. z)/(l-e is ~j of gj Then is z)J maps U onto a horizontal strip of 1 sup Jm I(z) -inf Jm I(z) ~S [sup arg( ) -inf arg( )Jdxs,.,. • zEU zEU Therefore containing 0 U U is contained in a horizontal strip of width I(U) 1(0) = O. The types of measures considered are dense I 'nl in the weak topology on the set of probability measures on Therefore by a normal family argument I(U) izontal strip of width I (0) = 0 bility measures TI ,.,. ~1'~2 containing = 1 • is contained in a horfor general proba- So there exists a constant c, \ c\ = 1, such that .r10g(1-'nZ)d(~1j.l2) -< loge (1 +cz)/(l- z)] , and the re1111 =1 suIt follows by exponentiation. THEOREM 2.26([B81). For k>2, VkC::C(~k-1)1 functions with boundary rotation at most of order are close-to-convex • ~k-l Proof. k that is, Let f E Vk . Since V k c: Vk I whenever k < k', we may assume that fl (z) Then ~l exp( {lOg(l-11Z)dO} 111 =1 (101 +0)/(k-2) and where ~2 rdo = -2 1111=1 and (Iol-o)/(k+ 2) Jd\ol = k l'nl=l are both 25 \ rd = 1 and rexp(-2 f1og(l-~Z)~2JdZ E V2 probability measures on ~(z) = z o 1,,1=1 - 2t-L2 ' we have ~ I (z) = exp( (~k-l)Jl09(l-"z)d(lJl-\J2l) • \,,1=1 ~ (fl/~/)l/(~k-l) By Theorem 2.25, is subordinate to some half-plane mapping, and the result follows from the definition of COROLLARY 2.27. C(~k-l) Functions of bounded boundary rotation • k s 4 are close-to-convex, hence univalent. We now determine the closed convex hull of terms of ~. at k ~ for k« 4 in ODC{B), which is known from Theorem 2.22. THEOREM 2.28«(B8]). For Vk Let "="0 (] For 4, ODC (~k-l) COVk and mass the Krein-Milman theorem PROBLEM. ~ be the signed measure on \,,1 = 1 with mass Then 4, we therefore have CoVkCODC(}zk-l) k EeoC (~k-l) ODe (~k-1) C C Vk ODVk I~ {do = -2 =1 by Theorem 2.22. By On the other hand, by Theorem 2.26. Determine and respects this is the more interesting case. resul ts to the case k ~ 4 • for 2<k<4. In many One would expect similar 26 We now solve the coefficient problem in method is due to Brannan, Clunie, and Kirwan THEOREM 2.29. C(S) and Vk • The [B8,B7]. The coefficients of functions in are dominated by the corresponding coefficients of C(6), e >0, k(zr-l,l,e) E C(e) • E.!QQi. ~e p > O. If > c , Icl =1, depending on 1, the coefficients of by the coefficients of [ (l+z)/(l-z) ,,2j.reio.p(z)6(1 +"z)/(l-T1Z)du. h\l =1 fl h were 0.,6, and eillp(z) 6 (l+nz)/(l-"z) is a probability measure and U e illp 8 cp 1 f' cp E K and By Theorem 2.11 some choice of 6+1 fEC(S), then I" Js + 1 J, = 1, 2" p. Since are dominated by Theorem 2.21. Since the same is true of Consequently, the coefficients of are dominated by the coefficients of I z2 j [(l+Z)/(1_z)JS+l (1+Z)S/(1-Z)6+ 2 =!.k(Zr-l,1,1l), J=O and the result follows. COROLLARY 2.30. k ;;, 2 2.29. 2.28. Vk for are dominated by the corresponding coefficients of Proof. for The coefficients of functions in k = 2. Since For V2 k >2 = K, the result Is contained in Theorem 2.12 the result follows from Theorems 2.26 and That k(zr-l,l,~-l) E V k was observed in the proof of Theorem

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